source: git/Singular/LIB/gradedModules.lib @ e9478b

//////////////////////////////////////////////////////////////////////////
version="version gradedModules.lib 4.0.1.1 Jan_2015 "; // $Id$
category="Commutative Algebra";
info="
LIBRARY: gradedModules.lib     Operations with graded modules/matrices/resolutions
AUTHORS:  Oleksandr Motsak <U@D>, where U=motsak, D=mathematik.uni-kl.de
@*        Hanieh Keneshlou <hkeneshlou@yahoo.com>
KEYWORDS: graded modules, graded homomorphisms, syzygies
OVERVIEW:
    The library contains several procedures for constructing and manipulating graded modules/matrices/resolutions.
    Basics about graded objects can be found in [DL].
    Throughout this library graded objects are graded maps, that is,
    matrices with polynomials, together with grading weights for source and
    destination. Graded modules are implicitly given as coker of a graded map.
    Note that in special cases we may also consider submodules in S^r generated
    by columns of a graded polynomial matrix (or a graded map).
NOTE:
    set assumeLevel to positive integer value in order to auto-check all assumptions.
    We denote the current basering by S.
REFERENCES:
[DL] Decker, W., Lossen, Ch.: Computing in Algebraic Geometry, Springer, 2006
PROCEDURES:
    grobj(M,w[,d])  construct a graded object (map) given by matrix M
    grtest(A)       check whether A is a valid graded object
    grdeg(M)        compute graded degrees of columns of the map M
    grview(M)       view the graded structure of map M
    grshift(M,d)    shift graded module coker(M) by +d
    grzero()        presentation of S(0)^1
    grtwist(r,d)    presentation of S(d)^r
    grtwists(v)     presentation of S(v[1])+...+S(v[size(v)])
    grsum(M,N)      direct sum of two graded modules coker(M) + coker(N)
    grpower(M,p)    direct p-th power of graded module coker(M)
    grtranspose(M)  un-ordered graded transpose of map M
    grgens(M)       try to compute submodule generators of coker(M)
    grpres(F)       presentation of submodule generated by columns of F
    grorder(M)      reorder cols/rows of M for correct graded-block-structure
    grtranspose1(M)  reordered graded transpose of map M
    TestGRRes(n,I)  compute/order/transpose a graded resolution of ideal I
    KeneshlouMatrixPresentation(v)  build some presentation with intvec v
    grsyz(M)        syzygy of Im(M)
    grres(M,l[,m])  resolution of Im(M) of length l... minimal?
    grlift(A,B)     graded lift, gens!
    grprod(A,B)     composition of graded maps (product of matrices?)
    grgroebner(M)   Groebner Basis of Im(M) as a graded object
    grconcat(M,N)   sum of maps into the same target module
    grrndmat(s,d[,p,b])   generate random matrix compatible with src and dst gradings
    grrndmap(S,D[,p,b])   generate random 0-deg homomorphism src(S) -> src(D)
    grrndmap2(S,D[,p,b])   generate random 0-deg homomorphism dst(S) -> dst(D)
    grlifting(A,B)     RND! chain lifting
    grlifting2(A,B)    RND! chain lifting
    mappingcone(M,N)   mapping cone?
    grlifting3(A,B)    RND! chain lifting? probably wrong one
    mappingcone3(A,B)  mapping cone3?
    grrange(M)         get the row-weightings
    grneg(A)           graded object given by -A
    matrixpres(a)      matrix presentation of direct sum of Omega^{a[i]}(i)
";

//    grisequal(A,B)  check whether A is exactly eqal to B? TODO: isomorphic!

LIB "matrix.lib"; // ?

//////////////////////////////////////////////////////////////////////////////////////////////////////////
// . view graded module/map
// . reorder graded resolution
// . transpose graded module/map?

// draw helpers
static proc repeat(int n, string c) { string r = ""; while( n > 0 ){ r = r + c; n--; } return(r); }
static proc pad(int m, string s, string c){ string r = s; while( size(r) < m ){ r = c + r; } return(r); }
static proc mstring( int m, string c){ if( m < 0 ) { return (c); }; return (string(m)); }

static proc grsumstr(string R, intvec v)
"direct sum_i=1^size R(-v[i]), for source and targets of graded objects"
{
  int n = size(v);
  if (n == 0) { return ("0"); }
  ASSUME(0, n > 0 );

  if (R == "")
  {
    R = nameof(basering);
  }

  ASSUME(0, defined(R) && (R != "") );

  v = -v; // NOTE: due to Mathematical meanings of Singular data


  int lst = v[1];
  int cnt = 1;

  string p = R;
  if( lst != 0 ) { p = p + "(" + string(lst) + ")"; }

  int k, d;
  for (k = 2; k <= n; k++ )
  {
    d = v[k];
    if( d == lst ) { cnt = cnt + 1; }
    else
    {
      if (cnt > 1){ p = p + "^" + string(cnt); }

      cnt = 1; lst = d;

      p = p + " + " + R;
      if( lst != 0 ) { p = p + "(" + string(lst) + ")"; }
    }
  }
  if (cnt > 1){ p = p + "^" + string(cnt); }

  return (p);
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  def E = grtwist(2, 0);
  def v = grrange(E); // grdeg(E);
  grsumstr("", v );
}

// view helper
static proc draw ( intmat D, int d )
{
//  print(D); return ();
  int nc = ncols(D); int nr = nrows(D);
  int s, r, c; int max = 0;
  // get maximum string-length among all {D[r,c]}
  for (r = nr; r > 0; r-- ) { for (c = nc; c > 0; c-- ) { s = size( string(D[r, c]) ); if( max < s ) { max = s; } } }
  max = max + 1;
  string head = ""; string foot = ""; string middle = "";
  for ( c = d+1; c < (nc-d); c++ )
  {
    head = head + pad(max, string(D[1 , c]), ".") + " ";
    foot = foot + pad(max, string(D[nr, c]), " ") + " ";
  }
  // last head/foot enties:
  head = head + pad(max, string(D[1 , c]), ".");
  foot = foot + pad(max, string(D[nr, c]), " ");
  // head/foot dash lines:
  string dash  = "-"; string dash2  = "=";
  dash = repeat( (nc - 2*d) - 1 , repeat(max, dash) + " " ) + repeat(max, dash) + " "; // dash  = repeat( (max + 1) * (nc - 2*d) , dash );
  dash2 = repeat( (nc - 2*d) - 1 , repeat(max, dash2) + " " ) + repeat(max, dash2) + " "; // dash2 = repeat( (max + 1) * (nc - 2*d) , dash2);
  for ( r = d+1; r <= (nr-d); r++ )
  {
    middle = middle + pad(max, string(D[r,1]), " ") + " :";
    for ( c = d+1; c < (nc-d); c++ ) { middle = middle + pad(max, mstring(D[r,c], "-"), " ") + " "; }
    middle = middle + pad(max, mstring(D[r,nc-d], "-"), " ") + " |" + pad(max, string(D[r,nc]), ".") + newline;
  }
  string corner_id = repeat(max, ".");
  string corner = repeat(max, " ");
  // print everything all at once:
  print                                     (
      corner + "  " + head  + " ." + corner_id + newline +
      corner + "  " + dash  + "+"  + corner_id + newline +
                      middle                          +
      corner + "  " + dash2 + " "  + corner + newline +
      corner + "  " + foot +  "  " + corner );
}

proc grview(N)
"USAGE:  grview(M), graded object M
RETURN:  nothing
PURPOSE: print the degree/grading data about the GRADED matrix/module/ideal/mapping object M
ASSUME:  M must be graded
EXAMPLE: example grview; shows an example
"
{
//  if( size(N) == 0 ) { return (); }
  string msg = "Graded";
  string lst;

  string arrow = " <- ";
  string R = nameof(basering);

  if( typeof( N ) == "list" ) // TODO: find a better DS for graded resolutions / chain map !?
  {
    int n = size(N); ASSUME(0, n > 0);

    string msg1 = "";
    if( size(R) >= 2 )
    {
      msg1 = msg1 + "(let R:="+R+")";
      R = "R"; // !!!
    }
    msg1 = msg1 + ": " ;



    list arr; arr[n] = list();
    int exact = (1==1);

    int i = 1;

    ASSUME(1, grtest(N[i]));

    string dst = grsumstr(R, grrange(N[i]));
    string src = grsumstr(R, grdeg(N[i]));

    arr[i] = list(dst,  src);

    i = i + 1;

    while( i <= n )
    {
      ASSUME(1, grtest(N[i]));

      dst = grsumstr(R, grrange(N[i]));

      if( exact && (src != dst) )
      {
//        "src: [" + src+ "] != [" + dst + "] :(!!";
        exact = (1==0);
      }

      src = grsumstr(R, grdeg(N[i]));

      arr[i] = list(dst,  src);

      i = i + 1;
    };

    string o = "";

    if( exact )
    { // complex?
      msg = msg + " resolution" + msg1;

      o = "d";

      for( i = 1; i <= n; i++ )
      {
        msg = msg + newline + arr[i][1] + " <-- "+o+"_" + string(i) + " --";
      };

      msg =  msg + newline + arr[n][2];
      msg = msg + ", given by maps: ";
    } else
    {
//      print(arr);

      msg = msg + "-object collection";
      o = "o";

//      for( i = 1; i <= n; i++ )
//      {
//        msg = msg + newline + arr[i][1] + " <-- "+o+"_" + string(i) + " -- " + arr[i][2];
//      };
      msg = msg + ", given by the following maps (named here as "+o+"_[1 .. "+string(n)+"]): ";
    }


    print(msg);

    for( i = 1; i <= size(N); i++ )
    {
      print( o+"_" + string(i) + " :" );
      grview( N[i] );
    };

    return ();
  }

//  typeof( N ) ;  attrib( N );  grrange(N);

  ASSUME(1, grtest(N) );

  msg = msg + " homomorphism";
  if( size(R) >= 2 )
  {
    msg = msg + "(let R:="+R+")" ;
    R = "R";
  }

  msg = msg + ": ";

  intvec gr = grrange(N); // grading weights?
  string dst = grsumstr(R, gr);

  intvec G = grdeg(N);
  string src = grsumstr(R, G);

  if( ncols(N) == 0 )
  {
    src = "0";
  }

  lst = msg;

  if( (size(lst) + size(dst) + size(src) + 4) > 80 )
  {
    if( (size(lst) + size(dst)) > 80 ) { msg = msg + newline; lst = ""; }

    msg = msg + dst + arrow;
    lst = lst + dst + arrow;

    if( (size(lst) + size(src)) > 80 ) { msg = msg + newline; lst = ""; }

    msg = msg + src;
    lst = lst + src;
  } else
  {
    msg = msg + dst + arrow + src;
    lst = lst + dst + arrow + src;
  }

  if( size(lst) > 70 ) { msg = msg + newline; } // lst = "";
  msg = msg + ", given by ";
//  lst = lst + ", given by ";


  int nc = ncols(N); int nr = nrows(N);

  if( size(N) == 0 )
  {
    msg = msg + "zero ("+ string(nr);
    if( nr == nc ) { msg = msg + "^2"; } else { msg = msg + " x " + string(nc); }
    print( msg+") matrix." );
  } else
  {
    ASSUME(0, nc > 0);

    matrix M = module(N);

    int r,c;
    int d = 1; // number of extra cols/rows for extra info around the central degree(N) block in D
    intmat D[nr+2*d][nc+2*d];

    for( c = nc; c > 0; c-- )
    {
      D[1, c+d] = c; // top row indeces
      D[nr+2*d, c+d] = G[c]; // deg(v) + gr[ leadexp(v)[m] ]; // bottom row with computed column induced degrees
    }

    for( r = nr; r > 0; r-- )
    {
      D[r+d, 1] = gr[r]; // left-most column with grading data
      for( c = nc; c > 0; c-- )
      {
        D[r+d, c+d] = deg(M[r, c]); // central block with degrees (-1 means zero entry)
      }
      D[r+d, nc+2*d] = r; // right-most block with indeces
    }

    if( nr == nc) // square matrix // detect diagonal?
    {
      for (c = nr; c > 0; c-- )
      {
        M[c,c] = 0;
      }

      if( size(module(M)) == 0 )
      {
        msg = msg + "a diagonal matrix";
      } else
      {
        msg = msg + "a square matrix";
      }

    } else
    {
      msg = msg + "a matrix";
    }

    print(msg + ", with degrees: " );
    draw(D, d); // print it nicely!
  }
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module A = grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0,0,1) );
  grview(A);

  module B = grobj( module([0,x,y]), intvec(15,1,1) );
  grview(B);

  module D = grsum( grsum(grpower(A,2), grtwist(1,1)), grsum(grtwist(1,2), grpower(B,2)) );
  grview(D);

  ring R = 0,(w,x,y,z), dp; def I = grobj( ideal(y2-xz, xy-wz, x2z-wyz), intvec(0) );
  list res1 = grres(I, 0); // non-minimal
  grview(res1);
  print(betti(res1,0), "betti");

  list res2 = grres(grshift(I, -10), 0, 1); //  minimal!
  grview(res2);
  print(betti(res2,0), "betti");
}

static proc issorted( intvec g, int s )
{
  g = s * g; //  "g: ", g;
  int i = size(g);

  for(; i > 1; i--)
  {
    if( (g[i] - g[i-1]) < 0 )
    {
      return (0);
    }
  }

  return (1);
}

static proc mysort( intvec gr, int s )
"
computes the permutation P of gr, such that (s*gr)[P] is ascendingly sorted
NOTE: looks like a bubble sort (was taken from sort) and modified to ensure stability!
TODO: replace with some kernel function if this turns out to be inefficient!!
"
{
  gr = s * gr;
  int m = size(gr);
  intvec pivot;

  int Bi;
  // compute reordering permutation pivot such that gr[pivot] is (stably) sorted
  for(Bi=m; Bi>0; Bi--) { pivot[Bi]=Bi; } // pivot = Id_m for starters

  int Bn,Bb; int P, D;

  int Bj = 0; int flag = 0;

  // Bi == 0
  while(Bj==0)
  {
    Bi++; Bj=1;
    for(Bn=1; Bn <= (m-Bi); Bn++)
    {
      D = (gr[pivot[Bn]] - gr[pivot[Bn+1]]); // sort gr
      P = (D > 0) or ( (D == 0) && ((pivot[Bn]-pivot[Bn+1]) > 0) ); // stability!?
      if(P)
      {
        Bb=pivot[Bn];
        pivot[Bn]=pivot[Bn+1];
        pivot[Bn+1]=Bb;

        Bj=0; flag = 1;
      }
    }
  }

  ASSUME(1, issorted(intvec(gr[pivot]), 1));

  return (pivot);
}

proc grdeg(M)
"USAGE:  grdeg(M), graded object M
RETURN:  intvec of degrees
PURPOSE: graded degrees of columns (generators) of M, describing the source of M
ASSUME:  M must be a graded object (matrix/module/ideal/mapping)
NOTE:    if M has zero cols it shoud have attrib(M,'degHomog') set.
EXAMPLE: example grdeg; shows an example
"
{
  ASSUME(1, grtest(M) );

  if ( typeof(attrib(M, "degHomog")) == "intvec" )
  {
    def t = attrib(M, "degHomog"); // graded degrees

    if( size(t) == 0 ){ return (t); } // ZERO!

    ASSUME(2, ncols(M) == size(t) );
    return (t);
  }

  if( ncols(M) == 0 ) { return (0:0); } // FIXME: Why???

  ASSUME(0, ncols(M) > 0);
  ASSUME(0, ncols(M) == size(M) );

  def w = grrange(M); // grading weights?

  if( size(M) == 0 ){ return (w); } // TODO: Q@Wolfram!???

  int m = ncols(M); // m > 0 in Singular!
  int n = nvars(basering) + 1; // index of mod. column in the leadexp

  module L = lead(M[1..m]); // leading module-terms for input column vectors
  intvec d = deg(L[1..m]); // their degrees
  intvec c = leadexp(L[1..m])[n]; // their module-components

//   w = intvec(-6665), w; // 0?????
  intvec gr = w[c]; //  + 1]; // weights?????

  gr = gr + d; // finally we compute their graded degrees

  return (gr);
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module A = grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0,0,1) );
  grview(A);

  module B = grobj( module([0,x,y]), intvec(15,1,1) );
  grview(B);

  module D = grsum(
                   grsum(grpower(A,2), grtwist(1,1)),
                   grsum(grtwist(1,2), grpower(B,2))
                  );

  grview(D);
  grdeg(D);

  def D10 = grshift(D, 10);

  grview(D10);
  grdeg(D10);
}

static proc reorder(def M, int s)
"
Reorder gens of M: compute graded degrees and the permutation to sort them
"
{
  // input should be graded:
  ASSUME(1, grtest(M) );

  intvec w = grrange(M); // grading weights

  intvec gr = grdeg( M );

//  intvec d = deg(M[1..nocls(M)]); // no need to deal with un-weighted degrees??!

  intvec pivot = mysort(gr, s);

  // grades & ordering permutation for N.  gr[pivot] should be sorted!
  ASSUME(1, issorted(gr[pivot], s));

  module N = grobj(module(M[pivot]), w, intvec(gr[pivot]));  // reorder the starting ideal/module

//  "reorder: "; grview(N);

  return (N, intvec(gr[pivot]));
}

proc grtranspose1(def M)
"USAGE:  grtranspose1(M), graded object or list M
RETURN:  same as input
PURPOSE: graded transpose of graded object or chain complex M
ASSUME:  M must be a graded object or a list of graded objects
EXAMPLE: example grtranspose1; shows an example
"
{
  if( typeof( M ) == "list" )
  {
    if( size(M) == 0 ) { return (); }

    int j = size(M);

    int i = 1;

    // TODO: extra grading argument???
    while( i < j )
    {
     if( size(M[i]) == 0 ){ break; }
     ASSUME(0, typeof(grrange(M[i])) == "intvec");
     i++;
    }

    if( size(M[i]) == 0 ) { i--; }

    j = i; i = 1;

    list L;
    while( j > 0 )
    {
//      grview(M[i]);
      L[j] = grtranspose1( grobj( M[i], grrange(M[i])) );
//      grview(L[j]);

      if( (i > 1) && (j > 0) )
      {
//        grview(L[j+1]);
        ASSUME(2, size( module( matrix(L[j])*matrix(L[j+1]) ) ) == 0 );
      };
//      grview(L[j]);
      j--; i++;
    };
    return (L); // ?
  }

//////
// "a";  grview(M);
  ASSUME(1, grtest(M) );

 intvec d; module N;

 (N,d) = reorder(M, -1);

 kill M; module M = grobj(transpose(N), -d, -grrange(N));

// "b";  grview(M);

 kill N,d; module N; intvec d;
 // reverse order:
 (N,d) = reorder(M, 1); kill M;

// "e"; grview( N );

 ASSUME(1, issorted( grrange(N), 1) );
 ASSUME(1, issorted(grdeg(N), 1) );


 return (N);
}
example
{ "EXAMPLE:"; echo = 2;

  "Surface Name: 'k3.d10.g9.quart2' in P^4";
  int @p=31991; ring R = (@p),(x,y,z,u,v), dp;
  ideal J = x3yz2+31/15x2y2z2-7231xy3z2+99/37y4z2+28/95x3z3+97/32x2yz3+13247xy2z3+12717y3z3-113/31x2z4-61/30xyz4-6844y2z4+104/3xz5-13849yz5+43/39z6+13061x3yzu-8463x2y2zu+94/69xy3zu-8/61y4zu-13297x3z2u+7217x2yz2u-7830xy2z2u-75/14y3z2u+2839x2z3u-14657xyz3u-52/7y2z3u-6/89xz4u-6169yz4u+44/7z5u-98/33x3yu2+41/30x2y2u2-65/98xy3u2+122/13y4u2+9906x3zu2-11587x2yzu2+17/53xy2zu2+6504y3zu2+49/106x2z2u2+11480xyz2u2+97/71y2z2u2+12560xz3u2-114/83yz3u2-13761z4u2-67/112x3u3-18/49x2yu3+21/67xy2u3-44/43y3u3+123/116x2zu3+4459xyzu3-13841y2zu3-805xz2u3-1382yz2u3-5293z3u3+133x2u4-122/79xyu4+9724y2u4+61/24xzu4+113/119yzu4-19/108z2u4+15893xu5+57/22yu5+4600zu5-618u6-27/53x3yzv+44/103x2y2zv-142xy3zv+19/84y4zv+105/8x3z2v+10532x2yz2v-75/74xy2z2v-70/19y3z2v+31/80x2z3v-481xyz3v+47/30y2z3v+14318xz4v+51/28yz4v-15/113z5v-46/17x3yuv-99/100x2y2uv-106/5xy3uv+14384y4uv+7/100x3zuv-15/64x2yzuv-6976xy2zuv+12051y3zuv-67/42x2z2uv-2627xyz2uv-49/104y2z2uv+77/16xz3uv+15766yz3uv+85/117z4uv-107/101x3u2v-6699x2yu2v+2443xy2u2v-27/28y3u2v+11945x2zu2v-14467xyzu2v-4873y2zu2v-63/124xz2u2v-8270yz2u2v+11900z3u2v+47/14x2u3v+53/8xyu3v-10/51y2u3v-87/119xzu3v+114/73yzu3v+86/57z2u3v+52/63xu4v-11587yu4v+1/18zu4v-121/109u5v+116/11x3yv2+19/108x2y2v2-31/3xy3v2-43/9y4v2-81/100x3zv2-7728x2yzv2-1037xy2zv2+24/101y3zv2-61/103x2z2v2-8/51xyz2v2+117/109y2z2v2+98/23xz3v2+1646yz3v2-3356z4v2+105/59x3uv2+117/31x2yuv2+519xy2uv2+12633y3uv2+25/6x2zuv2-963xyzuv2-49/23y2zuv2-116/25xz2uv2+14146yz2uv2+11480z3uv2-95/8x2u2v2-10928xyu2v2-51/23y2u2v2-12770xzu2v2-92/91yzu2v2+3872z2u2v2+3183xu3v2+6871yu3v2+90/37zu3v2+10019u4v2-69/88x3v3-1398x2yv3-97/72xy2v3-46/97y3v3+107/14x2zv3-20/89xyzv3-11367y2zv3+120/29xz2v3-86/81yz2v3+107/69z3v3-39/17x2uv3+83/11xyuv3+169y2uv3-11/71xzuv3-22/17yzuv3-14862z2uv3-13009xu2v3-101/12yu2v3+10617zu2v3+2567u3v3-23/85x2v4+27/50xyv4+113/51y2v4+97/16xzv4+4438yzv4-11857z2v4+14580xuv4-6426yuv4+9421zuv4-10585u2v4+14670xv5+1807yv5+10298zv5-116/53uv5+7869v6,x3yzu+31/15x2y2zu-7231xy3zu+99/37y4zu+28/95x3z2u+97/32x2yz2u+13247xy2z2u+12717y3z2u-113/31x2z3u-61/30xyz3u-6844y2z3u+104/3xz4u-13849yz4u+43/39z5u-124/43x3yu2-90/13x2y2u2-13244xy3u2-78/73y4u2+118/43x3zu2+37/67x2yzu2-10426xy2zu2+2412y3zu2-32/113x2z2u2+35/104xyz2u2+3952y2z2u2+9028xz3u2-1990yz3u2-59/109z4u2+15499x3u3+116/23x2yu3+95/58xy2u3+8/47y3u3+59/109x2zu3-29xyzu3+12412y2zu3+20/81xz2u3+2200yz2u3-13809z3u3+3889x2u4-8136xyu4+8922y2u4-4/121xzu4+82/113yzu4-65/23z2u4+101/53xu5+103/113yu5-99/118zu5-9524u6-2749x3yzv-7814x2y2zv+73/113xy3zv+9937y4zv-59/62x3z2v-23/12x2yz2v-10245xy2z2v+7130y3z2v-4427x2z3v+6656xyz3v+3448y2z3v-46/79xz4v+1611yz4v+8453z5v+12013x3yuv+49/17x2y2uv-4/115xy3uv-121/91y4uv-63/29x3zuv+64/7x2yzuv-8785xy2zuv-87/14y3zuv+36/121x2z2uv+9525xyz2uv+4215y2z2uv-17/13xz3uv-117/125yz3uv+101/122z4uv+42/37x3u2v-8747x2yu2v-105/79xy2u2v+10799y3u2v-58/49x2zu2v-8/75xyzu2v-67/49y2zu2v-38/11xz2u2v+53/27yz2u2v+113/52z3u2v+18/59x2u3v+71/106xyu3v+47/2y2u3v-4594xzu3v+95/4yzu3v-121/46z2u3v-55/62xu4v-101/72yu4v+40/53zu4v+15227u5v-15553x3yv2+29/94x2y2v2-4076xy3v2-7133y4v2+27/125x3zv2+33/29x2yzv2-63/95xy2zv2+9166y3zv2-480x2z2v2+9941xyz2v2+107/46y2z2v2+13018xz3v2+53/98yz3v2+92/35z4v2+17/30x3uv2+77/95x2yuv2+11/67xy2uv2+8262y3uv2+65/11x2zuv2+2567xyzuv2-33/94y2zuv2+85/92xz2uv2+103/25yz2uv2-27/100z3uv2+13210x2u2v2-109/90xyu2v2+141y2u2v2-124/51xzu2v2-3/109yzu2v2-4910z2u2v2+205xu3v2+14357yu3v2+85/57zu3v2-109/28u4v2-68/39x3v3+10545x2yv3-2176xy2v3-8743y3v3+15111x2zv3+25/119xyzv3+8/103y2zv3-6046xz2v3+8658yz2v3+106/5z3v3-31/126x2uv3-7762xyuv3+2315y2uv3+124/67xzuv3-77/104yzuv3+95/71z2uv3+69/119xu2v3+13069yu2v3-8620zu2v3+105/41u3v3-15772x2v4-11212xyv4-61/36y2v4+38/125xzv4-15860yzv4+8/63z2v4+7519xuv4-94/41yuv4+45/32zuv4+9417u2v4-71/35xv5-6287yv5+6481zv5+106/99uv5+3/41v6,-8036x3yu2+7966x2y2u2-151xy3u2-14/111y4u2-111/76x3zu2-102/11x2yzu2+7956xy2zu2-7397y3zu2-113/16x2z2u2-8049xyz2u2+7230y2z2u2+3978xz3u2-36/113yz3u2-8147z4u2-107/83x3u3+78/97x2yu3+12700xy2u3+11/72y3u3+88/31x2zu3-63/40xyzu3+101/35y2zu3-220xz2u3+3/103yz2u3-49/45z3u3-21/113x2u4+104/123xyu4+98/47y2u4-56/61xzu4-87/50yzu4+5913z2u4-120/17xu5+64/11yu5-109/80zu5+10371u6-118/25x3yzv+58/99x2y2zv-5/64xy3zv+7/46y4zv-49/103x3z2v-77/106x2yz2v-44/7xy2z2v-7559y3z2v-17/35x2z3v+948xyz3v-15043y2z3v-3576xz4v-2/109yz4v+74/11z5v+6436x3yuv+7316x2y2uv+29/5xy3uv-1326y4uv+34/49x3zuv-122/27x2yzuv-632xy2zuv+46/49y3zuv-13463x2z2uv-808xyz2uv-17/32y2z2uv-13149xz3uv-117/88yz3uv-45/79z4uv-65/94x3u2v+6/67x2yu2v+34/39xy2u2v-14026y3u2v+42/107x2zu2v-3287xyzu2v-70/43y2zu2v+29/104xz2u2v-47/18yz2u2v-11038z3u2v+6262x2u3v-5255xyu3v-7/10y2u3v+7065xzu3v+5608yzu3v+4675z2u3v-73/90xu4v-15822yu4v-71/63zu4v+110/97u5v-69/5x3yv2+4315x2y2v2-124/45xy3v2-79/16y4v2-10739x3zv2-93/46x2yzv2+12499xy2zv2-73/86y3zv2+6367x2z2v2-12876xyz2v2-306y2z2v2-89xz3v2-70/51yz3v2+13120z4v2+61/57x3uv2+14782x2yuv2-91/9xy2uv2-2625y3uv2+14747x2zuv2-5899xyzuv2-12944y2zuv2-47/14xz2uv2-4551yz2uv2-99/101z3uv2-12618x2u2v2+1507xyu2v2-11951y2u2v2+68/49xzu2v2+49/39yzu2v2-56/103z2u2v2-31/85xu3v2-32/49yu3v2-65/14zu3v2+15/7u4v2+5749x3v3-3667x2yv3-107/29xy2v3+11301y3v3+95/18x2zv3-121/74xyzv3+75/26y2zv3+101/98xz2v3-111/76yz2v3-11335z3v3-15923x2uv3-36/83xyuv3-4134y2uv3-87/118xzuv3-41/11yzuv3+104/61z2uv3+12583xu2v3-50/23yu2v3-31/44zu2v3-29/23u3v3+108/107x2v4-8216xyv4-5009y2v4+101/26xzv4-9779yzv4+71/74z2v4-3358xuv4+83/84yuv4-34/39zuv4+44/47u2v4-112/83xv5+113/74yv5+82/79zv5-115/99uv5+12/109v6,-x4y-31/15x3y2+7231x2y3-99/37xy4-28/95x4z-53/107x3yz-4623x2y2z+5300xy3z-41/111y4z+12205x3z2+113/120x2yz2+54/49xy2z2-85/63y3z2+104/89x2z3-52/121xyz3-22/49y2z3+14367xz4+71/93yz4+55/56z5-5/81x4u-67/81x3yu-83/13x2y2u+98/55xy3u+15289y4u-94/111x3zu+40/29x2yzu-16/59xy2zu-107/14y3zu+2965x2z2u-459xyz2u-2/47y2z2u+35/22xz3u+119/39yz3u-12180z4u-13679x3u2+1534x2yu2+11305xy2u2-62/9y3u2-68/39x2zu2+11/90xyzu2-36/101y2zu2-2896xz2u2-15114yz2u2-49/114z3u2+19/16x2u3-11401xyu3-109/3y2u3+67/80xzu3+53/92yzu3+2894z2u3+119/74xu4+407yu4-65/53zu4+95/94u5-9309x4v+21/40x3yv+1436x2y2v+2194xy3v+6994y4v-116/81x3zv+13/2x2yzv-12/13xy2zv-23/84y3zv-61/83x2z2v+2023xyz2v+19/40y2z2v+43/26xz3v-59/113yz3v-47/53z4v+15580x3uv+21x2yuv+113/97xy2uv-15419y3uv-15243x2zuv+5128xyzuv-34/47y2zuv+13206xz2uv-4833yz2uv+107/91z3uv-1693x2u2v+54/53xyu2v-86/67y2u2v+98/9xzu2v+86/17yzu2v+64/89z2u2v+25/113xu3v+7884yu3v+14089zu3v-12027u4v-9471x3v2-36/85x2yv2-21/13xy2v2+15888y3v2+76/109x2zv2+4547xyzv2+115/12y2zv2-11/107xz2v2+6764yz2v2-8321z3v2+84/101x2uv2-202xyuv2+3251y2uv2+91/4xzuv2+7124yzuv2-53/81z2uv2+47/84xu2v2-8833yu2v2+117/14zu2v2-3/113u3v2+126/97x2v3-78/115xyv3+68/63y2v3-34/109xzv3+5913yzv3+6226z2v3-2365xuv3+91/120yuv3+14120zuv3-69/8u2v3+71/12xv4-13094yv4-7262zv4-33uv4+5367v5,-9533x4y-318x3y2+8/49x2y3+83/29xy4+13129y5+221x4z+115/48x3yz+12508x2y2z+97/52xy3z+11479y4z+8941x3z2+104/109x2yz2+9191xy2z2+103/64y3z2+10584x2z3-7728xyz3+3979y2z3+15/82xz4+5409yz4-1326z5+3756x4u-57/62x3yu+63/47x2y2u-14600xy3u+159y4u-11/4x3zu-113/57x2yzu-26/125xy2zu-32/87y3zu-10/21x2z2u+12927xyz2u-73/62y2z2u+115/99xz3u-13/3yz3u-126/25z4u-3969x3u2-122/57x2yu2-5003xy2u2-100/117y3u2-71/30x2zu2+7356xyzu2-2211y2zu2+31/40xz2u2-6722yz2u2-139z3u2+4426x2u3+1/115xyu3-72/85y2u3+15260xzu3+7938yzu3+4/115z2u3-33/89xu4+31/108yu4-50/83zu4+14/107u5+24/95x4v-113/17x3yv+81/14x2y2v-9957xy3v-10075y4v-122/113x3zv+65/118x2yzv-96/29xy2zv-19/41y3zv+113/35x2z2v+121/31xyz2v-9/68y2z2v+91/45xz3v-23/116yz3v-67/99z4v-5355x3uv-3112x2yuv-12824xy2uv-58/123y3uv-13/22x2zuv-19/85xyzuv-121/24y2zuv-14093xz2uv+99/95yz2uv+89/50z3uv+13096x2u2v-109/120xyu2v+121/61y2u2v+80/41xzu2v-39yzu2v-8/99z2u2v+5/17xu3v+112/69yu3v+14346zu3v-7173u4v+125/13x3v2+43/53x2yv2-78/103xy2v2-109/111y3v2+33/13x2zv2-15333xyzv2+87/49y2zv2-7212xz2v2+7729yz2v2-86/123z3v2-119/103x2uv2-71/122xyuv2-81/113y2uv2+6133xzuv2+55/72yzuv2+69/31z2uv2+12828xu2v2+94/15yu2v2-7588zu2v2+21/41u3v2-8712x2v3+74/9xyv3-11/87y2v3+1446xzv3-3/95yzv3-87/55z2v3-717xuv3-110/97yuv3-13/113zuv3-95/81u2v3-37/68xv4+5112yv4-56/11zv4-6/115uv4+7910v5,25/42x4y-42/79x3y2-59/21x2y3+2736xy4-107/115x4z-203x3yz+47/101x2y2z+7686xy3z-63/64y4z+103/57x3z2-12082x2yz2+11/102xy2z2-83/43y3z2+13/49x2z3-2685xyz3+123/44y2z3+31/12xz4+126/83yz4+14745z5+83/37x4u+7362x3yu-14615x2y2u-14109xy3u+49/47y4u+1929x3zu+83/71x2yzu-13640xy2zu-97/58y3zu-11141x2z2u-61/49xyz2u-3745y2z2u-74/21xz3u+3493yz3u-7540z4u-103/118x3u2-43/32x2yu2-9200xy2u2-23/65y3u2+15895x2zu2-13924xyzu2-14291y2zu2-11039xz2u2-31/37yz2u2-101/93z3u2-39/83x2u3-4536xyu3-78/47y2u3+75/44xzu3-24/121yzu3-81/113z2u3-81/89xu4+15825yu4-4111zu4+5850u5-12534x4v-69/94x3yv-10076x2y2v+3952xy3v+25/12y4v+21/34x3zv+11002x2yzv-54xy2zv+20/23y3zv+4991x2z2v+549xyz2v+2687y2z2v-110/9xz3v+11359yz3v+49/24z4v+62/107x3uv-27/41x2yuv-17/52xy2uv-10972y3uv+12/103x2zuv-318xyzuv-77/40y2zuv-114/53xz2uv+17/28yz2uv-8084z3uv+85/36x2u2v+7/100xyu2v-5772y2u2v-89/114xzu2v-40/121yzu2v+3340z2u2v+36/113xu3v-38/93yu3v+2519zu3v-7084u4v+8136x3v2-55/23x2yv2+27/7xy2v2+74/39y3v2+63/16x2zv2-8661xyzv2+2/91y2zv2+3773xz2v2-75/122yz2v2+447z3v2-59/109x2uv2-119/9xyuv2-67/49y2uv2-11334xzuv2-10482yzuv2-60/91z2uv2+94/65xu2v2-108/17yu2v2-69/70zu2v2-23/20u3v2+8/115x2v3+29/41xyv3+8/15y2v3-95/6xzv3-9714yzv3+2550z2v3-121/80xuv3+67/18yuv3+43/5zuv3+23/124u2v3-12509xv4-104/79yv4-73/21zv4-1238uv4+9038v5,94/107x4y+47/14x3y2-6362x2y3-20/59xy4-43/120y5-3028x4z-15141x3yz-2028x2y2z+84/115xy3z-3024y4z+2811x3z2+47/45x2yz2+121/101xy2z2-100/57y3z2+8/115x2z3+1/101xyz3-13/112y2z3+3618xz4+88/67yz4-52/63z5+102/97x4u-12/89x3yu-102x2y2u-3846xy3u-61/86y4u+85/54x3zu+78/29x2yzu-13381xy2zu-49/95y3zu-77/2x2z2u-5784xyz2u+1557y2z2u-9163xz3u-114/121yz3u-57/103z4u+36/31x3u2-9062x2yu2-23/111xy2u2+7362y3u2-7671x2zu2+14945xyzu2+7901y2zu2+51/5xz2u2-109/48yz2u2+7696z3u2+11280x2u3-44/57xyu3-13736y2u3-13458xzu3-14723yzu3-707z2u3+899xu4-10381yu4+99/25zu4-7788u5-237x4v+45/43x3yv-7666x2y2v-4/109xy3v+4303y4v-13107x3zv-108/91x2yzv-7707xy2zv-73/47y3zv+61/118x2z2v-11/65xyz2v+2970y2z2v-104/37xz3v-15408yz3v-64/55z4v+47/113x3uv+2185x2yuv+7941xy2uv-61/37y3uv+6482x2zuv-11/70xyzuv+83/110y2zuv-109/83xz2uv-86/95yz2uv-7583z3uv+83/45x2u2v+89/38xyu2v-2/11y2u2v+3577xzu2v+124/125yzu2v-1151z2u2v+109/85xu3v+70/13yu3v+37/104zu3v-210u4v+51/29x3v2-104/111x2yv2+105/58xy2v2-13459y3v2-80/79x2zv2-3006xyzv2-115/16y2zv2+8208xz2v2+35/38yz2v2+49/27z3v2-1647x2uv2+10482xyuv2-34/93y2uv2+97/18xzuv2+101/20yzuv2+1711z2uv2+91/36xu2v2-96/23yu2v2+7006zu2v2+86/31u3v2-10734x2v3-43/18xyv3-4597y2v3-11174xzv3-7334yzv3+7/96z2v3+4/97xuv3-5/82yuv3-15600zuv3-69/94u2v3-71/25xv4+21/97yv4+117/23zv4-6557uv4-67/83v5,8164x4y+19/73x3y2-1592x2y3-28/87xy4-63/103x4z+11/42x3yz-52/67x2y2z-13766xy3z+11378y4z+10/37x3z2+115/41x2yz2+11/100xy2z2-49/40y3z2+86/111x2z3+124/5xyz3-25/79y2z3-14525xz4+11380yz4-53/42z5-12169x4u-14/51x3yu+68/33x2y2u-3/62xy3u-31/22y4u-74/93x3zu+12924x2yzu-103/123xy2zu-74/97y3zu-2789x2z2u-95/32xyz2u+45/13y2z2u+40/71xz3u+49/110yz3u+34/75z4u+9829x3u2-59/92x2yu2+106/65xy2u2+123/86y3u2+7133x2zu2-73/46xyzu2-7/29y2zu2-937xz2u2-65/67yz2u2-88/111z3u2-61/119x2u3+975xyu3-54/7y2u3-37/33xzu3+61/59yzu3+51/115z2u3+117/43xu4+8506yu4+13941zu4-14945u5-115/63x4v-14237x3yv-74/87x2y2v+104/47xy3v-95/104y4v+11535x3zv-119/75x2yzv-44xy2zv+11299y3zv-21/113x2z2v-2852xyz2v+95/77y2z2v-75/19xz3v-4864yz3v-79/88z4v+139x3uv-10068x2yuv+2049xy2uv+7515y3uv+97/56x2zuv+109/113xyzuv+7778y2zuv-71/11xz2uv-80/19yz2uv+55/59z3uv-69/98x2u2v-15679xyu2v+114/11y2u2v+69/65xzu2v+879yzu2v+45/104z2u2v+47/97xu3v-1373yu3v+15885zu3v+11121u4v-5042x3v2+4/25x2yv2-8607xy2v2-25/33y3v2+93/55x2zv2+68xyzv2-4167y2zv2+14180xz2v2-115/47yz2v2-81/67z3v2-12099x2uv2+34/107xyuv2+122/59y2uv2+775xzuv2-91yzuv2-85/96z2uv2-59/95xu2v2+174yu2v2+11/16zu2v2+66/37u3v2-121/36x2v3+6070xyv3-83/52y2v3-121/59xzv3-55/12yzv3+8088z2v3-20/29xuv3+76/125yuv3-10858zuv3+1833u2v3-103/50xv4+76/93yv4-119/18zv4+37/114uv4+51/7v5,85/56x4y-7839x3y2+12/37x2y3+6558xy4-8191x4z+115/7x3yz+81/23x2y2z-4121xy3z-1131y4z-23/37x3z2-71/32x2yz2+30/97xy2z2+5070y3z2-49/123x2z3+103/88xyz3-45/19y2z3+5132xz4+7277yz4+1896z5-103/75x4u-12020x3yu+12337x2y2u+6248xy3u+14290y4u-87/44x3zu-5364x2yzu-11801xy2zu-59/37y3zu+34/109x2z2u-14482xyz2u-10338y2z2u+118/73xz3u+7/8yz3u+158z4u+10590x3u2-5182x2yu2+83/62xy2u2+11557y3u2-92/119x2zu2-37/94xyzu2+5383y2zu2-365xz2u2+7/62yz2u2-7965z3u2-10/43x2u3+119/101xyu3-113/83y2u3-121/41xzu3+61/104yzu3+37/60z2u3-74/95xu4-113/66yu4-205zu4+4787u5-94/93x4v+14871x3yv-14723x2y2v+10730xy3v+112/17y4v-35/19x3zv-3487x2yzv-65/43xy2zv-7445y3zv-79/124x2z2v+7423xyz2v+91/2y2z2v+91/34xz3v-6970yz3v-50/113z4v+75/43x3uv-127x2yuv+11978xy2uv+48/113y3uv+113/62x2zuv-8941xyzuv-101/112y2zuv-5737xz2uv-31/123yz2uv+9490z3uv+19/92x2u2v-107/73xyu2v-23/121y2u2v+38/65xzu2v-672yzu2v+13/77z2u2v+46/119xu3v-103/18yu3v+107/59zu3v-52/21u4v-94/87x3v2-74/31x2yv2-9/22xy2v2-2896y3v2+113/3x2zv2-5386xyzv2-11391y2zv2+42/97xz2v2+77/64yz2v2-1610z3v2-102/43x2uv2+124/39xyuv2+14829y2uv2+88/113xzuv2-10411yzuv2-51/43z2uv2-36/121xu2v2+9487yu2v2-5589zu2v2+4335u3v2-5/91x2v3+6084xyv3-56/39y2v3-84/101xzv3-81/85yzv3-6521z2v3-2432xuv3+14317yuv3-43/82zuv3+121/8u2v3+14783xv4-92/45yv4+112/27zv4-8410uv4+31/105v5,-6691x4y-10158x3y2-5372x2y3+4132xy4+106/9y5+15600x4z-803x3yz+43/29x2y2z+9/91xy3z-92/61y4z+4807x3z2-12562x2yz2+14234xy2z2-91/17y3z2-91/30x2z3-10615xyz3-4206y2z3-29/45xz4-11/86yz4-115/9z5+125/112x4u+52/59x3yu+92/49x2y2u+121/85xy3u-51/14y4u-73/48x3zu-1/110x2yzu+12/65xy2zu+15045y3zu+12826x2z2u-123/89xyz2u+9465y2z2u-67/31xz3u-5080yz3u-7944z4u-107/72x3u2+1473x2yu2+7965xy2u2+15753y3u2-95/98x2zu2-9827xyzu2-25/53y2zu2-83/54xz2u2-13217yz2u2-117/110z3u2+230x2u3-12120xyu3+11/36y2u3-2071xzu3+109/59yzu3+6909z2u3-15/64xu4+45/82yu4-3091zu4-15711u5+5957x4v-45/86x3yv+26/29x2y2v-40/57xy3v+25/43y4v+126/37x3zv-38/33x2yzv+65/109xy2zv-33/68y3zv-7287x2z2v-4842xyz2v+35/118y2z2v+6157xz3v-97/89yz3v-91/50z4v-70/27x3uv+32/9x2yuv+78/125xy2uv+38/7y3uv-3214x2zuv-68/101xyzuv+87/55y2zuv-69/98xz2uv+5805yz2uv+41/102z3uv-43/54x2u2v-42/73xyu2v-13/49y2u2v+11864xzu2v+121/37yzu2v-100/109z2u2v-12609xu3v-9114yu3v-8746zu3v+11659u4v+3799x3v2-9581x2yv2+60/91xy2v2+2029y3v2+12075x2zv2+210xyzv2-1/22y2zv2+17/58xz2v2+1212yz2v2+118/27z3v2-3571x2uv2-3139xyuv2-23/100y2uv2-1240xzuv2+71/49yzuv2-21/103z2uv2-110/71xu2v2-40/77yu2v2-103/29zu2v2+10737u3v2+2828x2v3+14/39xyv3+7564y2v3+113/50xzv3+38/79yzv3+59/66z2v3+2726xuv3+91/94yuv3-15730zuv3-13408u2v3-97/42xv4+54/29yv4-33/73zv4+4823uv4+57/71v5,-14556x3yz-9751x2y2z-45/28xy3z+85/23y4z+5623x3z2+5369x2yz2-19/60xy2z2-36/5y3z2-95/36x2z3+5862xyz3-5/93y2z3+2949xz4+11357yz4-5679z5-52/45x3yu+4448x2y2u-9/22xy3u+2427y4u+3296x3zu+16/39x2yzu+53/57xy2zu+15/41y3zu+9473x2z2u+37xyz2u-58/69y2z2u-23/56xz3u-13/90yz3u-54/29z4u-41/67x3u2+10258x2yu2+23/44xy2u2-12952y3u2+2124x2zu2-1677xyzu2+12911y2zu2+22/45xz2u2+17/84yz2u2+5910z3u2+4782x2u3+119/39xyu3-17/84y2u3-120/91xzu3+35/59yzu3+17/77z2u3-4467xu4-77/4yu4-26/53zu4-3580u5-11977x3yv-118/77x2y2v+6040xy3v+9724y4v-47/5x3zv+59/101x2yzv+1212xy2zv-7/121y3zv+93/53x2z2v-56/23xyz2v-4470y2z2v+110/111xz3v-41/99yz3v-81/10z4v-71/24x3uv+26/115x2yuv+59/39xy2uv-10029y3uv+11748x2zuv+5749xyzuv+6887y2zuv+38/3xz2uv-116/61yz2uv-55/118z3uv+105/22x2u2v+70/87xyu2v-28/13y2u2v-109/123xzu2v-102/47yzu2v-52/71z2u2v+101/95xu3v+51/16yu3v+15/97zu3v-78/125u4v+35/46x3v2-9526x2yv2+10781xy2v2-119/44y3v2-23/10x2zv2+59/29xyzv2-15144y2zv2+29/120xz2v2-53/126yz2v2-93/85z3v2+53/8x2uv2-487xyuv2-12143y2uv2+13825xzuv2+55/6yzuv2-4250z2uv2+4237xu2v2-109/9yu2v2+67/53zu2v2+82/33u3v2+8660x2v3+15046xyv3-79/84y2v3-10310xzv3+110yzv3-7636z2v3+57/92xuv3-22/119yuv3-95/103zuv3+5138u2v3+123/49xv4-7587yv4+30/41zv4-124/121uv4+54/71v5,-29/60x4y-108/77x3y2-109/37x2y3-3619xy4+109/6x4z-37/67x3yz+53/45x2y2z+5291xy3z-2927y4z+34/5x3z2+87/17x2yz2+100/89xy2z2-114/29y3z2-4057x2z3-1/42xyz3-14/61y2z3-398xz4-122/73yz4+66/37z5+99/37x4u-5691x3yu-8778x2y2u+17/115xy3u+51/113y4u-71/101x3zu+85/91x2yzu-92/9xy2zu-3442y3zu+109/26x2z2u+50/37xyz2u+77/94y2z2u+16/35xz3u+9985yz3u+5/102z4u-5932x3u2+89/125x2yu2-895xy2u2-12455y3u2-630x2zu2-64/47xyzu2+25/9y2zu2+7906xz2u2+6827yz2u2+9808z3u2-113/118x2u3+79/8xyu3+9484y2u3+62/39xzu3+6/85yzu3-23/49z2u3-93/115xu4-11/93yu4-15177zu4-13/2u5-7623x4v-103/73x3yv-96/115x2y2v+39/76xy3v+80/79y4v+43/68x3zv+45/97x2yzv+101/87xy2zv+4632y3zv-918x2z2v+8248xyz2v-4276y2z2v+8853xz3v-39/61yz3v-121/87z4v+9968x3uv+473x2yuv+117/56xy2uv-19/21y3uv+121/119x2zuv+3/98xyzuv-65/42y2zuv-3723xz2uv+7/34yz2uv-112/87z3uv+103x2u2v+25/41xyu2v-14459y2u2v-56/41xzu2v-59/81yzu2v-109/102z2u2v-87/16xu3v-13011yu3v+49/123zu3v+106/89u4v-61/51x3v2+14107x2yv2+8035xy2v2-8853y3v2+5723x2zv2+123/53xyzv2-9727y2zv2-102/83xz2v2+1111yz2v2-15745z3v2+83/118x2uv2-57/35xyuv2-48/73y2uv2-28/37xzuv2-27/97yzuv2-27/58z2uv2+71/93xu2v2+117/8yu2v2+12344zu2v2-2497u3v2-118/71x2v3-11/19xyv3+21/104y2v3+32/113xzv3+15544yzv3+31/18z2v3+5909xuv3-67/58yuv3+27/35zuv3+115/9u2v3+79/13xv4+6722yv4-37/114zv4-71/124uv4+4657v5,-77/61x4y-88/101x3y2+93/88x2y3-11/70xy4+9806y5+7896x4z-4699x3yz+55/122x2y2z-63/122xy3z-125/74y4z+47/45x3z2+101/17x2yz2+92/47xy2z2+69/82y3z2+12402x2z3+113/98xyz3-101/33y2z3-15376xz4+47/71yz4-73/10z5+65/74x4u-14409x3yu-14478x2y2u+13593xy3u+102/97y4u+39/62x3zu-34/125x2yzu-83/9xy2zu+45/113y3zu+14484x2z2u-15293xyz2u-26/55y2z2u-958xz3u+67/35yz3u-93/19z4u+25/16x3u2+107/52x2yu2-4599xy2u2-86/51y3u2-9885x2zu2-77/47xyzu2+33/65y2zu2+90/109xz2u2-61/26yz2u2+6198z3u2-38/37x2u3-13935xyu3-142y2u3-64/5xzu3-7228yzu3+1251z2u3+1556xu4+117/121yu4-92/35zu4+99/92u5+13493x4v+12654x3yv+32/101x2y2v-11118xy3v+43/51y4v-575x3zv+103/21x2yzv+85/24xy2zv+1788y3zv+85/3x2z2v-64/25xyz2v+57/35y2z2v+37/120xz3v-69/110yz3v+48/49z4v+55/114x3uv-6439x2yuv+31/51xy2uv-90/49y3uv-45/104x2zuv-12018xyzuv+6/119y2zuv+40/63xz2uv+20/91yz2uv+50/43z3uv+1/26x2u2v-109/47xyu2v+99/7y2u2v+72/83xzu2v+61/118yzu2v+3530z2u2v+6146xu3v+117yu3v-9921zu3v-8708u4v-10/47x3v2-15294x2yv2-7336xy2v2+1/66y3v2-3057x2zv2+74/123xyzv2+146y2zv2-103/34xz2v2-117/76yz2v2+8472z3v2-7/92x2uv2+10033xyuv2+43/53y2uv2+4694xzuv2-49/2yzuv2-71/73z2uv2-125/17xu2v2-9817yu2v2+7218zu2v2+6897u3v2-19/90x2v3+11899xyv3-11779y2v3-5456xzv3+17/42yzv3+15340z2v3+12/7xuv3+9580yuv3-502zuv3-14069u2v3-4371xv4+14452yv4-9423zv4-117/122uv4+1126v5,49/108x4-39/4x3y-67/21x2y2-8/69xy3-9779y4+57/14x3z-11145x2yz+6928xy2z-7824y3z+1/79x2z2+5173xyz2-62/15y2z2-123/112xz3+88/79yz3+1/125z4+57/23x3u-11856x2yu-7444xy2u+115/8y3u-11133x2zu+71/73xyzu-7941y2zu+69/65xz2u+22/75yz2u+65/121z3u+9471x2u2+9167xyu2+51/59y2u2+12835xzu2+15047yzu2+11102z2u2-10059xu3+19/28yu3+65/21zu3-39/28u4-3/73x3v+94/61x2yv+8778xy2v-12922y3v-8711x2zv-37/97xyzv+14270y2zv+4487xz2v-59/112yz2v-14183z3v+15553x2uv+3579xyuv+114/91y2uv-4/97xzuv+13/85yzuv-89/15z2uv+58/75xu2v-34/7yu2v-90/61zu2v+90/101u3v-14673x2v2+90/19xyv2-45/37y2v2+23/49xzv2-71/11yzv2+119/8z2v2+89/10xuv2+109/91yuv2+36/49zuv2-7/31u2v2-40/113xv3-21/121yv3+9910zv3+33/14uv3-23/79v4,93/70x4-43/125x3y+9582x2y2+7565xy3-11511y4-3/79x3z-36/107x2yz-2038xy2z+879y3z-4700x2z2+103/14xyz2+102/79y2z2-67/68xz3-44/25yz3+105/79z4-29/24x3u-74/83x2yu+67/43xy2u+49/12y3u-115/11x2zu+23/67xyzu-61/27y2zu+12257xz2u+14068yz2u+23/15z3u+607x2u2+73/8xyu2+14237y2u2-13/33xzu2+110/71yzu2+41/101z2u2+5708xu3+88/67yu3+1460zu3-2472u4-1629x3v-51/70x2yv-88/73xy2v-36/97y3v+38/11x2zv+15899xyzv+54/19y2zv+9460xz2v-5150yz2v+3462z3v+5522x2uv-19/123xyuv+14871y2uv+53/5xzuv-7535yzuv-13430z2uv+107/47xu2v-8307yu2v-55/79zu2v-11945u3v-16/83x2v2+115/48xyv2+12389y2v2+11545xzv2-25/26yzv2-3755z2v2+4724xuv2-31/21yuv2+7872zuv2+89/45u2v2+87/47xv3+7625yv3+13494zv3-15376uv3-25/126v4;

  def I = grobj( groebner(J), intvec(0) ); // ASSUME: no zero entries in J!
  ASSUME(0, grtest(I));
  "Input degrees: "; grview(I);

  def RR = grres(I, 0, 1); list L = RR;

  " = Non-minimal betti numbers: "; print(betti(L, 0), "betti");

  "Graded (original) structure of 'res(Input,0)': "; grview(L);

  "Graded transpose of the previous resolution "; list LLL = grtranspose1( L ); grview( LLL );

  "Its non-minimal betti numbers: "; print(betti(LLL, 0), "betti");

}

proc grorder(def M)
"USAGE:  grorder(M), graded object or list M
RETURN:  same as input
PURPOSE: reorder/transform graded object or chain complex M into block form
ASSUME:  M must be a graded object or a list of graded objects
EXAMPLE: example grorder; shows an example
"
{
  if( typeof(M) == "list" )
  { // TODO: extra grading argument???
    if( size(M) == 0 ) { return (); }

    int j = size(M);  int i = 1;

    while( i < j )
    {
      if( size(M[i]) == 0 ){ break; }
      ASSUME(0, typeof(grrange(M[i])) == "intvec");
      i++;
    }

    if( size(M[i]) == 0 ) { i--; }

    list L; module Z = 0; L[i] = Z; j = i;

    while( i > 0 )
    {
//      "i: ", i;      "A"; grview(M[i]);
      L[i] = grorder( grobj( M[i], grrange(M[i])) );
//      "B"; grview(L[i]);
      if( i < j )
      {
        ASSUME(2, size( module( matrix(transpose(L[i+1]))*matrix(transpose(L[i])) ) ) == 0 );
      };

      i--;
    };

    return (L); // ?
  }

  ASSUME(1, grtest(M) );

// "a";  grview(M);

  intvec d; module N;

  (N,d) = reorder(M, 1); kill M;

  module M = grobj(transpose(N), -d, -grrange(N)); kill N,d;

// "b";  grview(M);

  module N; intvec d;
  // reverse order:
  (N,d) = reorder(M, -1); kill M;

  module M = grobj(transpose(N), -d, -grrange(N));

// "c";  grview(M);

  ASSUME(1, issorted(grrange(M), 1) );
  ASSUME(1, issorted(grdeg(M), 1) );

  return (M);
}
example
{ "EXAMPLE:"; echo = 2;

  "Surface Name: 'rat.d10.g9.quart2' in P^4";
  int @p=31991; ring R = (@p),(x,y,z,u,v), dp;
  ideal J = x3yu2-48/11x2y2u2-8356xy3u2+35/121y4u2+31/66x3zu2-54/83x2yzu2-61/18xy2zu2+11526y3zu2+7372x2z2u2-91/60xyz2u2-95/97y2z2u2-45/71xz3u2+71/115yz3u2+25/54z4u2-61/102x3u3-12668x2yu3+6653xy2u3+41/54y3u3+87/50x2zu3-5004xyzu3+13924y2zu3+2310xz2u3-93/14yz2u3-2/93z3u3-97/125x2u4-58/11xyu4+46/73y2u4-4417xzu4+60/101yzu4+56/75z2u4-113/118xu5+115/4yu5-40zu5-8554u6-54/83x3yuv-9770x2y2uv-590xy3uv+15/49y4uv+94/69x3zuv+121/105x2yzuv+95/88xy2zuv+3186y3zuv+11/6x2z2uv-44/81xyz2uv+637y2z2uv+109/121xz3uv-33yz3uv-94/115z4uv-49/95x3u2v-11/109x2yu2v+45/113xy2u2v+97/84y3u2v+5257x2zu2v+99/49xyzu2v+12584y2zu2v-4294xz2u2v+1137yz2u2v-58/69z3u2v-4749x2u3v+120/97xyu3v-31/103y2u3v+62/97xzu3v-107/74yzu3v+53/59z2u3v+91/33xu4v+1291yu4v+23/34zu4v+58/77u5v+16/17x3yv2-750x2y2v2+86/89xy3v2+123/46y4v2+53/123x3zv2-61/99x2yzv2+12389xy2zv2+10419y3zv2+43/11x2z2v2-146xyz2v2-116/51y2z2v2+13/62xz3v2-5524yz3v2-111/118z4v2-56/55x3uv2-3038x2yuv2+14/27xy2uv2-43/64y3uv2+3385x2zuv2+25/11xyzuv2+92/41y2zuv2+28/113xz2uv2-2049yz2uv2+89/37z3uv2-13094x2u2v2-2774xyu2v2+15474y2u2v2-15791xzu2v2-71/116yzu2v2+77/41z2u2v2-83/68xu3v2-33/106yu3v2+71/37zu3v2-41/17u4v2+12052x3v3+1906x2yv3+13825xy2v3+80/7y3v3-125/96x2zv3-9661xyzv3+85/116y2zv3-72/91xz2v3+13/112yz2v3-126/97z3v3-1637x2uv3+34/103xyuv3+3844y2uv3+77/10xzuv3+6359yzuv3-11185z2uv3-124/121xu2v3+66/91yu2v3-14636zu2v3-1051u3v3+9/64x2v4-12924xyv4-119/41y2v4+74/23xzv4+1622yzv4+73/37z2v4-60/101xuv4+111/22yuv4-45/124zuv4+59/37u2v4-66/37xv5-71/99yv5+12409zv5-113/64uv5-5267v6,-x4y-22/79x3y2-125/42x2y3-116/7xy4+98/111y5-31/66x4z-118/75x3yz+110/93x2y2z-43/92xy3z-788y4z-7372x3z2-2701x2yz2-67/124xy2z2-117/62y3z2+45/71x2z3-8396xyz3-10343y2z3-25/54xz4+30/59yz4+61/102x4u+11736x3yu+12726x2y2u+41/118xy3u-15832y4u-87/50x3zu-130x2yzu+41/8xy2zu-10300y3zu-2310x2z2u-101/5xyz2u+6205y2z2u+2/93xz3u+8679yz3u+97/125x3u2-43/37x2yu2-39/80xy2u2+12139y3u2+4417x2zu2+4294xyzu2+11/58y2zu2-56/75xz2u2+8338yz2u2+113/118x2u3-10190xyu3-37/16y2u3+40xzu3+74/23yzu3+8554xu4+115/22yu4-39/79x4v+61/72x3yv+8048x2y2v-9201xy3v+16/121y4v+113/93x3zv+109/75x2yzv+12700xy2zv-10607y3zv+50/11x2z2v+1223xyz2v-103/79y2z2v-123/58xz3v+31/26yz3v-15/122z4v+122/25x3uv-99/17x2yuv+1723xy2uv-38/121y3uv+11016x2zuv-25/102xyzuv-14970y2zuv-61/6xz2uv-14981yz2uv+15900z3uv+3268x2u2v-75/19xyu2v-1436y2u2v-1764xzu2v-57/41yzu2v+12741z2u2v-14615xu3v+119/61yu3v-115/119zu3v+10501u4v-8502x3v2-51/76x2yv2-6281xy2v2+17/49y3v2-106/7x2zv2+63/101xyzv2-27/95y2zv2-1606xz2v2+9245yz2v2+1912z3v2+11155x2uv2+223xyuv2-13/18y2uv2+110/43xzuv2+76/81yzuv2-6291z2uv2+1400xu2v2-95/23yu2v2-9701zu2v2+106/105u3v2+72/47x2v3-13118xyv3+14409y2v3+37/86xzv3+44/69yzv3-325z2v3+113/71xuv3+16/81yuv3+6/19zuv3-119/39u2v3-89/9xv4+72/53yv4+112/55zv4-8587uv4-6604v5,-x3y2+48/11x2y3+8356xy4-35/121y5-12750x3yz+100/111x2y2z+45/74xy3z+99/74y4z-6/7x3z2-47/67x2yz2+11465xy2z2-11865y3z2+7776x2z3+124/45xyz3-98/115y2z3+117/85xz4-59/120yz4-8748z5+61/102x3yu+12668x2y2u-6653xy3u-41/54y4u+13408x3zu-2185x2yzu-1240xy2zu+1161y3zu+44/27x2z2u-11164xyz2u-13388y2z2u-107/13xz3u+90/71yz3u+4204z4u+97/125x2yu2+58/11xy2u2-46/73y3u2+55/48x2zu2+121/31xyzu2+126/61y2zu2-55/69xz2u2+5988yz2u2+3755z3u2+113/118xyu3-115/4y2u3+3390xzu3-5762yzu3+30/61z2u3+8554yu4-14317zu4+99/116x3yv-113/119x2y2v+50/23xy3v-37/79y4v-8668x3zv+14049x2yzv+111/35xy2zv+61/28y3zv-10171x2z2v+68/21xyz2v+2023y2z2v-9/109xz3v+8520yz3v-2683z4v-13547x3uv+28/65x2yuv-5988xy2uv+61/111y3uv+12314x2zuv+29/44xyzuv+6141y2zuv+11280xz2uv+79/22yz2uv-38/111z3uv+19/51x2u2v+5093xyu2v-10291y2u2v-5009xzu2v-111/49yzu2v+3813z2u2v-61/37xu3v+15914yu3v-3218zu3v-12915u4v-118/101x3v2-7/57x2yv2+13128xy2v2+11606y3v2+42/101x2zv2-54/17xyzv2-43/49y2zv2-119/110xz2v2+9742yz2v2-43/4z3v2-55/8x2uv2-29/88xyuv2+12042y2uv2+101/37xzuv2-57/62yzuv2+106/97z2uv2+38/83xu2v2+8152yu2v2-5492zu2v2-47/79u3v2+15112x2v3+69/44xyv3-6/71y2v3+113/54xzv3-13210yzv3-707z2v3-119/8xuv3+3845yuv3-19/20zuv3+4852u2v3+15761xv4-12372yv4+74/69zv4-2100uv4-12833v5,-x3yz+48/11x2y2z+8356xy3z-35/121y4z-31/66x3z2+54/83x2yz2+61/18xy2z2-11526y3z2-7372x2z3+91/60xyz3+95/97y2z3+45/71xz4-71/115yz4-25/54z5+15/52x3yu+6039x2y2u+74/99xy3u-17/40y4u+29/50x3zu-7775x2yzu+6368xy2zu+14170y3zu+52/41x2z2u+7003xyz2u-5787y2z2u-101/37xz3u-23/28yz3u-20/63z4u+41/77x3u2+8650x2yu2-15922xy2u2-16/83y3u2+7278x2zu2+31/30xyzu2-2/107y2zu2+35/122xz2u2+85/58yz2u2-757z3u2+2/101x2u3+86/17xyu3+95/59y2u3+123/22xzu3-6869yzu3-9311z2u3-105/97xu4+5699yu4+15925zu4+13528u5-154x3yv+4187x2y2v+56/107xy3v-15932y4v-5137x3zv-37/56x2yzv+9401xy2zv+92/123y3zv-79/97x2z2v+9201xyz2v+19/53y2z2v+107/20xz3v+17/77yz3v-15306z4v+3215x3uv-79/117x2yuv-9/76xy2uv-6352y3uv+93/13x2zuv-65/89xyzuv-115/4y2zuv-34/57xz2uv+39/107yz2uv+31/9z3uv+107/48x2u2v+2632xyu2v+29/96y2u2v-125/89xzu2v+29/113yzu2v+3940z2u2v-116/111xu3v+6145yu3v-105/62zu3v+101/17u4v-9281x3v2-49/107x2yv2-12154xy2v2+4/19y3v2-114/71x2zv2-15/118xyzv2+4372y2zv2+45/121xz2v2+46/111yz2v2+6614z3v2+17x2uv2+10806xyuv2-10617y2uv2-25/111xzuv2-116/27yzuv2-7/58z2uv2-686xu2v2+3/13yu2v2-17/49zu2v2-40/107u3v2+47/90x2v3-83/43xyv3-6326y2v3+49/64xzv3+113/76yzv3-122/73z2v3+10232xuv3-116/109yuv3-1990zuv3+70/51u2v3-118/19xv4-27/55yv4+21/19zv4-23/57uv4-11721v5,-3399x4y+1849x3y2-3/29x2y3+28/87xy4+10/29y5-9788x4z-49/73x3yz+13829x2y2z+118/73xy3z+13129y4z-618x3z2+92/13x2yz2+101/117xy2z2-162y3z2+24/5x2z3-29/74xyz3+2687y2z3-74/39xz4+2/57yz4+68/73x4u-13787x3yu-11659x2y2u+14729xy3u+92/53y4u+15/71x3zu-62/15x2yzu+21/85xy2zu+4938y3zu-120/37x2z2u-77/102xyz2u-4785y2z2u-83/70xz3u-12128yz3u-13592z4u-123/20x3u2+2607x2yu2+40/19xy2u2+6361y3u2-3091x2zu2+89/113xyzu2+149y2zu2-2890xz2u2-8374yz2u2+11886z3u2-49/43x2u3-9854xyu3-6943y2u3+10743xzu3-122/45yzu3-13902z2u3-103/19xu4-48/59yu4+27/86zu4+46/35u5-117/17x4v-15/7x3yv+8409x2y2v-83/28xy3v+86/35y4v+37/45x3zv+4/3x2yzv+35/38xy2zv+4015y3zv-49/111x2z2v-1260xyz2v-25/33y2z2v+116/19xz3v+93/8yz3v+5755z4v-25/89x3uv-11669x2yuv-64/107xy2uv+2993y3uv+7767x2zuv-17/95xyzuv-103/80y2zuv-14576xz2uv+80/47yz2uv+25/107z3uv+103/2x2u2v+125/117xyu2v-2/89y2u2v-5298xzu2v-50/27yzu2v-71/53z2u2v+2652xu3v+15761yu3v+2124zu3v+11/82u4v+100/63x3v2+4180x2yv2+11/39xy2v2-1221y3v2+108/125x2zv2+97/126xyzv2-7698y2zv2+13984xz2v2+1342yz2v2-84/121z3v2-26/73x2uv2-14/15xyuv2-22/37y2uv2-71/82xzuv2+12430yzuv2+103/52z2uv2-13095xu2v2+10114yu2v2-8/73zu2v2-33/97u3v2+83/105x2v3+22/45xyv3-7961y2v3-9654xzv3-54/55yzv3-3/71z2v3-10148xuv3-117/98yuv3+101/102zuv3-606u2v3+97/43xv4-68/21yv4+63/16zv4+42/17uv4+5834v5,-3399x3y2-32/113x2y3+14/99xy4+15001y5-121/115x3yz+4604x2y2z+7/2xy3z+9532y4z-3267x3z2+97/118x2yz2-14238xy2z2-80/21y3z2-12332x2z3-19/69xyz3+116/15y2z3-103/32xz4+15340yz4+10509z5+112/109x3yu-97x2y2u-40/11xy3u+90/29y4u-95/106x3zu-114/67x2yzu+113/48xy2zu+12080y3zu-44x2z2u+18/17xyz2u-4814y2z2u-103/100xz3u-96/61yz3u-205z4u-87/82x3u2-97/108x2yu2+3230xy2u2+104/83y3u2+41/86x2zu2+116/49xyzu2-59/110y2zu2+14/59xz2u2-6962yz2u2-2185z3u2+59/91x2u3+2497xyu3+3/37y2u3-13010xzu3+6/83yzu3-11448z2u3+13/72xu4-69/62yu4-2869zu4+23/73u5-20/43x3yv+5074x2y2v+28/125xy3v-2706y4v+13010x3zv-17/109x2yzv+21/4xy2zv+59/93y3zv-2406x2z2v+117/11xyz2v-14978y2z2v+70/89xz3v-33/7yz3v-13676z4v-13690x3uv+9825x2yuv-117/107xy2uv+12760y3uv-93/98x2zuv-113/64xyzuv+113/103y2zuv-9748xz2uv+11016yz2uv-10729z3uv+90/13x2u2v-13/47xyu2v-11/39y2u2v+20/69xzu2v+5531yzu2v+125/49z2u2v-11025xu3v-9621yu3v+113/109zu3v+4710u4v-107/7x3v2+110/119x2yv2-10025xy2v2-6644y3v2-5041x2zv2+5/96xyzv2+11472y2zv2-5128xz2v2+2927yz2v2+121/18z3v2-125/89x2uv2+12936xyuv2-71/47y2uv2+34/47xzuv2-75/103yzuv2-2654z2uv2-2350xu2v2-7707yu2v2+47/72zu2v2-952u3v2-21/67x2v3+58/37xyv3-8757y2v3+3615xzv3+44/123yzv3-13027z2v3-9/10xuv3+75/43yuv3+115/18zuv3+8071u2v3-26/3xv4-67/65yv4+14186zv4-41/122uv4+33/28v5,-3399x3yz-32/113x2y2z+14/99xy3z+15001y4z-9788x3z2+37/96x2yz2+7743xy2z2+31/55y3z2-618x2z3-8171xyz3+82/109y2z3+24/5xz4+88/85yz4-74/39z5-13165x3yu+3407x2y2u-12509xy3u-23/45y4u-11774x3zu-10/67x2yzu+69/79xy2zu-10/123y3zu-7636x2z2u+83/32xyz2u+51/112y2z2u+19/8xz3u+9309yz3u-44/49z4u+4089x3u2-374x2yu2-919xy2u2+98/107y3u2+2776x2zu2+85/26xyzu2+31/13y2zu2-103/82xz2u2+35/76yz2u2+59/45z3u2+2950x2u3+27/44xyu3+88/71y2u3+7/114xzu3-72/77yzu3+12917z2u3-34/67xu4-85/82yu4-55/84zu4+4690u5+11/42x3yv-19/125x2y2v-8288xy3v+9199y4v-12929x3zv+13357x2yzv-4903xy2zv-584y3zv-10/33x2z2v+59/113xyz2v+103/92y2z2v+101/69xz3v+8708yz3v-8/7z4v+13560x3uv-43/49x2yuv-121/98xy2uv+75/79y3uv-39x2zuv-88/69xyzuv-89/78y2zuv+110/67xz2uv+61/4yz2uv-98/45z3uv+82/7x2u2v-85/41xyu2v+6548y2u2v+9367xzu2v-59/81yzu2v-14408z2u2v+2363xu3v-80/11yu3v-50/17zu3v-14799u4v-53/21x3v2+9437x2yv2-117/80xy2v2+81/85y3v2-8/45x2zv2-6428xyzv2+15126y2zv2+68/89xz2v2+7/122yz2v2+9639z3v2+113/4x2uv2-8678xyuv2-104/45y2uv2-79/90xzuv2+39/101yzuv2-7234z2uv2-28/43xu2v2+1251yu2v2-97/56zu2v2+17/41u3v2+107/24x2v3+2747xyv3+9933y2v3-4199xzv3+53/83yzv3+6364z2v3-5456xuv3+618yuv3-123/55zuv3+2375u2v3+63/76xv4-115/106yv4-8811zv4-31/75uv4+10/109v5,13/89x4y+77/31x3y2+36/83x2y3-11411xy4+6936y5-12223x4z+7400x3yz+33/118x2y2z-12146xy3z+108/79y4z+82/99x3z2-9877x2yz2-79/70xy2z2-19/123y3z2-1491x2z3+7953xyz3-43/126y2z3+60/17xz4+98/57yz4-13317x4u-77/27x3yu-6811x2y2u-69/61xy3u+6144y4u+5404x3zu+121/120x2yzu-91/23xy2zu-71/106y3zu+1435x2z2u-120/13xyz2u-12019y2z2u-68/7xz3u-113/82yz3u+11526z4u-8706x3u2-89/53x2yu2-14804xy2u2+120/107y3u2+71/94x2zu2-1/70xyzu2+1532y2zu2+4470xz2u2+13/60yz2u2-115/102z3u2-82/21x2u3+27/121xyu3-4439y2u3-101/47xzu3-3186yzu3-106/101z2u3-10169xu4+19/58yu4-96/73zu4-7959u5-10526x4v-107/92x3yv+47/6x2y2v-23/43xy3v-69/62y4v+59/65x3zv-28/95x2yzv+5479xy2zv-39/77y3zv+11/69x2z2v-11713xyz2v+43/79y2z2v-15602xz3v+16/73yz3v-13952z4v+61/82x3uv-2219x2yuv-91/106xy2uv+5/37y3uv-148x2zuv+31/51xyzuv+18/101y2zuv+97/68xz2uv-73/32yz2uv+47/2z3uv+2/41x2u2v-13009xyu2v-7/60y2u2v+15779xzu2v+72/7yzu2v-11/73z2u2v-119/44xu3v-9067yu3v+3249zu3v+61/51u4v+12525x3v2-118/9x2yv2-3270xy2v2-4/25y3v2-5075x2zv2+77/40xyzv2-89/65y2zv2+17/58xz2v2-15609yz2v2+95/54z3v2-75/79x2uv2-4907xyuv2+12418y2uv2-57/17xzuv2-8746yzuv2+13/95z2uv2-124/67xu2v2+16/13yu2v2+28/23zu2v2-10847u3v2-645x2v3+106/75xyv3+6/115y2v3-8495xzv3+58/35yzv3-9398z2v3-101/72xuv3-71/20yuv3-124/65zuv3-8971u2v3+27/28xv4+12/29yv4-4276zv4+10858uv4+29/12v5,13/89x3y2+12068x2y3-15543xy4-77/79y5+6626x3yz+64/53x2y2z-6/23xy3z-47/125y4z+14403x3z2-43/78x2yz2-31/115xy2z2+94/59y3z2-118/117x2z3-11229xyz3+2268y2z3-116/85xz4+25/58yz4+3085z5+59/27x3yu+67/82x2y2u+11/6xy3u+103/47y4u-63/80x3zu-81/47x2yzu+7760xy2zu-115/56y3zu-10/17x2z2u+101/5xyz2u+15634y2z2u+1/107xz3u-9282yz3u+43/62z4u+62/55x3u2+100/113x2yu2-9205xy2u2-46/13y3u2+43/96x2zu2+10159xyzu2+692y2zu2+859xz2u2-19/74yz2u2+123/47z3u2-9/20x2u3-11391xyu3-2375y2u3+109/24xzu3-57/53yzu3-925z2u3-82/45xu4+97/34yu4+13/82zu4-108/29u5+63/10x3yv+38/17x2y2v-19/115xy3v+3150y4v+22/69x3zv+26/57x2yzv+110/27xy2zv+87/77y3zv+85/18x2z2v+39/47xyz2v-48/17y2z2v-7/27xz3v-13/100yz3v-11662z4v-17/8x3uv+37/11x2yuv+29/11xy2uv-109/88y3uv-2817x2zuv-61/44xyzuv+10/31y2zuv+10010xz2uv+51/86yz2uv-97/83z3uv-89/96x2u2v+4030xyu2v-58/77y2u2v-114/43xzu2v-37/10yzu2v-2011z2u2v+14483xu3v-109/101yu3v+121/102zu3v-79/92u4v+15113x3v2+10781x2yv2-14259xy2v2-113/48y3v2-7/94x2zv2-17/74xyzv2-5/117y2zv2-59/75xz2v2+13188yz2v2+103/43z3v2+4/125x2uv2-52/59xyuv2+85/92y2uv2-1/46xzuv2-9106yzuv2-83/11z2uv2-23/94xu2v2+6742yu2v2-35/107zu2v2-14596u3v2-117/43x2v3+1026xyv3+90/19y2v3+14671xzv3-101/100yzv3+6962z2v3+61/68xuv3+108/37yuv3-4157zuv3-3974u2v3+15677xv4+8661yv4+8459zv4-16/23uv4-37/119v5,13/89x3yz+12068x2y2z-15543xy3z-77/79y4z-12223x3z2-13941x2yz2+115/84xy2z2+13/98y3z2+82/99x2z3+7751xyz3+122/17y2z3-1491xz4+1327yz4+60/17z5+15363x3yu+9780x2y2u+19/117xy3u-1924y4u-14600x3zu+46/41x2yzu-5466xy2zu-73/12y3zu+10838x2z2u-8302xyz2u-89/113y2z2u+53/69xz3u-9224yz3u+47/33z4u-7399x3u2+89/77x2yu2+9312xy2u2-41/80y3u2-732x2zu2-6781xyzu2-8608y2zu2-9270xz2u2-117/58yz2u2-115/68z3u2-48/31x2u3-9067xyu3+97/107y2u3+73/57xzu3-2719yzu3-110/59z2u3-37/86xu4-15796yu4-61/4zu4-115/72u5+6161x3yv+4134x2y2v+677xy3v-8375y4v+1150x3zv+1551x2yzv+4157xy2zv+112/87y3zv+8171x2z2v+6040xyz2v+15651y2z2v-7/66xz3v-47/61yz3v+77/64z4v+14848x3uv+48/119x2yuv-9534xy2uv-117/95y3uv+5/4x2zuv+122xyzuv+90/31y2zuv-41/26xz2uv+31/30yz2uv-10428z3uv-9896x2u2v-71/21xyu2v-55/38y2u2v-29/22xzu2v-11092yzu2v+39/122z2u2v+93/73xu3v+22/49yu3v-21/106zu3v+56u4v+8565x3v2-1695x2yv2+2/17xy2v2+1/78y3v2-113/71x2zv2-41/100xyzv2+55/14y2zv2+15286xz2v2+17/53yz2v2+126/71z3v2-79/87x2uv2+109/97xyuv2-28/31y2uv2-6533xzuv2+22/5yzuv2-10449z2uv2+10830xu2v2-15516yu2v2+28/57zu2v2-81/22u3v2+4198x2v3+5667xyv3-7133y2v3-8408xzv3+11066yzv3-26/125z2v3-808xuv3+95/54yuv3-64/17zuv3-5267u2v3-15333xv4+42/89yv4+63/85zv4+119/113uv4-2011v5,5583x4y+1725x3y2-8652x2y3-91/25xy4-8495x4z-13731x3yz+9298x2y2z-41/111xy3z-15503y4z-13805x3z2+3962x2yz2-2/63xy2z2+3314y3z2+2522x2z3-10/87xyz3-408y2z3+7/16xz4+69/22yz4-7254z5-59/21x4u+115/7x3yu-1718x2y2u+7851xy3u+2632y4u-82/3x3zu+37/86x2yzu+101/113xy2zu+6747y3zu-109/113x2z2u+7399xyz2u+24/103y2z2u+89/9xz3u-14630yz3u+15066z4u-12561x3u2+113/115x2yu2+87/97xy2u2-126/67y3u2-48/7x2zu2+123/103xyzu2-11/107y2zu2-2747xz2u2+8158yz2u2-3/107z3u2+41/6x2u3+12767xyu3+3873y2u3+74/83xzu3-55/119yzu3-24/83z2u3+55xu4-7/95yu4+57/44zu4+2/101u5-6928x4v-121/57x3yv+111/104x2y2v+946xy3v-29y4v+3057x3zv-14/25x2yzv+43/31xy2zv-105/2y3zv+2336x2z2v+61/77xyz2v-7880y2z2v+5/58xz3v+10593yz3v+7094z4v+63/59x3uv-5/69x2yuv-11/81xy2uv-4157y3uv+73/65x2zuv-1676xyzuv-2376y2zuv-85/63xz2uv-95/2yz2uv-14903z3uv-119/110x2u2v-115/24xyu2v+125/9y2u2v+106/87xzu2v-13/12yzu2v-4/19z2u2v+7838xu3v-43/111yu3v+7/113zu3v-12500u4v+7743x3v2-2023x2yv2-85/83xy2v2+49/41y3v2+20/87x2zv2+3932xyzv2-77/6y2zv2+47/90xz2v2-15580yz2v2+39/4z3v2-61/8x2uv2+2518xyuv2+29/98y2uv2+11057xzuv2-18/107yzuv2+708z2uv2+14720xu2v2-3175yu2v2-113/59zu2v2-14735u3v2+7/69x2v3-4029xyv3+54/91y2v3+12372xzv3+67/2yzv3+8856z2v3-2178xuv3+995yuv3+64/95zuv3+4039u2v3-37/44xv4+23/17yv4-3035zv4-103/124uv4+69/64v5,-5583x3y2-1725x2y3+8652xy4+91/25y5+6201x3yz-73/49x2y2z-3844xy3z+10548y4z-11057x3z2-105/122x2yz2+31/53xy2z2+79/89y3z2-24/101x2z3+107/119xyz3-126y2z3+8164xz4+2/77yz4-51/8z5-14941x3yu-106x2y2u+8695xy3u+125/62y4u+4328x3zu+29/117x2yzu-6249xy2zu-2791y3zu+67/49x2z2u-38/29xyz2u+122/41y2z2u+10603xz3u-3029yz3u+5578z4u+14754x3u2-108/79x2yu2+4408xy2u2-12401y3u2-1426x2zu2-1741xyzu2-83/86y2zu2+79/95xz2u2+122/121yz2u2+81/2z3u2-1172x2u3-41/68xyu3-70/3y2u3+24/107xzu3+120/79yzu3+18/119z2u3-65/122xu4+1018yu4+22/107zu4+15189u5+5/8x3yv-12060x2y2v+3/62xy3v-227y4v+60/41x3zv-123/115x2yzv+110/123xy2zv+12864y3zv-86/121x2z2v-69/94xyz2v+14/79y2z2v+118/45xz3v+10842yz3v-37/58z4v+100/69x3uv-47/65x2yuv-7/67xy2uv-93/100y3uv-6262x2zuv-4/75xyzuv+2082y2zuv-9117xz2uv+12450yz2uv-84/67z3uv+123/26x2u2v-51/89xyu2v+19/74y2u2v-104/77xzu2v+318yzu2v+12402z2u2v+95/8xu3v-81/26yu3v-4486zu3v+3872u4v+72/91x3v2-83/63x2yv2+93/92xy2v2-15924y3v2-53/62x2zv2+6046xyzv2+1408y2zv2+60/107xz2v2-1150yz2v2-126/19z3v2-7429x2uv2+2554xyuv2+3602y2uv2+10738xzuv2-57/64yzuv2+86/69z2uv2+8172xu2v2+91/113yu2v2+92/65zu2v2+118/37u3v2+47/83x2v3+12750xyv3+10851y2v3+4216xzv3+6/101yzv3-108z2v3+2920xuv3-101/102yuv3-157zuv3+7742u2v3-7234xv4-2/111yv4+59/33zv4-93/91uv4+24/19v5,1592x4y+75/121x3y2+40/19x2y3-2651xy4+9934x4z+245x3yz+11665x2y2z+30/41xy3z+1823y4z+89/88x3z2-105/46x2yz2+79/58xy2z2-4191y3z2-76/61x2z3-21/32xyz3-9516y2z3-14896xz4-85/77yz4+51/109z5+61/30x4u-10/101x3yu+11796x2y2u+76/101xy3u+123/88y4u-5932x3zu-11857x2yzu+7128xy2zu-45/79y3zu+119/18x2z2u+9/74xyz2u+7042y2z2u-1114xz3u-11/82yz3u-1466z4u-6/85x3u2+27/106x2yu2+14246xy2u2-6216y3u2+47/6x2zu2-45/59xyzu2+89/41y2zu2+41/80xz2u2-7583yz2u2-75/113z3u2-14808x2u3-10873xyu3-90/67y2u3-11081xzu3-7369yzu3-7131z2u3-1402xu4-15386yu4-108/73zu4-5039u5+120/113x4v+10617x3yv-50/87x2y2v-2395xy3v-20/69y4v-8587x3zv+12960x2yzv-41/50xy2zv-13844y3zv-65/32x2z2v-77/122xyz2v-85/66y2z2v+13/100xz3v-20/51yz3v-13676z4v+76/97x3uv+1046x2yuv-8059xy2uv-117/59y3uv-29/105x2zuv+7287xyzuv-107/119y2zuv-35/118xz2uv+79/86yz2uv-2211z3uv+5448x2u2v+62/35xyu2v-2275y2u2v+29/121xzu2v-1674yzu2v-56/43z2u2v-3377xu3v-43/110yu3v+23/10zu3v-24/61u4v+121/53x3v2-4745x2yv2-57/64xy2v2+9554y3v2-12741x2zv2+10449xyzv2+37/108y2zv2+8621xz2v2-11/57yz2v2+1566z3v2+125/49x2uv2-121/118xyuv2+109/84y2uv2-335xzuv2+10167yzuv2-59/109z2uv2-103/119xu2v2+43/13yu2v2-73/87zu2v2+2037u3v2+13002x2v3+83/48xyv3-10713y2v3+1026xzv3-105/64yzv3-37/6z2v3+14779xuv3-6448yuv3+19/69zuv3-1/110u2v3+10010xv4+79/12yv4+12/19zv4-35/61uv4-11/57v5,-1592x3y2-75/121x2y3-40/19xy4+2651y5+39/121x3yz+122/77x2y2z-114/31xy3z+1544y4z+2/3x3z2-10271x2yz2-8373xy2z2+56/61y3z2+55/48x2z3-116xyz3-25/7y2z3-108/113xz4-34/53yz4+5548z5-122x3yu-9690x2y2u+43/87xy3u-5/19y4u+97/54x3zu-17/19x2yzu+4355xy2zu+12/5y3zu-1/100x2z2u+12754xyz2u+13600y2z2u+17/45xz3u-12091yz3u+5145z4u-63/64x3u2-84/31x2yu2-97/41xy2u2+7/13y3u2-79/62x2zu2-80/103xyzu2-69/14y2zu2+119/4xz2u2-35/87yz2u2-13840z3u2+14101x2u3+7952xyu3-1857y2u3-9861xzu3+3180yzu3+75/107z2u3-250xu4-15134yu4+4717zu4-2/41u5+22/27x3yv-8983x2y2v+10520xy3v-113/2y4v+10/73x3zv-1986x2yzv-110/13xy2zv+1550y3zv+32/111x2z2v-111/35xyz2v+101/98y2z2v+8045xz3v-2/89yz3v+2924z4v-79/11x3uv-15178x2yuv+10874xy2uv+54/11y3uv-8950x2zuv+70/53xyzuv-2403y2zuv-8249xz2uv+6935yz2uv+20/89z3uv+885x2u2v-76/71xyu2v-4/17y2u2v-31/52xzu2v-4/99yzu2v+10333z2u2v-93/104xu3v+82/101yu3v-71/37zu3v+9397u4v-15/112x3v2-6614x2yv2+119/2xy2v2+88/119y3v2+306x2zv2+2790xyzv2+10992y2zv2-115/74xz2v2-14711yz2v2+11612z3v2-1788x2uv2-75/97xyuv2+79/30y2uv2+99/59xzuv2-11439yzuv2-121/113z2uv2+108/37xu2v2+37/36yu2v2-3/65zu2v2-55/42u3v2+13/100x2v3-209xyv3-1272y2v3-117/68xzv3+63/94yzv3+32/59z2v3+1013xuv3-3463yuv3+6946zuv3-37/86u2v3+67/117xv4+85/28yv4-3024zv4-82/9uv4-32/65v5,-35/52x4y-12140x3y2+23/83x2y3+69/5xy4-80/79y5+120/43x4z-11865x3yz-3487x2y2z+53/59xy3z+53/102y4z-14083x3z2-14430x2yz2-2442xy2z2-33/104y3z2-91/38x2z3+4/87xyz3-26/57y2z3+4097xz4-9/122yz4+6364z5+9634x4u-97/95x3yu-46/99x2y2u+3847xy3u+121/106y4u+12765x3zu-5292x2yzu+1607xy2zu-67/121y3zu-12/35x2z2u+4/55xyz2u-17/27y2z2u+91/122xz3u-23/31yz3u+65/49z4u+73/46x3u2-124/27x2yu2-9933xy2u2+46/75y3u2+53/114x2zu2+3503xyzu2-14147y2zu2-11283xz2u2+11889yz2u2+99/104z3u2+3117x2u3+12624xyu3-10060y2u3+2193xzu3-80/47yzu3-77/13z2u3+11/31xu4-47/90yu4+49/48zu4-2/105u5-92/61x4v+7443x3yv+35/76x2y2v+114/67xy3v-73/126y4v+97/107x3zv+9464x2yzv+10869xy2zv+15718y3zv-37/33x2z2v+124/13xyz2v-11/26y2z2v-61/40xz3v+91/100yz3v-18/103z4v+60/29x3uv+21/125x2yuv-11117xy2uv+11748y3uv-16/117x2zuv+18/103xyzuv-1711y2zuv+1872xz2uv-109/123yz2uv-18/113z3uv-26/103x2u2v+14140xyu2v+11065y2u2v+8686xzu2v-5/111yzu2v+30/101z2u2v-10501xu3v-36/113yu3v-73/74zu3v+12753u4v-43/52x3v2-76/15x2yv2-5793xy2v2+18/13y3v2+1/79x2zv2+84/23xyzv2-172y2zv2+86/77xz2v2+15/37yz2v2+11835z3v2-6482x2uv2+94/113xyuv2+10727y2uv2-102/41xzuv2+15914yzuv2-12973z2uv2-9038xu2v2-13107yu2v2+1533zu2v2+12549u3v2-13528x2v3+903xyv3+23/114y2v3-123/64xzv3-81/5yzv3+111/103z2v3+4734xuv3-33/20yuv3-7954zuv3-2478u2v3+15518xv4-6723yv4-14/31zv4-3482uv4+10919v5,-3/94x4y-12936x3y2+2/11x2y3+32/23xy4-15921y5+61/93x4z+82/111x3yz-93/2x2y2z-6659xy3z-97/90y4z+402x3z2-14586x2yz2-121/39xy2z2+68/7y3z2+1212x2z3-2980xyz3+49/52y2z3-72/89xz4+92/47yz4+8478z5+2733x4u-103/89x3yu+1166x2y2u-7/53xy3u-106/23y4u+677x3zu+907x2yzu+7891xy2zu-9014y3zu+76/47x2z2u+49/116xyz2u-49/78y2z2u+12261xz3u+118/105yz3u-126/13z4u-8812x3u2-97/120x2yu2-9534xy2u2+92/5y3u2-54/71x2zu2+94/103xyzu2+2256y2zu2+4182xz2u2-5798yz2u2-31/115z3u2-73/98x2u3+15822xyu3+1004y2u3-578xzu3+9494yzu3-6779z2u3+14506xu4+10/121yu4+58/27zu4-2817u5-19/119x4v+7128x3yv+75/64x2y2v-65/109xy3v+5129y4v-53/55x3zv+54/125x2yzv-3009xy2zv+6144y3zv+15601x2z2v+123/55xyz2v-58/77y2z2v-56/61xz3v+121/10yz3v-103/86z4v-93/25x3uv+94/123x2yuv-25/107xy2uv+14807y3uv+65/7x2zuv+87/44xyzuv+6605y2zuv+23/99xz2uv-413yz2uv-17/15z3uv-79/46x2u2v+15240xyu2v-42/67y2u2v+8932xzu2v-5888yzu2v-4204z2u2v+7002xu3v-36/97yu3v-1634zu3v+61/102u4v-14/33x3v2-6520x2yv2+9004xy2v2-67/36y3v2-7/8x2zv2-24/11xyzv2-9373y2zv2+1556xz2v2-79/74yz2v2-6691z3v2+108x2uv2-76/61xyuv2+220y2uv2-1191xzuv2-4/9yzuv2+4546z2uv2+12205xu2v2+9/22yu2v2+64/93zu2v2-44/125u3v2+292x2v3+41/74xyv3+16/79y2v3-15892xzv3+5733yzv3+6796z2v3-42/55xuv3+71/79yuv3-19/104zuv3-38/15u2v3+6436xv4+28/15yv4+87/55zv4+2270uv4-30/41v5,-117/4x3y+97/122x2y2-3618xy3+6566y4+97/113x3z-12634x2yz+9865xy2z-1764y3z+114/31x2z2+5006xyz2+7/44y2z2-15040xz3+8/125yz3+11134z4-12980x3u-79/41x2yu-79/98xy2u+89/65y3u-1217x2zu+89/87xyzu+83/66y2zu+115/11xz2u+123/107yz2u+10920z3u-86/73x2u2-11/94xyu2-14054y2u2+6752xzu2-123/124yzu2+12129z2u2-13310xu3-52/63yu3+12847zu3-1545u4-11064x3v+11499x2yv-37/64xy2v+50/103y3v+123/94x2zv-126xyzv-111/44y2zv+95/14xz2v+113/83yz2v-77/103z3v+41/64x2uv+91/90xyuv-4932y2uv+103/31xzuv+62/63yzuv+1161z2uv-99/106xu2v-3181yu2v-11741zu2v-33/8u3v-3/118x2v2-9369xyv2+527y2v2-113/39xzv2-88/49yzv2-113/101z2v2+95/68xuv2-5930yuv2-20/43zuv2+7/41u2v2+109/93xv3-107/61yv3-8352zv3-5255uv3+12021v4,-2159x4-94/3x3y-4602x2y2+1609xy3+10721y4+28/9x3z-99/35x2yz+1/110xy2z+113/114y3z-118/75x2z2-103/93xyz2-68/67y2z2+13687xz3-1531yz3+61/107z4+6076x3u+9004x2yu+2211xy2u+110/53y3u+47/102x2zu+8495xyzu-9238y2zu+57/121xz2u-8543yz2u+8/19z3u-13527x2u2-13293xyu2+1138y2u2+26/115xzu2+78/53yzu2-12556z2u2+7299xu3+70/19yu3-14687zu3+13559u4+113/9x3v-85/126x2yv-83/3xy2v-3/46y3v+1814x2zv+28/79xyzv+103/51y2zv+78/31xz2v-14387yz2v+1/88z3v+116/75x2uv-101/59xyuv-70/3y2uv+109/71xzuv+13/88yzuv-147z2uv-113/76xu2v-9661yu2v+13855zu2v-6162u3v-1857x2v2-8208xyv2-4634y2v2-6178xzv2-7352yzv2-8247z2v2-113/15xuv2+99/40yuv2+21/97zuv2+11/37u2v2-6605xv3+8964yv3+35/121zv3+8543uv3-6008v4;

  def I = grobj( groebner(J), intvec(0) ); // ASSUME: no zero entries in J!
  ASSUME(0, grtest(I));

  "Input degrees: "; grview(I);

  def RR = grres(I, 0, 1);
  list L = RR;

  " = Non-minimal betti numbers: ";  print(betti(L, 0), "betti");
  "Graded reordered structure of 'res(Input,0)': ";  grview(grorder(L));
}

proc TestGRRes(Name, J)
"USAGE:  TestGRRes(name, I), string name, ideal I
RETURN:  nothing
PURPOSE: compute/test/output/order/transpose a graded resolution of I
EXAMPLE: example TestGRRes; shows an example
"
{
  "==============================================";
  "";
  "=== Example: [", Name, "]";
  " = Ring: ", string(basering);

  def I = grobj( groebner(J), intvec(0) ); // ASSUME: no zero entries in J!
  ASSUME(0, grtest(I));
//  " = Input degrees: "; grview(I);

  " ! Resolution via 'grres': ";
  def R = grres(I, 0, 1); // sres, lres: no grading! // nres, mres - graded (with attrib(, "isHomog"))

  " = Non-minimal betti numbers: ";
  print(betti(R, 0), "betti");

  list L = R ; // SRES_list(R); //  " = Degrees of maps: "; grview(L); // MUST BE GRADED!!!

  " = Degrees of (ordered) maps: ";
  def LL = grorder(L); // MUST BE GRADED

  // ordres(L, intvec(0)); // ?

  grview( LL ); " = TRANSPOSE'd complex: %%%%%%%%%%%%%%";
  list LLL = grtranspose1( LL ); //  resolution RR = LLL;
  print(betti(LLL, 0), "betti");

  grview( LLL ); ""; //  "==============================================";

  kill L, R;
}
example
{ "EXAMPLE:"; echo = 2;
//  if( defined(assumeLevel) ){ int assumeLevel0 = assumeLevel; } else { int assumeLevel; export(assumeLevel); }; assumeLevel = 5; // store the state of aL

  // note: data from random generation 2
  string Name = "castelnuovo"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = 5153xy2-98/23y3-101/51xyz+33/41y2z+99/79xz2+7136yz2-106/111z3+119/53xyu+34/57y2u-77/92xzu+84/73yzu-109/78z2u-27/56xu2+10023yu2+82/103zu2-34/25u3+3/2xyv-68/25y2v+12721xzv+4/63yzv-73/21z2v-7291xuv-91/53yuv-4/79zuv-34/91u2v-122/53xv2+123/70yv2-64/73zv2+44/65uv2+14/31v3,xy2-15202y3+10613xyz+13640y2z-107/103xz2+5292yz2+19/119z3-10042xyu+2770y2u+7957xzu+14008yzu+92/121z2u-92/51xu2+1178yu2+1/117zu2-12726u3+82/101xyv-92/17y2v-107/56xzv+14233yzv+79/28z2v+51/50xuv-31/5yuv+95/91zuv+19/108u2v+12151xv2-69/110yv2+37/89zv2-63/116uv2-88/23v3,-5153x2+37/23xy+8706y2-13160xz+68/115yz+5548z2-22/61xu-113/98yu+11818zu+2114u2-101/97xv+89/22yv-3355zv-113/5uv-5521v2;TestGRRes(Name, I); kill R, Name, @p;  "";

  string Name = "ell.d8.g7"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = x2y2-47/69xy3+6059y4+78/85x2yz+55/124xy2z+13641y3z+8/17x2z2+7817xyz2-2746y2z2+85/124xz3+87yz3+13182z4+109/93x2yu-69/17xy2u+12089y3u+8769x2zu-53/36xyzu-14834y2zu+123/23xz2u+103/77yz2u-2344z3u-43/104x2u2-6198xyu2+47/115y2u2-39/19xzu2-29/24yzu2+51/89z2u2-65/37xu3-95/94yu3+11302zu3-53/57u4-2874x2yv+4347xy2v-25/77y3v+13819x2zv+29/34xyzv+474y2zv+33/107xz2v-3517yz2v+10617z3v+1834x2uv+54/113xyuv-8751y2uv+111/70xzuv-66/61yzuv+9195z2uv-14289xu2v-13/110yu2v+103/9zu2v+5113u3v+116/89x2v2+15142xyv2+13078y2v2-38/41xzv2-13/113yzv2-12824z2v2-57/11xuv2-114/17yuv2-125/31zuv2+11939u2v2+44/13xv3+56/69yv3+12/125zv3+643uv3+3530v4,-3454x2y-1285xy2-6182y3-8/69x2z+9/19xyz+64/49y2z+98/67xz2-13809yz2+21/44z3+77/47x2u+748xyu-41/77y2u+7318xzu+4217yzu+12562z2u-98/69xu2-14/85yu2+119/46zu2-61/121u3+5582x2v+108/77xyv-93/4y2v-65/49xzv-4135yzv+2477z2v+11114xuv+85/14yuv+51/125zuv-7572u2v-115/52xv2-7647yv2+4647zv2-5684uv2-1/55v3,3454x3-6645x2y-43/34xy2+14590y3+8/11x2z-117/112xyz+109/54y2z+6566xz2+23/57yz2-13078z3+95/61x2u+67/40xyu-4544y2u-95/72xzu-8/103yzu+100/77z2u+23/63xu2+69/61yu2-94/105zu2+8619u3+68/123x2v+8/117xyv+101/77y2v+124/125xzv+17/84yzv+23/67z2v+18/59xuv+3216yuv-77/59zuv-9/50u2v+96/109xv2-2491yv2+14089zv2+14067uv2-56/113v3;TestGRRes(Name, I); kill R, Name, @p;  "";

  string Name = "ell.d7.g6"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = 4971xy3+3/101y4-12318xy2z-12835y3z+97/98xyz2+63y2z2-8056xz3+23/91yz3-9662z4-7398xy2u+69/71y3u-53/68xyzu-49/67y2zu-113/122xz2u-9/61yz2u+71/88z3u+11358xyu2-38/29y2u2-10232xzu2+14490yzu2+2274z2u2+3501xu3+10427yu3-109/38zu3-99/5u4-6605xy2v-1555y3v-648xyzv-2083y2zv-61/41xz2v+75/17yz2v-69/55z3v-6104xyuv-9582y2uv+69/2xzuv-12551yzuv+47/49z2uv-118/13xu2v+34/105yu2v+105/41zu2v+6533u3v+122/25xyv2+2/43y2v2+16/61xzv2+11524yzv2+113/99z2v2-71/26xuv2+7809yuv2-4865zuv2-2122u2v2+53/118xv3-13209yv3-11106zv3-49/79uv3+3006v4,xy3+15492y4-13742xy2z+112/117y3z+6/47xyz2+28/41y2z2+71/111xz3+49/57yz3-61/44z4-11759xy2u+4242y3u-109/18xyzu+2260y2zu-6873xz2u-41/112yz2u+12574z3u-10939xyu2+119/38y2u2-62/33xzu2-3699yzu2+2651z2u2-13194xu3-15185yu3-11/116zu3-61/83u4-10094xy2v+13/4y3v-74/73xyzv+43/20y2zv-11547xz2v+53/43yz2v-92/93z3v+32/41xyuv+118/33y2uv-121/39xzuv-15913yzuv+53/11z2uv+97/76xu2v+85/29yu2v-5183zu2v+8520u3v+121/28xyv2+64/51y2v2-15810xzv2+1/43yzv2-6160z2v2+13988xuv2+9/40yuv2+123/4zuv2+15024u2v2+73/95xv3+80/97yv3+57/25zv3-109/81uv3-121/87v4,-4971x2+14389xy+1607y2+59/119xz+12020yz+103/122z2+8894xu+7091yu+54/19zu-50/77u2+28/25xv-113/56yv+68/29zv-14620uv+79/107v2;TestGRRes(Name, I); kill R, Name, @p;  "";

  string Name = "k3.d7.g5"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = -97/108x2y-31/118xy2-73/61y3-79/14x2z-15930xyz-2324y2z+1842xz2+656yz2-8852z3-89/38x2u-102/43xyu+14719y2u+70/67xzu+7335yzu+27/56z2u-10744xu2-55/83yu2+120/73zu2+120/61u3-126/125x2v+691xyv-15385y2v+117/16xzv-17/97yzv+80/121z2v-48/119xuv+21/34yuv-103/65zuv-49/32u2v-41/42xv2+11/75yv2-502zv2-7583uv2+26/69v3,97/108x3+77/114x2y+71/21xy2+13679y3-1645x2z-1/33xyz-79/7y2z-52/53xz2+11940yz2-5800z3+109/13x2u-115/64xyu-125/56y2u-2365xzu+2103yzu+56/87z2u-84/79xu2+107/106yu2-79/70zu2-419u3+5354x2v+92/53xyv-32/19y2v+11/74xzv+4193yzv+45/79z2v-113/72xuv+17/71yuv+11164zuv-17/33u2v+103/66xv2+55/79yv2+118/15zv2-2646uv2+57/106v3,x3-61/113x2y-64/21xy2-107/8y3-13/60x2z+43/35xyz+41/114y2z-13683xz2-5829yz2+71/38z3+90/17x2u-39/29xyu+42/5y2u-61/55xzu+111/77yzu-87/100z2u+10735xu2-83/91yu2-4884zu2-7965u3-65/12x2v+109/86xyv+10606y2v-14164xzv-6678yzv+83/18z2v-93/10xuv+120/49yuv-1592zuv-8710u2v-73/57xv2+10762yv2-2956zv2-89/63uv2-12/7v3;TestGRRes(Name, I); kill R, Name, @p;  "";

  string Name = "rat.d8.g6"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = -19/125x2y2-87/119xy3-97/21y4+36/53x2yz+2069xy2z-59/50y3z-65/33x2z2-14322xyz2+79/60y2z2-9035xz3-14890yz3+87/47z4-23/48x2yu+45/44xy2u+1972y3u+79/118x2zu-5173xyzu+115/121y2zu+1239xz2u-115/17yz2u-15900z3u-78/95x2u2+67/101xyu2-12757y2u2+12752xzu2+68/21yzu2+103/90z2u2-12917xu3+97/92yu3-24/49zu3-13/79u4-51/61x2yv-3103xy2v+77/117y3v+73/115x2zv-79/33xyzv+123/110y2zv+11969xz2v-31/95yz2v-123/95z3v-105/124x2uv+12624xyuv+2/63y2uv+6579xzuv+13/62yzuv+4388z2uv-12747xu2v-26/105yu2v-78/61zu2v-125/53u3v-5/71xyv2+62/77y2v2+21/44xzv2-9806yzv2+3/91z2v2+361xuv2+568yuv2+2926zuv2+53/38u2v2-14523yv3+2082zv3+113/115uv3,108/73x2y2+4028xy3+38/43y4-1944x2yz+39/80xy2z+8/109y3z+52/27x2z2+103/45xyz2+5834y2z2+63/101xz3+107/80yz3+1178z4-1/6x2yu+78/25xy2u-21/43y3u+50/71x2zu-14693xyzu+15074y2zu+9/103xz2u-7396yz2u-14493z3u+93/25x2u2+61/4xyu2-11306y2u2-79/81xzu2+59/82yzu2-5/106z2u2+89/71xu3-34/11yu3+15/103zu3-115/52u4-54/65x2yv+67/16xy2v-7/68y3v-10/13x2zv+32/85xyzv+1/91y2zv+107/118xz2v+7594yz2v-98/103z3v+9919x2uv-965xyuv+53/34y2uv+119/11xzuv-3400yzuv-8329z2uv+75/98xu2v-24yu2v+55/87zu2v-82/71u3v-73/115x2v2+85/19xyv2-213y2v2-7704xzv2-15347yzv2+14960z2v2+15065xuv2-125/17yuv2+32/83zuv2-14/73u2v2-21/44xv3+79/2yv3-61/32zv3+46/119uv3-2082v4,9/20x2y2+113/71xy3-88/65y4+9983x2yz-6722xy2z+87/68y3z+1893x2z2+65/32xyz2+51/55y2z2-102/53xz3+58/5yz3-7187z4-96/7x2yu-14/87xy2u-3532y3u+95/54x2zu+19/65xyzu-6728y2zu+31/121xz2u+73/106yz2u-91/5z3u-12928x2u2+707xyu2-55/48y2u2-96/25xzu2+15869yzu2-20/107z2u2-10030xu3-13786yu3-122/9zu3+19/59u4-7/52x2yv+101/74xy2v+83/6y3v-91/55x2zv-5266xyzv+85/61y2zv+126/95xz2v+56/51yz2v+13073z3v-50/21x2uv-13553xyuv-116/53y2uv+68/71xzuv-111/98yzuv-11037z2uv+68/121xu2v-124/53yu2v+54/55zu2v+5862u3v+12318x2v2-119/29xyv2+101/17y2v2-51/40xzv2-82/33yzv2-30/41z2v2-29/52xuv2+7817yuv2+8121zuv2-28/99u2v2+1125xv3-73/55yv3-14141zv3+8742uv3-1203v4,x2y2+11357xy3+295y4+144x2yz-31/54xy2z+89/119y3z+1/46x2z2+29/26xyz2+1384y2z2+1461xz3+113/91yz3+9494z4-7/32x2yu+12850xy2u-3626y3u-33/106x2zu-7/60xyzu-5935y2zu-8597xz2u+5527yz2u+1708z3u+6182x2u2-15780xyu2+4669y2u2-38/69xzu2+8412yzu2+9265z2u2-5679xu3-67/18yu3-34/67zu3-7178u4+113/56x2yv-3669xy2v+17/113y3v-87/35x2zv-4871xyzv-111/11y2zv-1131xz2v-72/13yz2v+838z3v-115/4x2uv+3395xyuv-43/68y2uv-82/13xzuv+7042yzuv-88/119z2uv+100/19xu2v+24/11yu2v+89/3zu2v+7395u3v-119/109x2v2+1/104xyv2+18/25y2v2+700xzv2-59/9yzv2-92/87z2v2+2486xuv2-67/103yuv2+1469zuv2-101/91u2v2-79/33xv3+10838yv3+81/4zv3-11843uv3+7204v4,19/125x3-15698x2y-22/117xy2-95/107y3+2027x2z-7750xyz+85/104y2z-15326xz2+31/101yz2+67/81z3-7879x2u-112/115xyu+124/81y2u+99/61xzu-7458yzu+40/33z2u-1502xu2+6591yu2-7/73zu2-42/95u3+93/83x2v-15/112xyv-84/95y2v+35/36xzv+5/24yzv-12768z2v+13232xuv-76/103yuv-79/52zuv-7217u2v+75/92xv2-49/64yv2+17/14zv2-6109uv2+1695v3;TestGRRes(Name, I); kill R, Name, @p; "";

  string Name = "k3.d14.g19"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = x4y2+52/25x3y3+73/79x2y4+33/83xy5-116/65y6+23/39x4yz-45/11x3y2z-45/16x2y3z+73/8xy4z+107/70y5z+115/8x4z2-4619x3yz2+13504x2y2z2+9/118xy3z2-113/64y4z2+10731x3z3-7/13x2yz3-26/17xy2z3+3/59y3z3-4602x2z4+71/44xyz4-66/43y2z4-37/70xz5-6988yz5-123/29z6-6158x4yu+9171x3y2u+71/122x2y3u-119/73xy4u-15409y5u+85/3x4zu+104/59x3yzu-3336x2y2zu-50/107xy3zu-10232y4zu-8965x3z2u-2736x2yz2u+4/61xy2z2u+49/92y3z2u+13261x2z3u+64/31xyz3u-89/70y2z3u+11080xz4u+10139yz4u+13653z5u+34/25x4u2-5678x3yu2+43/13x2y2u2+4119xy3u2+72/109y4u2-77/64x3zu2+15044x2yzu2+97/53xy2zu2+9036y3zu2-10599x2z2u2-107/126xyz2u2+43/12y2z2u2-4690xz3u2-43/75yz3u2-15886z4u2-113/7x3u3-242x2yu3+85/19xy2u3-20/57y3u3+109/37x2zu3+6103xyzu3-51/43y2zu3-89/92xz2u3-14/87yz2u3+59/111z3u3-40/63x2u4+113/86xyu4-17/109y2u4-11039xzu4+107/31yzu4+121/47z2u4-123/5xu5+83/97yu5+110/23zu5-122/83u6-15/17x4yv+96/65x3y2v-1491x2y3v+3885xy4v+26/87y5v-51/89x4zv-123/88x3yzv-2315x2y2zv-4119xy3zv-3390y4zv-68/5x3z2v-57/67x2yz2v-116/77xy2z2v+109/54y3z2v+23/89x2z3v+19/77xyz3v+32/107y2z3v-85/121xz4v+9325yz4v+87/32z5v-65/81x4uv-39/19x3yuv+4671x2y2uv+54/77xy3uv+8/47y4uv+1194x3zuv+5/14x2yzuv+1151xy2zuv+90/103y3zuv-55/8x2z2uv-44/101xyz2uv-4713y2z2uv-1/7xz3uv+13344yz3uv-11451z4uv-101/108x3u2v+42/55x2yu2v+4810xy2u2v-3/40y3u2v+4699x2zu2v+114/53xyzu2v+49/116y2zu2v-9/74xz2u2v-84/109yz2u2v-119/97z3u2v+15282x2u3v-109/71xyu3v+2490y2u3v-119/47xzu3v+43/40yzu3v-105/86z2u3v-101/70xu4v+10041yu4v-93/53zu4v+4204u5v+83/43x4v2+15199x3yv2+26/43x2y2v2-6627xy3v2+59/113y4v2+10482x3zv2-8354x2yzv2-78/79xy2zv2+7126y3zv2+68/25x2z2v2+94/59xyz2v2+83/45y2z2v2-59/60xz3v2+15768yz3v2+15/29z4v2-43/100x3uv2-11939x2yuv2+1832xy2uv2+29/6y3uv2+115/76x2zuv2+51/106xyzuv2+7/23y2zuv2+19/107xz2uv2-13797yz2uv2+16/109z3uv2-7519x2u2v2+3472xyu2v2-28/107y2u2v2-11283xzu2v2+12707yzu2v2+54/47z2u2v2+1841xu3v2-3037yu3v2+9239zu3v2+3729u4v2+7476x3v3+59/81x2yv3-49/85xy2v3-91/60y3v3+20/77x2zv3-2761xyzv3+4245y2zv3+97/75xz2v3+42/121yz2v3+1822z3v3-97/16x2uv3-47/10xyuv3-11222y2uv3-2194xzuv3+116/21yzuv3+118/47z2uv3-89/23xu2v3+7317yu2v3-10812zu2v3+70/13u3v3-5855x2v4-9755xyv4+14875y2v4+59/117xzv4-33/109yzv4+48/59z2v4+62/57xuv4-56/73yuv4-3601zuv4+2155u2v4-77/107xv5+12478yv5+7028zv5-3652uv5+81/28v6,-50/87x4yz-2298x3y2z+9266x2y3z-55/16xy4z+31/76y5z+13/46x4z2-11/100x3yz2+90/119x2y2z2-65/54xy3z2-102/77y4z2-76/109x3z3-6558x2yz3-60/19xy2z3-17/113y3z3-74/43x2z4+8092xyz4-13313y2z4-13/107xz5-47/82yz5+5501z6+39/17x4yu-11/13x3y2u-4407x2y3u+89/117xy4u+118/101y5u-37/57x4zu+100/33x3yzu-99/46x2y2zu-2772xy3zu-75/94y4zu-6429x3z2u-7628x2yz2u+10112xy2z2u+85/32y3z2u-4068x2z3u+81/95xyz3u+97/89y2z3u-12700xz4u+19/113yz4u+14306z5u+13878x4u2-22/21x3yu2+107/41x2y2u2-16/97xy3u2-2426y4u2-12247x3zu2+71/41x2yzu2+16/69xy2zu2+37/113y3zu2-12195x2z2u2-8354xyz2u2-6177y2z2u2-10139xz3u2+9118yz3u2-113/115z4u2-8042x3u3+37/97x2yu3+7/30xy2u3+90/73y3u3-14676x2zu3+630xyzu3+7824y2zu3-5947xz2u3-12019yz2u3+99/19z3u3+17/117x2u4+1943xyu4-47/45y2u4-79/98xzu4+46/119yzu4-98/25z2u4-20/69xu5+45/28yu5-123/14zu5-20/29u6+83/41x4yv+32/19x3y2v+5/107x2y3v+9/95xy4v+35/33y5v+14385x4zv+2237x3yzv+109/11x2y2zv-6489xy3zv-96/107y4zv+1749x3z2v+23/64x2yz2v+105/17xy2z2v+11/70y3z2v-4677x2z3v-4/119xyz3v+8243y2z3v-7538xz4v-45/58yz4v+1768z5v-10804x4uv-102/89x3yuv-80/17x2y2uv-103/66xy3uv-14/55y4uv-5981x3zuv+2/35x2yzuv+24/71xy2zuv-17/62y3zuv-59/52x2z2uv+468xyz2uv+61/110y2z2uv-1265xz3uv-11317yz3uv+15610z4uv+61/116x3u2v+1972x2yu2v+26/121xy2u2v+9787y3u2v-91/62x2zu2v+101/65xyzu2v-85/111y2zu2v+99/95xz2u2v-125/51yz2u2v+10324z3u2v+95/73x2u3v-705xyu3v-6165y2u3v+2215xzu3v-109/12yzu3v+72/95z2u3v-149xu4v+14154yu4v+412zu4v+2747u5v+113/25x4v2+5988x3yv2-119/107x2y2v2-70/41xy3v2+2891y4v2+85/103x3zv2-9398x2yzv2-116/117xy2zv2-12527y3zv2-8408x2z2v2-76/47xyz2v2+13729y2z2v2-65/77xz3v2+23/94yz3v2+61/77z4v2+18/121x3uv2-8520x2yuv2+103/34xy2uv2+17/40y3uv2+23/104x2zuv2+7/29xyzuv2-5908y2zuv2-7791xz2uv2-12032yz2uv2+125/82z3uv2-89/126x2u2v2+22/75xyu2v2-118/67y2u2v2+34/29xzu2v2+30/109yzu2v2-64/37z2u2v2-113/87xu3v2+110/109yu3v2+81/23zu3v2-40/99u4v2-14204x3v3+5939x2yv3-7749xy2v3+2924y3v3-3662x2zv3-51/122xyzv3-7218y2zv3-13482xz2v3+12673yz2v3-9675z3v3+6567x2uv3-14008xyuv3+14242y2uv3-4310xzuv3+757yzuv3-9110z2uv3+27/14xu2v3-81/40yu2v3-42/29zu2v3+7287u3v3+119/94x2v4-11926xyv4-82/117y2v4+4138xzv4-67/70yzv4+79/49z2v4-113/50xuv4+102/113yuv4-4791zuv4-11/97u2v4-19/18xv5+6866yv5+63/101zv5+4646uv5-374v6,109/29x4yz+94/43x3y2z-13879x2y3z-53/73xy4z+9074y5z+846x4z2-107/44x3yz2+62/71x2y2z2-121/68xy3z2+14200y4z2-78/89x3z3+51/94x2yz3+123/5xy2z3-731y3z3-15645x2z4-91/25xyz4+39/17y2z4+10343xz5-14671yz5+101/71z6+10467x4yu-3/64x3y2u+75/31x2y3u-43/66xy4u-118/55y5u+7/25x4zu-59/95x3yzu+11830x2y2zu-76/121xy3zu+95/72y4zu-121/53x3z2u+120/7x2yz2u-7033xy2z2u+89/2y3z2u+86/47x2z3u-13/99xyz3u-1447y2z3u+6253xz4u-9082yz4u-4518z5u+31/111x4u2+68/79x3yu2-3532x2y2u2+61/94xy3u2+1210y4u2-11697x3zu2-9386x2yzu2-8263xy2zu2+11759y3zu2+113/118x2z2u2-6488xyz2u2+41/30y2z2u2-15179xz3u2-10848yz3u2-14/51z4u2-13/123x3u3-21/10x2yu3-4018xy2u3-11558y3u3-68/13x2zu3+41/119xyzu3-94/31y2zu3+97/101xz2u3+2551yz2u3+51/73z3u3-10862x2u4-44/125xyu4+113/13y2u4-5704xzu4+65/116yzu4-9578z2u4-95/58xu5+8033yu5-79/124zu5-107/76u6+7293x4yv+15260x3y2v+32/35x2y3v+8519xy4v-50/83y5v+83/40x4zv-6104x3yzv+83/121x2y2zv+230xy3zv+9/50y4zv+1507x3z2v-75/112x2yz2v-20/7xy2z2v-12689y3z2v-75/112x2z3v-8673xyz3v+79/69y2z3v-12453xz4v-4805yz4v-125/11z5v-22/97x4uv-11396x3yuv+67/88x2y2uv-75/122xy3uv-30/29y4uv-837x3zuv-71/12x2yzuv+37/77xy2zuv+78/59y3zuv-13742x2z2uv+13080xyz2uv+643y2z2uv-7517xz3uv-15577yz3uv+52/75z4uv+12922x3u2v-14629x2yu2v-10188xy2u2v-5113y3u2v+67/51x2zu2v-15273xyzu2v-13388y2zu2v+121/109xz2u2v-64/83yz2u2v+69/20z3u2v-117/44x2u3v-7091xyu3v-118/45y2u3v+73/26xzu3v-117/38yzu3v+1448z2u3v+71/19xu4v-12698yu4v+107/81zu4v+21/43u5v-94/79x4v2-1660x3yv2-15179x2y2v2-10/71xy3v2+9523y4v2+6/115x3zv2-15x2yzv2+15142xy2zv2+6157y3zv2+5/79x2z2v2+126/47xyz2v2+14/107y2z2v2+35/11xz3v2-65/51yz3v2-7246z4v2-2652x3uv2-79/69x2yuv2-21/59xy2uv2-14050y3uv2-89/81x2zuv2+91/55xyzuv2-7501y2zuv2-7688xz2uv2-73/102yz2uv2-27/13z3uv2-2186x2u2v2+120/31xyu2v2+45/43y2u2v2-101/16xzu2v2-69/83yzu2v2-12658z2u2v2-7658xu3v2-53/103yu3v2+11620zu3v2+672u4v2-10795x3v3+10972x2yv3-7178xy2v3+81/2y3v3+67/43x2zv3-113/12xyzv3-3947y2zv3-17/5xz2v3-102/49yz2v3-67/26z3v3+5/6x2uv3-95/4xyuv3+2582y2uv3+72/23xzuv3+8490yzuv3-46/111z2uv3+224xu2v3+658yu2v3-98/89zu2v3+7954u3v3+87/35x2v4+7260xyv4+91/40y2v4+11611xzv4+9076yzv4+6444z2v4-8/95xuv4-12845yuv4+86/61zuv4-59/113u2v4-115/84xv5-116/41yv5-119/62zv5+15120uv5-37/51v6,21/122x4yz-9662x3y2z-11556x2y3z+67/18xy4z-10712y5z-4891x4z2+113/118x3yz2-94/83x2y2z2+61/90xy3z2+108/101y4z2+9905x3z3+12/29x2yz3+13047xy2z3-55/46y3z3+59/88x2z4+59/78xyz4+11951y2z4-1357xz5-75/77yz5+15673z6+40/113x4yu+9/55x3y2u-62/83x2y3u+65/111xy4u+4184y5u-9578x4zu+124/17x3yzu+91/85x2y2zu-1/66xy3zu+4184y4zu+101/77x3z2u-12238x2yz2u-5394xy2z2u+82/9y3z2u-73/113x2z3u+3736xyz3u+52/81y2z3u+10426xz4u+11/120yz4u+89/123z5u-99/74x4u2-9/46x3yu2-57/29x2y2u2-24/125xy3u2+115/119y4u2-10/103x3zu2-66/17x2yzu2-662xy2zu2-31/17y3zu2-56/27x2z2u2-4728xyz2u2-86/59y2z2u2-19/34xz3u2+61/13yz3u2-14093z4u2-9068x3u3-115/94x2yu3+15912xy2u3-79/66y3u3-3631x2zu3+1074xyzu3-113/105y2zu3-89/109xz2u3-80/91yz2u3+119/12z3u3+19/87x2u4-42/25xyu4+116/55y2u4+10/77xzu4+74/79yzu4-77/81z2u4-90/121xu5-43/31yu5+107/122zu5+76/113u6+60/17x4yv-6269x3y2v-124/75x2y3v+89/48xy4v-69/2y5v+8832x4zv+13984x3yzv+35/29x2y2zv-65/88xy3zv+53/74y4zv-55/79x3z2v-104/105x2yz2v+50/29xy2z2v-118/119y3z2v-25/88x2z3v+69/82xyz3v-56/69y2z3v-10495xz4v-73/50yz4v+1872z5v-62/19x4uv+99/89x3yuv+3156x2y2uv+5804xy3uv+85/91y4uv+10928x3zuv-4/83x2yzuv+2839xy2zuv+11/38y3zuv+55/126x2z2uv+15613xyz2uv+69/28y2z2uv+75/17xz3uv+115/51yz3uv-111/68z4uv-9781x3u2v+2/105x2yu2v-29/10xy2u2v+90/53y3u2v-12840x2zu2v+85/71xyzu2v-91/80y2zu2v+15904xz2u2v-82/69yz2u2v-32/75z3u2v-91/2x2u3v-77/61xyu3v-9757y2u3v-97/52xzu3v-32/9yzu3v-7457z2u3v-113/100xu4v-13367yu4v-16zu4v+17/53u5v+90/103x4v2-9338x3yv2-42/61x2y2v2+57/124xy3v2-17/6y4v2+6201x3zv2+75/8x2yzv2+13205xy2zv2-21/23y3zv2+6724x2z2v2-1646xyz2v2-3/41y2z2v2+13206xz3v2+14595yz3v2+3100z4v2-94/107x3uv2+106/99x2yuv2+53/24xy2uv2-10113y3uv2+13103x2zuv2+121/124xyzuv2-104/103y2zuv2+59/62xz2uv2+13343yz2uv2-73/72z3uv2-35/123x2u2v2+91/33xyu2v2+75/58y2u2v2-69/73xzu2v2-15760yzu2v2+684z2u2v2-12551xu3v2-99/79yu3v2+74/87zu3v2+9255u4v2-9727x3v3+1222x2yv3+31/115xy2v3+37/50y3v3-86/125x2zv3-5/82xyzv3+7/2y2zv3+69/88xz2v3-25/119yz2v3-120/101z3v3-48/113x2uv3-25/97xyuv3-14896y2uv3+13431xzuv3+13246yzuv3+7556z2uv3-103/111xu2v3+13/108yu2v3+9471zu2v3+31/114u3v3-121/23x2v4-65/69xyv4+66/95y2v4+30/59xzv4-111/40yzv4+55/4z2v4+114/121xuv4+7610yuv4-9205zuv4+85/81u2v4-88/59xv5-4248yv5+95/91zv5+156uv5-71/90v6,7698x4y-93/32x3y2+95/37x2y3+29/104xy4+10/23y5-11774x4z+4544x3yz-9/85x2y2z-45/49xy3z+110/41y4z-44/91x3z2-6083x2yz2+116/111xy2z2+47/68y3z2-11603x2z3-4229xyz3-13462y2z3-31/19xz4+4222yz4-700z5-34/11x4u-20/17x3yu-2471x2y2u-11235xy3u+13259y4u-111x3zu-109x2yzu-89/61xy2zu-28/11y3zu+74/97x2z2u+5554xyz2u+75/47y2z2u-68/77xz3u+15754yz3u-7/51z4u-53/98x3u2+9699x2yu2-9/104xy2u2+64/87y3u2-95/4x2zu2-595xyzu2+4/19y2zu2-18/95xz2u2-13449yz2u2+2931z3u2-11155x2u3-83/29xyu3+7830y2u3+108/91xzu3+13161yzu3+37/42z2u3-16/79xu4-10604yu4-15832zu4+40/39u5-6020x4v+910x3yv+13/110x2y2v+7/86xy3v-97/101y4v-3286x3zv+80/91x2yzv+12467xy2zv+115/99y3zv+79/60x2z2v-8/19xyz2v+105/71y2z2v+60/119xz3v-71/15yz3v+15272z4v-7397x3uv+125/7x2yuv-9507xy2uv+5301y3uv-5605x2zuv-32/35xyzuv-5523y2zuv+67/88xz2uv+15144yz2uv+3/8z3uv-33/56x2u2v+37/29xyu2v+49/19y2u2v-10604xzu2v+37/44yzu2v-8754z2u2v+4184xu3v-56/89yu3v-23/32zu3v+74/101u4v-23/123x3v2-13803x2yv2+54/95xy2v2+9751y3v2+55/119x2zv2+75/32xyzv2+10091y2zv2+1108xz2v2-13283yz2v2+98/111z3v2+5/109x2uv2+28/79xyuv2-95/6y2uv2-7880xzuv2-12659yzuv2+14820z2uv2+4279xu2v2+79/51yu2v2-67/49zu2v2+11207u3v2+85/54x2v3-78/43xyv3-3/95y2v3-86/65xzv3+1/114yzv3+27/74z2v3-102/31xuv3-11/59yuv3-33/29zuv3-3/110u2v3-8455xv4+77/100yv4-6225zv4-70/9uv4-10939v5,-7698x5-16/39x4y+51/61x3y2+59/33x2y3+12841xy4-11040y5+14163x4z-79/55x3yz-83/18x2y2z+121/67xy3z+52/35y4z+26/15x3z2+12770x2yz2-67/15xy2z2-12895y3z2-11/16x2z3-82/45xyz3+2446y2z3+116/9xz4+6/61yz4+83/98z5+9973x4u+67/101x3yu+25/19x2y2u+7517xy3u-59/117y4u-1/86x3zu-10/119x2yzu+2552xy2zu-2448y3zu-112/45x2z2u+35/71xyz2u+12328y2z2u+124/65xz3u-15531yz3u-39/16z4u+4678x3u2-44/103x2yu2-9303xy2u2+59/20y3u2-45/97x2zu2+2707xyzu2+65/61y2zu2+75/68xz2u2+2853yz2u2-12748z3u2-17/18x2u3-115/121xyu3+72/71y2u3-12194xzu3-14204yzu3-63/17z2u3+5772xu4-99/16yu4-51/43zu4+49/43u5-2588x4v+89/44x3yv+32/107x2y2v-117/76xy3v-84/115y4v+113/30x3zv-13/68x2yzv+15120xy2zv-59/28y3zv+61/52x2z2v+12390xyz2v+11436y2z2v+109/40xz3v+40/61yz3v+65/31z4v+12764x3uv+15885x2yuv-11299xy2uv-113/66y3uv+2887x2zuv-918xyzuv+12579y2zuv+39/10xz2uv-119/53yz2uv-62/115z3uv-10887x2u2v+115/122xyu2v-8863y2u2v-30/79xzu2v-26/5yzu2v+15294z2u2v-15701xu3v-11/19yu3v+25/14zu3v-48/55u4v+1341x3v2-4973x2yv2+55/117xy2v2-1787y3v2-115/57x2zv2+28/29xyzv2-184y2zv2+11738xz2v2-8375yz2v2-5962z3v2+52/55x2uv2+17/48xyuv2-103/52y2uv2-53/25xzuv2-101/3yzuv2-123/35z2uv2-14815xu2v2-103/14yu2v2-68/81zu2v2-81/22u3v2-121/56x2v3-12609xyv3+5555y2v3+8/17xzv3-741yzv3-73/103z2v3-12550xuv3-17/78yuv3+7817zuv3+6534u2v3-15384xv4+1807yv4-4677zv4-101/115uv4-83/19v5,-53/83x5+107/75x4y+26/51x3y2+45/109x2y3-3009xy4-27/61y5+16/85x4z-14859x3yz-20/27x2y2z+6326xy3z-4508y4z-10006x3z2-11979x2yz2+8579xy2z2+14669y3z2+67/79x2z3-2551xyz3-61/124y2z3+83/10xz4+12698yz4+15/113z5+123/86x4u-77/6x3yu+15113x2y2u+79/117xy3u-115/88y4u+101/95x3zu+56/13x2yzu-25/62xy2zu+1955y3zu+70/33x2z2u+7470xyz2u-2148y2z2u-14263xz3u-3962yz3u+47/35z4u+441x3u2+14944x2yu2-2/77xy2u2+68/23y3u2-121/43x2zu2-14321xyzu2-35/32y2zu2+34/21xz2u2+11645yz2u2+7131z3u2-3615x2u3-2748xyu3-15200y2u3+81/101xzu3+39/64yzu3-3967z2u3-10346xu4+55/18yu4-8/9zu4+33/68u5-13957x4v+31/116x3yv+58/81x2y2v+36/71xy3v-5706y4v-95/48x3zv+3214x2yzv+14729xy2zv+71/109y3zv+15365x2z2v-5109xyz2v-20/107y2z2v-6/11xz3v+74/55yz3v-76/11z4v+41/72x3uv+7215x2yuv-18/59xy2uv+1741y3uv+7698x2zuv-7299xyzuv+12127y2zuv+7/93xz2uv+71/8yz2uv-123/73z3uv+13657x2u2v-98/13xyu2v+11818y2u2v+22/23xzu2v-3038yzu2v+68/61z2u2v-7173xu3v-7460yu3v+3540zu3v+27/20u4v-37/41x3v2+20x2yv2+107/82xy2v2-2237y3v2+9827x2zv2+124/27xyzv2-18/5y2zv2-77/24xz2v2-10231yz2v2-32/7z3v2-11980x2uv2-36/35xyuv2+8618y2uv2+3174xzuv2-123/2yzuv2-117/38z2uv2+117/115xu2v2+70/3yu2v2-3144zu2v2+815u3v2-116/85x2v3+98/41xyv3-648y2v3-38/5xzv3-9/125yzv3+8710z2v3+48/31xuv3+101/109yuv3+70/11zuv3-51/4u2v3-76/59xv4-93/52yv4+15291zv4+4/55uv4+64/59v5,568x5+4355x4y+48/5x3y2+33/7x2y3-53/111xy4+1749y5-23/4x4z-98/69x3yz-47/56x2y2z-8519xy3z-5/113y4z+11488x3z2+79/21x2yz2+11/89xy2z2-64/83y3z2-15697x2z3-67/86xyz3+11545y2z3+3336xz4-106/39yz4+15466z5+15202x4u-34/63x3yu-121/72x2y2u+1/52xy3u+10800y4u+4993x3zu-55/112x2yzu-26/51xy2zu-114/125y3zu+113/2x2z2u-87/88xyz2u-91/107y2z2u-65/6xz3u+15415yz3u+1373z4u-27/86x3u2-76/93x2yu2+9/22xy2u2+16/91y3u2+10326x2zu2-61/84xyzu2-28/99y2zu2+87/14xz2u2-88/45yz2u2+60/59z3u2-13/60x2u3-10824xyu3-121/119y2u3+14919xzu3-81/25yzu3+11233z2u3+14676xu4-8474yu4+12211zu4+32/83u5+57/52x4v+10/13x3yv-277x2y2v-6961xy3v-4594y4v-13439x3zv-1/30x2yzv-118/43xy2zv-62/15y3zv+76/15x2z2v+3805xyz2v-26/15y2z2v+3081xz3v+662yz3v+13856z4v+107/30x3uv+6063x2yuv+100/37xy2uv+110/107y3uv-10346x2zuv-67/44xyzuv-93/29y2zuv+17/89xz2uv-57/104yz2uv-68/91z3uv+3804x2u2v-75/107xyu2v-11842y2u2v-103/57xzu2v-37/18yzu2v+10795z2u2v-90/31xu3v+14200yu3v+97/124zu3v+5256u4v+52/101x3v2-94/107x2yv2-12841xy2v2+77/72y3v2+93/74x2zv2+7033xyzv2+87/76y2zv2-15415xz2v2-15164yz2v2-14749z3v2+86/53x2uv2+14707xyuv2+9443y2uv2+118/5xzuv2-81/2yzuv2+43/57z2uv2+59/83xu2v2-121/79yu2v2+4449zu2v2-50/63u3v2+79/31x2v3+95/32xyv3+125/107y2v3-9165xzv3+3151yzv3+5006z2v3+45/19xuv3-5194yuv3-82/11zuv3+121/15u2v3-10265xv4-99/118yv4-3162zv4-16/29uv4-37/4v5;TestGRRes(Name, I); kill R, Name, @p;  "";

  string Name = "k3.d11.g11.ss0"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = 38/41x4y+116/67x3y2+6547x2y3-50/71xy4+13/89y5+63/22x4z-12151x3yz-3400x2y2z-52/45xy3z+9027y4z+5420x3z2+6983x2yz2-3285xy2z2-47/2y3z2-137x2z3+40/51xyz3+111/40y2z3-97/93xz4+918yz4-7492z5-122/49x4u-123/47x3yu-95/32x2y2u-83/36xy3u-125/77y4u-9824x3zu-51/115x2yzu-83/18xy2zu-19/20y3zu-117/98x2z2u+885xyz2u-20/97y2z2u+69/55xz3u-11/82yz3u+93/47z4u-97/113x3u2-83/48x2yu2+12868xy2u2-4932y3u2-97/114x2zu2-121/116xyzu2-79/108y2zu2-27/68xz2u2+116/19yz2u2+10019z3u2-6268x2u3+5/31xyu3+13810y2u3-120/37xzu3-33/124yzu3-31/41z2u3+79/50xu4+71/59yu4+110/81zu4-27/53u5-12764x4v-7/22x3yv+15253x2y2v-67/29xy3v+15620y4v-1202x3zv+56/57x2yzv-99xy2zv-29/28y3zv-15010x2z2v-101/16xyz2v+11110y2z2v+7300xz3v+58/95yz3v+10228z4v+65/57x3uv+9/2x2yuv+116/27xy2uv+7960y3uv-11/8x2zuv+59/23xyzuv+19/26y2zuv+14327xz2uv-14671yz2uv+126/101z3uv+69/83x2u2v+14041xyu2v-73/90y2u2v+11/108xzu2v-5492yzu2v-3858z2u2v-2840xu3v+15365yu3v+28/113zu3v+149u4v-94/85x3v2-32/123x2yv2-1409xy2v2-8233y3v2+851x2zv2-7458xyzv2-10713y2zv2-18/11xz2v2-15/47yz2v2-32/97z3v2+39/68x2uv2-10832xyuv2+39/59y2uv2-12211xzuv2+66/71yzuv2+121/115z2uv2-65/101xu2v2+82/11yu2v2-78/103zu2v2-10128u3v2-100/33x2v3-29/28xyv3+61/29y2v3-5266xzv3+1953yzv3+5799z2v3+13/77xuv3-2680yuv3-57/98zuv3+56/125u2v3+121/42xv4+47/111yv4+2590zv4-60/53uv4+43/61v5,8057x5+3453x4y-71/57x3y2+14017x2y3+89/28xy4-15/44y5+10785x4z-14385x3yz-112/31x2y2z+29/42xy3z+162y4z-12837x3z2+31/33x2yz2+105/74xy2z2+97/59y3z2-23/61x2z3-246xyz3+65/93y2z3+1/54xz4+6/25yz4+96x4u-41/74x3yu+15120x2y2u-1465xy3u-47/42y4u+11/80x3zu-84/55x2yzu+12120xy2zu+84/65y3zu+496x2z2u-29/24xyz2u+3753y2z2u-2608xz3u+11/123yz3u+7955z4u-11390x3u2+37/79x2yu2+3230xy2u2-72/83y3u2-87/74x2zu2-11620xyzu2+8276y2zu2+608xz2u2-7002yz2u2-7897z3u2-65/97x2u3-37/3xyu3+7/43y2u3+37/20xzu3+97/28yzu3+98/19z2u3-47/45xu4-49/109yu4-53/89zu4-66/125u5+7516x4v-1097x3yv+15928x2y2v-12128xy3v-12988y4v-23/7x3zv+5146x2yzv+9/28xy2zv-1816y3zv+115/42x2z2v+11840xyz2v+40/119y2z2v-66/85xz3v-8269yz3v-59/45z4v-114/37x3uv-126/55x2yuv-31/24xy2uv-19/9y3uv+77/3x2zuv+14268xyzuv-4133y2zuv-11603xz2uv-55/83yz2uv-5/61z3uv-13069x2u2v+4387xyu2v+94/77y2u2v-111/79xzu2v-31/61yzu2v-26/53z2u2v-13/103xu3v-27/91yu3v-100/57zu3v-104/111u4v+77/15x3v2+100/69x2yv2-1664xy2v2+14971y3v2-67/45x2zv2-55/27xyzv2+13/98y2zv2-958xz2v2+3/68yz2v2+34/65z3v2+10771x2uv2+19/106xyuv2-84/125y2uv2+4723xzuv2-10707yzuv2+50/103z2uv2+1766xu2v2+110/3yu2v2-58/51zu2v2-16/105u3v2-70/111x2v3+83/103xyv3-104/83y2v3-2394xzv3+91/109yzv3-11779z2v3+111/80xuv3+41/109yuv3+71/77zuv3+110/31u2v3+8933xv4-109/113yv4-56/87zv4-120/7uv4-107/57v5,-38/41x5-116/67x4y-6547x3y2+50/71x2y3-13/89xy4-43/79x4z+1342x3yz+14316x2y2z+737xy3z+15/44y4z+124/79x3z2-67/13x2yz2+79/104xy2z2-162y3z2-109/120x2z3+4540xyz3-97/59y2z3+3772xz4-65/93yz4-6/25z5-13041x4u-90/89x3yu+107/36x2y2u+43/46xy3u-11755y4u+12017x3zu-97/58x2yzu+5592xy2zu-11967y3zu+24/79x2z2u-63/83xyz2u-7367y2z2u-62/79xz3u+55/64yz3u-67/18z4u-672x3u2+19/77x2yu2-100/63xy2u2-23/56y3u2+61/78x2zu2+120/67xyzu2+48/107y2zu2+5855xz2u2+9877yz2u2+6940z3u2+12754x2u3+12989xyu3-123/106y2u3+59/88xzu3-3400yzu3-8976z2u3+43/21xu4-86/111yu4+2/29zu4+57/31u5+79/48x4v+95/56x3yv+3056x2y2v-13681xy3v+13735y4v-122/91x3zv+79/74x2yzv+21/47xy2zv+12060y3zv-12314x2z2v+64/17xyz2v-57/52y2z2v-57/86xz3v+15436yz3v+9387z4v+3345x3uv+2/109x2yuv-13978xy2uv+5604y3uv+11645x2zuv+2633xyzuv+15505y2zuv+10/99xz2uv+14409yz2uv+8127z3uv+99/104x2u2v+14440xyu2v+37/43y2u2v-9707xzu2v+9171yzu2v-656z2u2v+3723xu3v+4/19yu3v+109/76zu3v-8740u4v+2121x3v2+34/11x2yv2-89/61xy2v2-26/21y3v2+111/22x2zv2+88/41xyzv2+119/99y2zv2+108/77xz2v2+9279yz2v2-45/116z3v2+71/107x2uv2+25/76xyuv2-112/121y2uv2-18/71xzuv2-84/125yzuv2+496z2uv2+96/113xu2v2-964yu2v2-73/4zu2v2+103/95u3v2+9962x2v3+38/39xyv3-15120y2v3+11747xzv3+2/123yzv3+14829z2v3-6538xuv3-52/63yuv3-83/107zuv3+73/37u2v3-4183xv4+11/103yv4+83/119zv4+94/31uv4-19/98v5,-12092x5-2411x4y+3922x3y2+117/41x2y3+95/22xy4+45/19x4z-5881x3yz+97/80x2y2z-4981xy3z-21/67y4z+9068x3z2+1/11x2yz2-109/65xy2z2+22/117y3z2-7823x2z3-76/75xyz3-34/25y2z3-79/5xz4-60/37yz4-87/100z5+58/83x4u-36/61x3yu+79/76x2y2u+502xy3u+4988y4u+43/38x3zu-4/107x2yzu-48/11xy2zu+11685y3zu-15002x2z2u-97/17xyz2u+93/2y2z2u-35/71xz3u+43/83yz3u-69/26z4u+119/6x3u2-87/14x2yu2-29/109xy2u2-36/97y3u2-116/119x2zu2+4610xyzu2-33/58y2zu2+121/25xz2u2-125/121yz2u2-35/121z3u2-8975x2u3+9014xyu3+14845y2u3-10277xzu3+75/124yzu3-10/83z2u3-73/69xu4+46/85yu4-4971zu4-113/4u5+9321x4v+109/19x3yv+11/17x2y2v+6672xy3v+16/99y4v+8092x3zv+1725x2yzv+80/41xy2zv+2445y3zv+4/99x2z2v+69/101xyz2v+13182y2z2v-10090xz3v+3817yz3v+106/79z4v-71/89x3uv+110/107x2yuv-56/19xy2uv-708y3uv-108/97x2zuv-11889xyzuv+9744y2zuv-24/121xz2uv-10711yz2uv+3182z3uv-22/35x2u2v+81/44xyu2v-40/31y2u2v-13494xzu2v+47/80yzu2v+71/52z2u2v-6214xu3v-5144yu3v-115/49zu3v-109/19u4v+8621x3v2-62/79x2yv2-102/103xy2v2-8174y3v2+13689x2zv2+15544xyzv2-107/73y2zv2+120/73xz2v2+19/56yz2v2-4544z3v2-824x2uv2-2/17xyuv2+67/78y2uv2-54/85xzuv2+31/51yzuv2-59/19z2uv2+50/7xu2v2+5762yu2v2+79/64zu2v2-3729u3v2+12212x2v3+1833xyv3+12543y2v3+11974xzv3-11/17yzv3-75/74z2v3+26/3xuv3-37/36yuv3-7683zuv3+14069u2v3+12261xv4+12489yv4+1657zv4+10781uv4-46/3v5,12092x4z+2411x3yz-3922x2y2z-117/41xy3z-95/22y4z+11164x3z2+6625x2yz2+112/75xy2z2+1533y3z2+60/101x2z3-173xyz3+15913y2z3+14954xz4-5022yz4+12391z5-38/41x4u-116/67x3yu-6547x2y2u+50/71xy3u-13/89y4u-12/55x3zu-10086x2yzu+6782xy2zu-111/89y3zu-52/85x2z2u+9744xyz2u+9553y2z2u-15590xz3u+76/13yz3u+5938z4u+42/11x3u2-25/116x2yu2+2554xy2u2-12842y3u2-3/88x2zu2+102/113xyzu2-10298y2zu2-32/69xz2u2+9709yz2u2+8775z3u2-9937x2u3+5128xyu3-56/75y2u3+12493xzu3-7/39yzu3-41/24z2u3-125/49xu4-15745yu4-1005zu4+16/55u5-7566x3zv+14010x2yzv+51/23xy2zv+37/113y3zv+9/14x2z2v+2251xyz2v+10076y2z2v-106/69xz3v-4060yz3v+3379z4v-11120x3uv-9744x2yuv+50/17xy2uv+7065y3uv-70/37x2zuv+10016xyzuv+47/86y2zuv+6928xz2uv-11190yz2uv-11611z3uv+15042x2u2v-52/105xyu2v+2185y2u2v-76/67xzu2v-57/104yzu2v-7610z2u2v-3912xu3v+1/48yu3v-24/109zu3v+3287u4v+17/96x2zv2+29/76xyzv2+15768y2zv2+8410xz2v2-116/41yz2v2+89/48z3v2+15119x2uv2-11840xyuv2-89/43y2uv2+10115xzuv2+93/101yzuv2+33/62z2uv2+11864xu2v2-14582yu2v2+95/53zu2v2-8816u3v2+35/43xzv3+75/107yzv3+13515z2v3-38/67xuv3+83/71yuv3+113/51zuv3-80/111u2v3+49/5zv4+13850uv4,14959x5-3/13x4y+97/37x3y2+14586x2y3-71/120xy4-10158x4z+59/50x3yz-101/96x2y2z+118/47xy3z-65/83y4z+49/66x3z2-107/18x2yz2-2/47xy2z2+104/63y3z2-121/43x2z3+9552xyz3-113/36y2z3+4699xz4+55/53yz4+16/109z5-3576x4u-103/52x3yu-8299x2y2u+1585xy3u+3377y4u+15336x3zu-119/24x2yzu-19/68xy2zu-65/11y3zu-17/117x2z2u+788xyz2u+73/115y2z2u-46/93xz3u-108/7yz3u+3774z4u-14034x3u2-15420x2yu2-5128xy2u2+83/55y3u2+17/5x2zu2+13098xyzu2+96/29y2zu2-54/115xz2u2+4/5yz2u2+2747z3u2+56/83x2u3-1545xyu3+14384y2u3+7787xzu3-69/64yzu3-27/20z2u3-17/15xu4+46/13yu4-44/41zu4-14876u5+10090x4v+184x3yv+125/44x2y2v-13987xy3v+104/123y4v+24/35x3zv+77/95x2yzv+162xy2zv+31/113y3zv-94/51x2z2v-14750xyz2v-110/101y2z2v+3224xz3v-12389yz3v-77/57z4v-7340x3uv+48/79x2yuv-55/71xy2uv-65/57y3uv+13981x2zuv+76/73xyzuv-41/22y2zuv-10847xz2uv+3230yz2uv+37/14z3uv+64/47x2u2v-89/97xyu2v-608y2u2v-93/112xzu2v-22yzu2v+7158z2u2v-120/31xu3v-13481yu3v-97/37zu3v-113/87u4v-35/19x3v2+79/53x2yv2-1037xy2v2-29/63y3v2+5990x2zv2-3380xyzv2-17/126y2zv2-40/121xz2v2-15041yz2v2+3779z3v2+13583x2uv2+11/73xyuv2+2762y2uv2-16/49xzuv2-40/93yzuv2-37/75z2uv2+13312xu2v2+4407yu2v2+5449zu2v2+3013u3v2+86/73x2v3-37/66xyv3-73/15y2v3+75/107xzv3-63/31yzv3-97/99z2v3+11234xuv3+37/92yuv3+47/27zuv3+121/26u2v3-15690xv4+73/40yv4-12281zv4+6014uv4-92/83v5,-12092x4y-2411x3y2+3922x2y3+117/41xy4+95/22y5+14327x4z+3699x3yz+18/85x2y2z-56/53xy3z-31/16y4z+5371x3z2+73/69x2yz2+65/34xy2z2-59/49y3z2+23/37x2z3-9067xyz3+79/37y2z3-17/50xz4+8412yz4+110/39z5-36/125x4u-20/51x3yu-28/123x2y2u-4482xy3u+4217y4u+8/19x3zu+9/77x2yzu-10526xy2zu+76/79y3zu-12494x2z2u-50/13xyz2u-98/123y2z2u-23/19xz3u+1491yz3u-50/109z4u-67/62x3u2+68/47x2yu2-38/99xy2u2-4891y3u2+68/113x2zu2-73/35xyzu2-2384y2zu2+71/109xz2u2-68/39yz2u2-94/125z3u2+120/121x2u3-20/39xyu3+435y2u3-23/14xzu3-39/97yzu3+75/23z2u3+71/30xu4-8426yu4+125/7zu4+11/46u5-2334x4v+9113x3yv-1060x2y2v+12839xy3v+9876y4v-8395x3zv-79/82x2yzv+89/79xy2zv+805y3zv-7473x2z2v+80/89xyz2v+105/52y2z2v-54/103xz3v-2102yz3v+19/117z4v-1326x3uv-8963x2yuv-12/13xy2uv+11798y3uv-80/7x2zuv+519xyzuv-58/67y2zuv+14572xz2uv+8426yz2uv-66/47z3uv-574x2u2v+12480xyu2v-41/89y2u2v-111/83xzu2v-58/37yzu2v-15255z2u2v+31/97xu3v-89/113yu3v+15475zu3v-3982u4v+51/8x3v2+9547x2yv2-8060xy2v2-195y3v2-5/21x2zv2-2411xyzv2+65/12y2zv2-5410xz2v2+3844yz2v2+3/61z3v2-62/101x2uv2+53/19xyuv2-6567y2uv2+11302xzuv2+123/37yzuv2-107/118z2uv2-29/79xu2v2-91/40yu2v2+40/99zu2v2-12274u3v2-47/124x2v3+12752xyv3+11039y2v3-115/43xzv3+44/79yzv3-37/28z2v3+5364xuv3+14510yuv3+3024zuv3-40/3u2v3+71/122xv4+12024yv4+10/93zv4-4183uv4+11229v5,-51/74x4y-99/61x3y2-10442x2y3+6943xy4-15040y5+126/97x4z+101/41x3yz+4018x2y2z+98/61xy3z+87/34y4z+75/52x3z2-109/70x2yz2-81/71xy2z2+8231y3z2+78/49x2z3+53/69xyz3+70/71y2z3+48/23xz4-6678yz4+14913z5-32/97x4u+78/55x3yu+11669x2y2u-58/75xy3u-61/64y4u+118/35x3zu-125/28x2yzu-125/79xy2zu-49/23y3zu-14966x2z2u-34/19xyz2u-53/126y2z2u+13854xz3u-2177yz3u+9962z4u+15886x3u2+47/38x2yu2+10992xy2u2-111/31y3u2-79/21x2zu2-14524xyzu2-3442y2zu2-45/86xz2u2+53/3yz2u2+8738z3u2+9062x2u3+3872xyu3+56/61y2u3-6255xzu3-104/107yzu3+9858z2u3+7891xu4-11114yu4+67/37zu4-15307u5-108/7x4v+11586x3yv+19/4x2y2v+14404xy3v-99/116y4v+124/33x3zv-73/17x2yzv+77/115xy2zv-4537y3zv+9949x2z2v+101/111xyz2v+6634y2z2v-11818xz3v+23/52yz3v-63z4v+7/6x3uv-13686x2yuv+15879xy2uv+4731y3uv+107/82x2zuv+104/119xyzuv-56/67y2zuv-21/5xz2uv-24/41yz2uv-125/108z3uv+58/77x2u2v+79/49xyu2v+82/19y2u2v-82/117xzu2v-8/61yzu2v+210z2u2v+27/25xu3v+71/111yu3v-14500zu3v-15/2u4v-113/108x3v2+40/31x2yv2+89/69xy2v2-8271y3v2+111/20x2zv2+1/31xyzv2-51/101y2zv2+3967xz2v2+79/32yz2v2-6893z3v2-1060x2uv2+38/5xyuv2+21/20y2uv2-79/34xzuv2+15/118yzuv2+25/106z2uv2+97/20xu2v2+6/67yu2v2+113/120zu2v2+13/3u3v2+1800x2v3-97/15xyv3-12712y2v3+49/118xzv3+5/63yzv3+234z2v3-97/123xuv3+13/88yuv3+8/103zuv3-38/97u2v3-15634xv4+18/25yv4-7517zv4+115/103uv4-56/23v5,-123/61x5+4/87x4y-15558x3y2+78/5x2y3-31/83xy4+9742x4z+23/125x3yz+1865x2y2z+56/125xy3z-4/33y4z-54/71x3z2-113/84x2yz2+37/75xy2z2+6/23y3z2-28/123x2z3+31/112xyz3-15736y2z3-106/111xz4+46/109yz4-29/10z5-28/117x4u+15892x3yu+8096x2y2u-91/29xy3u+57/92y4u+96/31x3zu+166x2yzu+2654xy2zu-11662y3zu+1565x2z2u+9017xyz2u+121/4y2z2u-29/115xz3u+9828yz3u-15873z4u-5354x3u2+118/19x2yu2+88/95xy2u2+68/19y3u2-7/110x2zu2-109/37xyzu2-45/74y2zu2+12579xz2u2-6659yz2u2+10257z3u2+7229x2u3+39/67xyu3-108/19y2u3-58/9xzu3-29/81yzu3-1461z2u3+23/43xu4-8/19yu4-44zu4+100/109u5+121/51x4v-20/39x3yv-85/121x2y2v+73/61xy3v+103/50y4v+7334x3zv+52/125x2yzv-14/55xy2zv+106/103y3zv-4079x2z2v+109/57xyz2v+123/28y2z2v-21/83xz3v+3/31yz3v-9826z4v+89/77x3uv-79/35x2yuv+8767xy2uv+83/11y3uv+113/36x2zuv-57/68xyzuv+1675y2zuv+10873xz2uv+84/11yz2uv+89/110z3uv-5834x2u2v-121/97xyu2v+49/66y2u2v-103/95xzu2v+14863yzu2v+119/94z2u2v+44/35xu3v-15429yu3v-9302zu3v-73/12u4v-77/86x3v2-121/46x2yv2+48/13xy2v2+68/115y3v2+58/63x2zv2+58/117xyzv2-16/7y2zv2-76/39xz2v2+41/103yz2v2-8042z3v2-59/5x2uv2+3277xyuv2-120/107y2uv2-5/84xzuv2-30/103yzuv2+31/71z2uv2-1593xu2v2-4668yu2v2+5749zu2v2+38/43u3v2+26/17x2v3+1894xyv3+62/17y2v3-4851xzv3-10816yzv3+62/55z2v3+1366xuv3-70/11yuv3+90/43zuv3-115/57u2v3-10886xv4+1987yv4-78/25zv4+110/29uv4+4694v5,47/12x5+64/29x4y-113/70x3y2+15185x2y3-52/81xy4+67/28x4z-4220x3yz-9239x2y2z+10/117xy3z-5609y4z-4439x3z2-61/91x2yz2+15282xy2z2+8165y3z2-19/109x2z3+123/94xyz3+62/49y2z3+77/2xz4-1/37yz4-1202z5-14940x4u+13396x3yu-6/5x2y2u-91/50xy3u+115/118y4u-455x3zu-2755x2yzu+14353xy2zu+11/53y3zu+115/57x2z2u+86/49xyz2u+43y2z2u+5247xz3u+4438yz3u+109/16z4u-3113x3u2-15629x2yu2-15664xy2u2-115/57y3u2+3/103x2zu2-3723xyzu2+89/17y2zu2-59/56xz2u2+19/123yz2u2+43/14z3u2+103/14x2u3+106/31xyu3-13591y2u3-71/43xzu3+74/119yzu3-36/113z2u3-73/96xu4-31/120yu4-105/76zu4-37/34u5-58/23x4v-11111x3yv+5888x2y2v+14867xy3v+8608y4v-2829x3zv-31/120x2yzv+6306xy2zv+5913y3zv+35/6x2z2v+75/104xyz2v+33/41y2z2v-15317xz3v-31/49yz3v-7300z4v+2214x3uv-91/46x2yuv+10949xy2uv-3/82y3uv-16/27x2zuv-14175xyzuv-13/19y2zuv-21/65xz2uv+148yz2uv+14615z3uv-43/6x2u2v-4133xyu2v+33/49y2u2v-1299xzu2v+96/53yzu2v-12667z2u2v-86/113xu3v+10437yu3v-53/13zu3v-15678u4v-53/121x3v2-20/49x2yv2+3813xy2v2+10006y3v2+15644x2zv2-99/7xyzv2-42/13y2zv2-124/57xz2v2+73/70yz2v2+110/107z3v2-97/22x2uv2-125/116xyuv2+92/93y2uv2-12120xzuv2+143yzuv2-15344z2uv2+13225xu2v2+17/104yu2v2+1212zu2v2-7685u3v2-81/4x2v3+10328xyv3-21/19y2v3+13407xzv3-1/118yzv3+59/88z2v3+965xuv3-93/43yuv3-76/85zuv3-81/86u2v3-6949xv4+9205yv4-7777zv4+15020uv4+8845v5,3079x5-28/11x4y+117/88x3y2-89/115x2y3+40/13xy4+13407y5+3942x4z+30/121x3yz-2922x2y2z-71/74xy3z-103/114y4z-89/59x3z2+12255x2yz2+44/109xy2z2+13778y3z2+103/110x2z3-73/95xyz3+8049y2z3+10042xz4-101/35yz4-1636z5-53/27x4u+130x3yu-5989x2y2u+67/109xy3u+4601y4u-12943x3zu-86/75x2yzu+5837xy2zu+107/104y3zu+2396x2z2u-25/58xyz2u+29/110y2z2u+10526xz3u-7/45yz3u-9105z4u-64/35x3u2+64/105x2yu2+113/24xy2u2-5955y3u2+20/91x2zu2+119/78xyzu2+83/8y2zu2+125/103xz2u2+83/101yz2u2+14895z3u2-118/93x2u3-19/23xyu3+100/53y2u3-92/59xzu3+34/89yzu3+9417z2u3-6626xu4-36/65yu4+119/82zu4+8761u5-15340x4v-89/115x3yv+89/85x2y2v+33/74xy3v-8088y4v+29/16x3zv-6581x2yzv-88/85xy2zv-12/11y3zv-58/19x2z2v+12219xyz2v-100/119y2z2v+116/53xz3v-67/94yz3v+13/5z4v+236x3uv+107/96x2yuv+115/113xy2uv-7860y3uv+15064x2zuv+29/59xyzuv-7/103y2zuv-71/66xz2uv-103/79yz2uv-83/4z3uv-77/23x2u2v+7/37xyu2v+106/103y2u2v+3/50xzu2v+125/17yzu2v-15620z2u2v+6424xu3v-12832yu3v+58/53zu3v-3893u4v+25/124x3v2-99/61x2yv2-61/10xy2v2-103/39y3v2-9764x2zv2+123/23xyzv2-83/126y2zv2+13541xz2v2+73/106yz2v2+1172z3v2-14333x2uv2+7/72xyuv2-41/25y2uv2-42/31xzuv2+7/12yzuv2-9744z2uv2-15518xu2v2-121/5yu2v2-7/115zu2v2+75/14u3v2-9/67x2v3+10248xyv3+5738y2v3+71/13xzv3-43/24yzv3+9/94z2v3-113/16xuv3-27/25yuv3-50/11zuv3+41/49u2v3+10896xv4-46/43yv4-58/15zv4-5671uv4-61/9v5,6254x5-95/34x4y-7133x3y2+111/38x2y3-43/19xy4-104/111y5+92/51x4z+11826x3yz+3453x2y2z+5703xy3z+13099y4z-73/79x3z2-69/11x2yz2-15727xy2z2+98/115y3z2-10548x2z3-14452xyz3-5138y2z3-138xz4+46/109yz4-5419z5+73/6x4u-63/5x3yu+31/60x2y2u-11963xy3u+6161y4u-2311x3zu+104/105x2yzu+10873xy2zu+8165y3zu-17/83x2z2u-18/91xyz2u+13544y2z2u+36/121xz3u-8704yz3u+46/7z4u+11392x3u2+133x2yu2+7460xy2u2+8831y3u2+144x2zu2-79/47xyzu2-70/113y2zu2+7369xz2u2-11234yz2u2-19/52z3u2+9830x2u3+13/41xyu3+1985y2u3-22/15xzu3-5449yzu3+5967z2u3+9321xu4+13078yu4-97/3zu4+4079u5-5700x4v+5013x3yv-10774x2y2v-13183xy3v-20/109y4v-106/65x3zv+49/117x2yzv-67/17xy2zv-13917y3zv+9457x2z2v+95/67xyz2v-108/65y2z2v+65/77xz3v-11235yz3v+125/104z4v+17/117x3uv+6355x2yuv+5/36xy2uv-117/31y3uv-1225x2zuv+117/53xyzuv-7/26y2zuv+87/46xz2uv+33/76yz2uv-82/73z3uv-34/39x2u2v+9/22xyu2v+83/11y2u2v+13740xzu2v-108/11yzu2v-9627z2u2v-32/95xu3v-7966yu3v-12400zu3v-111/91u4v+47/11x3v2-10/27x2yv2-5257xy2v2-85/19y3v2-3/62x2zv2+12660xyzv2-37/110y2zv2+10980xz2v2+73/29yz2v2+87/92z3v2-73/90x2uv2-77/25xyuv2+66/31y2uv2-23/96xzuv2-3169yzuv2+125/71z2uv2+9/11xu2v2+6550yu2v2-15163zu2v2+10810u3v2+96/79x2v3+196xyv3-95/94y2v3+15146xzv3+14695yzv3+97/80z2v3+13047xuv3-44/79yuv3-43/41zuv3+40/57u2v3-8369xv4+12/5yv4+77/100zv4-53/79uv4-53/86v5,-122/49x5-3959x4y-7277x3y2+41/93x2y3+39/59xy4-91/68x4z+37/11x3yz+11561x2y2z-5297xy3z+53/14y4z-17/122x3z2+5470x2yz2+14889xy2z2-123/49y3z2+131x2z3+13/43xyz3-82/37y2z3-106/91xz4+11066yz4-72/13z5+15904x4u+44/47x3yu+10450x2y2u+86/41xy3u-40/41y4u+17/92x3zu-81/29x2yzu-40/101xy2zu+103/59y3zu-29/72x2z2u+17/123xyz2u+38/3y2z2u+3358xz3u-69/97yz3u+7627z4u+15451x3u2+5371x2yu2-9009xy2u2-62/35y3u2-44/75x2zu2+13188xyzu2+50/51y2zu2+53/93xz2u2-6796yz2u2-13/6z3u2+39/62x2u3-22/71xyu3-32/97y2u3+43/14xzu3-9/37yzu3-48/125z2u3+100/103xu4+13286yu4+6246zu4-26/107u5-15634x4v+112/9x3yv+54/109x2y2v+23/93xy3v+101/13y4v+79/84x3zv+5290x2yzv-52/29xy2zv-109/111y3zv+107/88x2z2v+10186xyz2v+122/97y2z2v-8370xz3v-108/47yz3v+6311z4v-1126x3uv+51/10x2yuv+9201xy2uv-111/119y3uv+105/58x2zuv-74/23xyzuv-130y2zuv-99/28xz2uv+24/65yz2uv-111/49z3uv-4961x2u2v-23/42xyu2v-32/101y2u2v+48/83xzu2v-13001yzu2v-31/86z2u2v+85/91xu3v+4019yu3v-10/43zu3v+11/105u4v+9790x3v2+73/78x2yv2+63/50xy2v2+35/2y3v2-52/119x2zv2+13680xyzv2-46/55y2zv2+89/12xz2v2-23/82yz2v2-107/85z3v2-20/79x2uv2-85/111xyuv2+31/95y2uv2-13xzuv2+5661yzuv2+9399z2uv2+73/75xu2v2+4782yu2v2-5440zu2v2+45/64u3v2-1484x2v3+1/54xyv3+43/115y2v3+139xzv3-47/85yzv3-103/43z2v3+48/97xuv3+67/15yuv3-69/95zuv3-67/90u2v3+6540xv4+6276yv4-9756zv4-10/121uv4+118/63v5,x5+13/23x4y-10277x3y2-81/107x2y3+18/97xy4+193x4z-206x3yz+1473x2y2z-3907xy3z+9620y4z-11603x3z2+7430x2yz2-3745xy2z2-23/55y3z2-31/13x2z3-115/51xyz3+54/5y2z3+107/6xz4+8432yz4-23/71z5-21/79x4u-7130x3yu-55/46x2y2u+61/113xy3u+11/95y4u-31/125x3zu+47/80x2yzu+12/79xy2zu-51/112y3zu-28/75x2z2u+1962xyz2u-12942y2z2u-93/37xz3u-7/9yz3u+81/7z4u-79/62x3u2+98/29x2yu2-113/36xy2u2-59/51y3u2-5931x2zu2+8/29xyzu2-22/117y2zu2-12146xz2u2+12607yz2u2-8748z3u2-11878x2u3-6xyu3-11798y2u3+97/111xzu3+122/3yzu3+10228z2u3-99/67xu4-8058yu4+116/43zu4-6801u5-7565x4v-21/58x3yv-23/90x2y2v-111/49xy3v+62/3y4v-118/109x3zv-91/27x2yzv-2256xy2zv-6909y3zv+47/126x2z2v+397xyz2v-65/103y2z2v+106/43xz3v-43/9yz3v+4502z4v+5383x3uv-19/48x2yuv+91/92xy2uv-15545y3uv+11204x2zuv+7609xyzuv-125/44y2zuv+15/7xz2uv-4157yz2uv+27/10z3uv+70/67x2u2v+5/121xyu2v-12337y2u2v-12417xzu2v+46/113yzu2v+27/89z2u2v-9419xu3v-59/47yu3v+52/97zu3v+71/56u4v+56/123x3v2+122/73x2yv2+100/103xy2v2+43/8y3v2+99/98x2zv2-14242xyzv2-93/38y2zv2+61/109xz2v2-21/125yz2v2+80/23z3v2+11603x2uv2-1734xyuv2-55/73y2uv2+5/89xzuv2+77/100yzuv2-71/38z2uv2+41/47xu2v2+981yu2v2+11/41zu2v2-8983u3v2+98/59x2v3+105/97xyv3-6372y2v3+12829xzv3+79/96yzv3+110/57z2v3+8495xuv3-11228yuv3-2396zuv3-101/9u2v3-71/39xv4-11746yv4-43/96zv4+64/55uv4-769v5,122/49x4y+3959x3y2+7277x2y3-41/93xy4-39/59y5+32/97x4z+97/96x3yz-43/116x2y2z+19/30xy3z+18/17y4z+39/122x3z2-98/99x2yz2+23/65xy2z2-50/103y3z2-11445x2z3+111/119xyz3-82/67y2z3+49/15xz4+59/74yz4+87/124z5+97/113x3yu+59/113x2y2u-17/16xy3u-1892y4u-15886x3zu+75/124x2yzu-13/108xy2zu+14203y3zu+13871x2z2u+24xyz2u+1483y2z2u-34/91xz3u+51/29yz3u+88/107z4u+6268x2yu2+13776xy2u2+1935y3u2-9062x2zu2-4534xyzu2+9553y2zu2-8429xz2u2-11141yz2u2-89/88z3u2-79/50xyu3-121/15y2u3-7891xzu3-27/94yzu3+96/71z2u3+27/53yu4+15307zu4-775x4v-2648x3yv-2095x2y2v+7594xy3v+12989y4v-17/29x3zv+11801x2yzv+32/45xy2zv-55/73y3zv+9756x2z2v+6886xyz2v+5939y2z2v+39/23xz3v+2632yz3v+77/6z4v+36/121x3uv-76/55x2yuv+5296xy2uv-32/47y3uv-125/84x2zuv-30/67xyzuv-14914y2zuv+73/23xz2uv+123/101yz2uv+49/43z3uv-76/71x2u2v+79/50xyu2v+38/7y2u2v-5262xzu2v-5211yzu2v+2/95z2u2v-3570xu3v-9280yu3v+32/91zu3v+1089u4v-73/33x3v2-3931x2yv2-21/113xy2v2+191y3v2-93/2x2zv2-75/47xyzv2+16/115y2zv2+120/23xz2v2+23/114yz2v2-115/108z3v2-103/12x2uv2+20/107xyuv2+28/11y2uv2+23/42xzuv2-15290yzuv2+3651z2uv2-9566xu2v2-113/68yu2v2+7705zu2v2-12261u3v2+9882x2v3-8520xyv3+14858y2v3+200xzv3+14/73yzv3+45/53z2v3-4/113xuv3-11327yuv3+59/51zuv3-13869u2v3-33/29xv4-13026yv4+4236zv4+10782uv4+2817v5,-123/61x4y+4/87x3y2-15558x2y3+78/5xy4-31/83y5+10267x4z-29/21x3yz+2197x2y2z-57/98xy3z+121/31y4z-1310x3z2+9386x2yz2-37/50xy2z2-1767y3z2-489x2z3-117/31xyz3+13576y2z3-50/9xz4-47/77yz4+53/89z5-57/11x4u-8374x3yu+104/103x2y2u+29/12xy3u+7368y4u+102/37x3zu+10508x2yzu+28xy2zu-14484y3zu-240x2z2u-115/114xyz2u+97/74y2z2u+105/11xz3u+125/46yz3u-6746z4u+3454x3u2+14656x2yu2-82/111xy2u2+13130y3u2+121/58x2zu2-42/11xyzu2-89/81y2zu2+68/49xz2u2+8243yz2u2-11615z3u2-40/87x2u3+80/39xyu3+14735y2u3+10706xzu3-2448yzu3+64/51z2u3+4514xu4+658yu4-39/49zu4+21/23u5+11/105x4v-108x3yv-31/98x2y2v-73/64xy3v+205y4v-15348x3zv-13734x2yzv-52/11xy2zv+5897y3zv+106/27x2z2v+9613xyz2v+81/59y2z2v+1172xz3v+3716yz3v-3581z4v+50x3uv+81/38x2yuv+84/109xy2uv-34/69y3uv-58/35x2zuv-23/105xyzuv+43/78y2zuv-57/31xz2uv+98/37yz2uv+11/96z3uv+147x2u2v+11414xyu2v+5/96y2u2v+19/77xzu2v+1274yzu2v+4/63z2u2v-4396xu3v+7060yu3v+5044zu3v-102/121u4v+67/25x3v2-13601x2yv2+68/65xy2v2+82/77y3v2-8394x2zv2-13318xyzv2+71/104y2zv2-75/107xz2v2+39/100yz2v2+11229z3v2+113/84x2uv2-2988xyuv2-9616y2uv2-41/50xzuv2+3/34yzuv2-12322z2uv2-120/11xu2v2+4/75yu2v2+12562zu2v2-92/39u3v2+1497x2v3+80/103xyv3+15/64y2v3-16/109xzv3-85/81yzv3-52/81z2v3-23/49xuv3+1295yuv3+13460zuv3+52/73u2v3-24/5xv4+11079yv4-25/57zv4-119/47uv4-7/120v5,123/61x4z-4/87x3yz+15558x2y2z-78/5xy3z+31/83y4z-2966x3z2+73/89x2yz2+75/112xy2z2+51/29y3z2-46/47x2z3+3812xyz3-8592y2z3+7582xz4+113/92yz4-11815z5+34/111x3zu-25/63x2yzu+115/79xy2zu+36/115y3zu-17/38x2z2u+7906xyz2u-108/71y2z2u+7541xz3u-12683yz3u-111/110z4u-107/24x2zu2-70/9xyzu2+42/113y2zu2-75xz2u2+64/101yz2u2+65/106z3u2-118/97xzu3+12960yzu3-918z2u3-72/113zu4+17/73x4v+15/17x3yv-22/91x2y2v+78/121xy3v+9/77y4v-91/102x3zv+15814x2yzv-12480xy2zv+1177y3zv-7/40x2z2v+47/69xyz2v-2847y2z2v+987xz3v-1954yz3v+43/97z4v-7692x3uv-109/44x2yuv-94/111xy2uv+84/17y3uv+13682x2zuv+64/7xyzuv-13069y2zuv+92/67xz2uv-109/84yz2uv-93/56z3uv-5368x2u2v-93/101xyu2v-118/43y2u2v-40/57xzu2v-58/55yzu2v+84/19z2u2v+4/9xu3v-540yu3v+11588zu3v-93/25u4v-85/89x3v2-1800x2yv2-40/51xy2v2-7293y3v2+15927x2zv2-33/106xyzv2+25/98y2zv2-13657xz2v2+43/37yz2v2+79/115z3v2-74/11x2uv2+29/114xyuv2-50/69y2uv2+252xzuv2-65/69yzuv2-88/5z2uv2+4418xu2v2-6634yu2v2-9854zu2v2-20/73u3v2+94/11x2v3+1430xyv3+1/41y2v3+21/101xzv3-68/59yzv3-9515z2v3+55/8xuv3-12014yuv3+57/73zuv3-97/58u2v3-124/91xv4+5575yv4-5470zv4+15340uv4+123/7v5,-14327x4y-14863x3y2+3913x2y3+4595xy4+75/77y5-5371x3yz+86/123x2y2z-84/115xy3z-117/100y4z-23/37x2yz2-82/125xy2z2+2426y3z2+17/50xyz3+11188y2z3-110/39yz4-126/97x4u-85/41x3yu-22/25x2y2u-10619xy3u+3473y4u-75/52x3zu-121/37x2yzu+48/125xy2zu+7246y3zu-78/49x2z2u+106/3xyz2u-50/59y2z2u-48/23xz3u+79/111yz3u-14913z4u+31/87x3u2+101/84x2yu2-14270xy2u2-116/83y3u2-15/86x2zu2-3946xyzu2+17/23y2zu2+11/2xz2u2-6992yz2u2-118/77z3u2+68/87x2u3-101/70xyu3-5015y2u3-134xzu3-13479yzu3-103/19z2u3-24/61xu4+35/83yu4-15310zu4-10610u5-22/41x4v-120/91x3yv-10964x2y2v-44/31xy3v+65/122y4v-29/47x3zv+2889x2yzv+43/63xy2zv-47/9y3zv+14/41x2z2v-58/57xyz2v-67/9y2z2v+20/33xz3v+32/87yz3v-113/114z4v+31/19x3uv-7367x2yuv+11612xy2uv-27/62y3uv-47/2x2zuv-28/15xyzuv-6682y2zuv+14672xz2uv+4043yz2uv-2529z3uv-39/115x2u2v+17/4xyu2v+49/39y2u2v+15437xzu2v+91/40yzu2v+14140z2u2v-18/107xu3v-49/5yu3v-88/115zu3v-77/82u4v+1398x3v2+7/111x2yv2+1081xy2v2+6822y3v2-19/100x2zv2-19/9xyzv2+116/83y2zv2+123xz2v2-37/54yz2v2+83/32z3v2-23/82x2uv2-17/45xyuv2-10206y2uv2+20/11xzuv2-29/43yzuv2-61/77z2uv2-15878xu2v2-39/44yu2v2+19/41zu2v2+35/88u3v2+2955x2v3-87/115xyv3-16/17y2v3-107/34xzv3+80/91yzv3+112/69z2v3+109/107xuv3+85/28yuv3-68/121zuv3+2654u2v3-64/55xv4-37/60yv4+69/7zv4+66/85uv4+5754v5,-36/35x4u+12883x3yu+49/120x2y2u-4628xy3u-75/118y4u+65/93x3zu+5754x2yzu-41/62xy2zu+328y3zu-43/49x2z2u+98/107xyz2u+9/16y2z2u+15504xz3u-94/39yz3u-7028z4u-74/81x3u2+3486x2yu2+110/31xy2u2+7609y3u2-17/113x2zu2-71/78xyzu2-49/102y2zu2-2012xz2u2-4935yz2u2+459z3u2+6/53x2u3-12342xyu3-4432y2u3-95/23xzu3+11567yzu3+23/84z2u3-78/19xu4+57/10yu4+37/92zu4+11049u5-70/83x4v-34/69x3yv-9309x2y2v+10247xy3v-119/89y4v-13641x3zv+7018x2yzv+33/95xy2zv+8545y3zv+8410x2z2v-119/33xyz2v-31/44y2z2v-15/79xz3v+49/2yz3v+15796z4v-8/61x3uv-4248x2yuv+1419xy2uv-7061y3uv-12022x2zuv+5656xyzuv+101/51y2zuv-6255xz2uv+8208yz2uv-25/31z3uv-55/64x2u2v+35/88xyu2v+7627y2u2v-15213xzu2v-125/47yzu2v+716z2u2v-13163xu3v-12319yu3v+9945zu3v-122/123u4v-55/112x3v2-14498x2yv2-8389xy2v2+31/24y3v2-48/61x2zv2-10781xyzv2-51/43y2zv2-47/38xz2v2+73/88yz2v2+3185z3v2+41/11x2uv2+53/15xyuv2+35/94y2uv2-2/81xzuv2-109/55yzuv2-12412z2uv2+8/67xu2v2-79/101yu2v2+61/76zu2v2-119/47u3v2+100/7x2v3-56/81xyv3+21/82y2v3+87/52xzv3-85/57yzv3-13804z2v3-106/31xuv3+12534yuv3-93/7zuv3-14/27u2v3+85/11xv4-13639yv4-119/76zv4+59/111uv4-3220v5,975x4y+104/33x3y2-106/65x2y3+66/23xy4+59/86y5+73/72x3yz+33/92x2y2z+41/19xy3z-14775y4z-99/23x2yz2+3753xy2z2+71/41y3z2-92/5xyz3+6199y2z3+125yz4-5129x4u+112/123x3yu+19/2x2y2u-2647xy3u-35/89y4u-31/54x3zu+72/37x2yzu-121/52xy2zu-111/68y3zu+3697x2z2u+128xyz2u+14200y2z2u-27/53xz3u+59/78yz3u-22/45z4u-123/38x3u2+102/53x2yu2-49/57xy2u2+5449y3u2-2572x2zu2+3/2xyzu2+9486y2zu2+6/125xz2u2+45/2yz2u2-19/55z3u2-14525x2u3+83/18xyu3+123/56y2u3-12494xzu3-15240yzu3-28/117z2u3-20/93xu4-10552yu4-79/50zu4+107/33u5-11945x4v-86/85x3yv-115/62x2y2v+1874xy3v+96/13y4v+11797x3zv+15569x2yzv+6118xy2zv-106/103y3zv+117/40x2z2v+13729xyz2v-4329y2z2v+2102xz3v-77/27yz3v-73/25z4v+79/50x3uv+115/58x2yuv-1/111xy2uv-14589y3uv-10733x2zuv-83/65xyzuv+1735y2zuv-104/31xz2uv+99/97yz2uv+23/60z3uv+4021x2u2v+15801xyu2v+4183y2u2v-173xzu2v-100/51yzu2v-79/123z2u2v-8262xu3v-9744yu3v+28/37zu3v+5/111u4v-91/124x3v2+100/123x2yv2+113/96xy2v2-67/50y3v2+101/6x2zv2+3452xyzv2+9281y2zv2+53/22xz2v2-4861yz2v2+8899z3v2+91/23x2uv2+13924xyuv2-4/75y2uv2+59/4xzuv2-214yzuv2-114/49z2uv2+14285xu2v2+85/27yu2v2-99/5zu2v2-74/33u3v2+8313x2v3-96/71xyv3+41/69y2v3-97/31xzv3-6935yzv3-102/29z2v3+67/82xuv3+35/9yuv3-33/95zuv3+6316u2v3-6800xv4-93/4yv4-6367zv4+75/56uv4-13460v5;TestGRRes(Name, I); kill R, Name, @p; "";

  string Name = "ell.d10.g9"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = -44/83x3y2-51/22x2y3+104/11xy4+38/119y5+22/87x3yz-27/61x2y2z-49/121xy3z-60/43y4z+124/65x3z2+39/107x2yz2-10379xy2z2-86/125y3z2+33/8x2z3+3501xyz3+53/9y2z3-69/106xz4+441yz4-19/126z5+12584x3yu+21/104x2y2u+12937xy3u-113/108y4u+13758x3zu+4991x2yzu+5360xy2zu-11946y3zu-2442x2z2u-9336xyz2u+9027y2z2u-112/37xz3u+14035yz3u-8230z4u+58/83x3u2+9675x2yu2+29/114xy2u2+10729y3u2+11/23x2zu2+3609xyzu2+7065y2zu2-2477xz2u2+32/37yz2u2-109/89z3u2-68/109x2u3-107/45xyu3-65/53y2u3+12249xzu3-11480yzu3+4/119z2u3+4418xu4+11875yu4-6961zu4+2678u5-76/71x3yv-87/4x2y2v+119/27xy3v+20/31y4v+122/65x3zv-116/61x2yzv-10976xy2zv-5720y3zv-12559x2z2v-109/102xyz2v-123/121y2z2v-18/73xz3v-89/124yz3v-50/109z4v-113/50x3uv+31/79x2yuv-14121xy2uv+11/123y3uv-3754x2zuv-1676xyzuv-117/113y2zuv-11336xz2uv-35/44yz2uv-116/21z3uv+10245x2u2v+17/4xyu2v-6432y2u2v-12632xzu2v+80/47yzu2v-9/115z2u2v-2607xu3v-79/82yu3v-71/63zu3v+71/102u4v-5212x3v2-2079x2yv2+104/107xy2v2+50/19y3v2+14719x2zv2-7/115xyzv2-113/6y2zv2+123/100xz2v2+22/9yz2v2+15772z3v2-10/111x2uv2-35/52xyuv2-8041y2uv2-17/38xzuv2-11677yzuv2+54/67z2uv2+7941xu2v2-8500yu2v2-147zu2v2-101/68u3v2+7864x2v3-120/61xyv3+13984y2v3+81/101xzv3-79/20yzv3-95/7z2v3-19/89xuv3-5837yuv3+112/95zuv3+119/87u2v3-51/50xv4-85/16yv4+3/44zv4+5109uv4-9320v5,44/83x4y+7230x3y2-110/69x2y3+5570xy4+87y5-103/18x4z+93/70x3yz+75/37x2y2z-61xy3z+7367y4z+5/31x3z2+9111x2yz2-97/44xy2z2+28/55y3z2+89/12x2z3+1832xyz3-15162y2z3+81/100xz4+52/9yz4-3/17z5+23/118x4u+101/55x3yu-33/47x2y2u-149xy3u+7961y4u-7891x3zu-81/14x2yzu+10246xy2zu+51/125y3zu+43/64x2z2u-68/37xyz2u+11/5y2z2u+5578xz3u-5440yz3u+61/118z4u-4318x3u2-69x2yu2-15141xy2u2+12/13y3u2+12180x2zu2-67/6xyzu2-18/121y2zu2+738xz2u2-91/115yz2u2+105/8z3u2-14085x2u3-5451xyu3+91/79y2u3-43/113xzu3-36/11yzu3-89/122z2u3+101/59xu4-125/117yu4-113/25zu4+63/118u5+13/106x4v+11326x3yv+2583x2y2v+14291xy3v+11795y4v-9923x3zv+10823x2yzv+30/31xy2zv-6558y3zv+89/20x2z2v-111/109xyz2v+10420y2z2v-81/44xz3v-76/3yz3v-31/55z4v+3423x3uv+47/80x2yuv-7843xy2uv-15365y3uv+978x2zuv+73/24xyzuv-2/39y2zuv-28/75xz2uv+1849yz2uv-10885z3uv+120/91x2u2v+9293xyu2v-8837y2u2v+109/105xzu2v+7/93yzu2v+4076z2u2v-106/5xu3v+8053yu3v-90zu3v+8598u4v+81/76x3v2-103/42x2yv2-6873xy2v2+49/73y3v2+8447x2zv2+3846xyzv2+107/126y2zv2+2145xz2v2+53/105yz2v2-62/75z3v2-11/35x2uv2+7066xyuv2+69/59y2uv2-53/64xzuv2+8587yzuv2+58/125z2uv2-15386xu2v2+9/121yu2v2+14966zu2v2-2518u3v2-37/56x2v3-11840xyv3-23/47y2v3+110/81xzv3-81yzv3+2719z2v3-65/118xuv3-1221yuv3-33/82zuv3-106/15u2v3+39/112xv4+13479yv4-126/109zv4-2251uv4+8907v5,-10723x3yz+16/79x2y2z+62/49xy3z-5204y4z-49/3x3z2+91/71x2yz2+77/3xy2z2-88/43y3z2-13949x2z3-9045xyz3+109/16y2z3-89/62xz4-15674yz4-10313z5-6773x3yu+124/61x2y2u-7483xy3u+18/23y4u-5136x3zu-3/31x2yzu-11796xy2zu-56/57y3zu-68/95x2z2u-7917xyz2u+1349y2z2u-17/56xz3u+14713yz3u-7938z4u+3104x3u2+14197x2yu2-15411xy2u2-14704y3u2+24/77x2zu2+78/19xyzu2-23/51y2zu2+98/61xz2u2+60/71yz2u2-103/51z3u2-103/42x2u3+15/79xyu3+101/62y2u3-15165xzu3+83/5yzu3-61/60z2u3+7/78xu4-3907yu4+6635zu4-12205u5-388x3yv+63/59x2y2v+88/109xy3v-13978y4v+80/53x3zv+645x2yzv+117/25xy2zv-81/53y3zv+43/72x2z2v+27/8xyz2v-10620y2z2v-13547xz3v-8282yz3v+13426z4v-58/119x3uv+480x2yuv-15697xy2uv-25/58y3uv-115x2zuv-46/123xyzuv-22/13y2zuv-88/105xz2uv-111/62yz2uv+3415z3uv-117/77x2u2v+18/29xyu2v-80/91y2u2v+7851xzu2v-11079yzu2v+39/50z2u2v+14917xu3v+1131yu3v+13259zu3v-122/25u4v-109/37x3v2+7/78x2yv2-73/13xy2v2-5623y3v2+11389x2zv2+63/68xyzv2+28/27y2zv2+69/112xz2v2-10851yz2v2+15829z3v2+86/51x2uv2+2298xyuv2+17/103y2uv2-97/2xzuv2+1/5yzuv2-104/119z2uv2+2/95xu2v2+11655yu2v2-42/65zu2v2+37/26u3v2-12355x2v3+35xyv3+20/7y2v3+8032xzv3+9572yzv3+11508z2v3+10858xuv3+115/81yuv3+10973zuv3+10832u2v3+33/101xv4-71/88yv4+67/18zv4-15473uv4+11/107v5,-28/81x3y2-126/61x2y3-70/23xy4+14521y5-3/55x3yz-100/107x2y2z-10724xy3z+2013y4z+1241x3z2+61/104x2yz2+3577xy2z2-2748y3z2-15718x2z3+106/5xyz3-41/93y2z3-95/61xz4-1/30yz4-73/4z5-59/56x3yu-14867x2y2u-11840xy3u+7444y4u-38/41x3zu-153x2yzu-2654xy2zu+73/83y3zu+6940x2z2u-1241xyz2u+79/17y2z2u-39/35xz3u-4590yz3u+8/109z4u+73/34x3u2+15609x2yu2+12721xy2u2-95/31y3u2+33/14x2zu2+77/65xyzu2-14476y2zu2+73/101xz2u2-73/101yz2u2+92/5z3u2-71/94x2u3-97/33xyu3+125/82y2u3+10213xzu3-6039yzu3-87/61z2u3-14084xu4-43/7yu4+3169zu4+4248u5+100/101x3yv+624x2y2v-4023xy3v-87/65y4v-2971x3zv-8/21x2yzv+8171xy2zv-7776y3zv-55/104x2z2v+96/113xyz2v-7318y2z2v+9436xz3v+42/41yz3v-115/14z4v-2550x3uv+82/29x2yuv+3830xy2uv+71/97y3uv-98/67x2zuv-4319xyzuv+33/49y2zuv-123/49xz2uv-35/62yz2uv-85/6z3uv-10726x2u2v+98/85xyu2v+24/43y2u2v-1382xzu2v+39/109yzu2v-36/71z2u2v+15911xu3v-38/35yu3v+2079zu3v+123/29u4v+4369x3v2-59/126x2yv2-11066xy2v2+110/117y3v2+14815x2zv2-19/49xyzv2-29/10y2zv2+41/38xz2v2-96/55yz2v2+686z3v2+67/30x2uv2+123/23xyuv2-52/25y2uv2-55xzuv2-23/89yzuv2-89/77z2uv2+9906xu2v2-12766yu2v2-104/7zu2v2+6533u3v2-65/97x2v3+57/2xyv3-11084y2v3+13/81xzv3-23/7yzv3-122/111z2v3+224xuv3+5112yuv3+103/30zuv3+78/43u2v3+4811xv4+6171yv4-3422zv4-1589uv4-5541v5,28/81x4y-39/88x3y2-101/87x2y3+11/10xy4+29/44y5-21/94x4z-100/81x3yz+123/73x2y2z+5440xy3z-15190y4z+14140x3z2+74/27x2yz2+92/15xy2z2+8048y3z2-95/64x2z3-60/89xyz3-11/118y2z3+56/15xz4+102/31yz4-1571z5-2598x4u-4426x3yu+2148x2y2u-95/11xy3u-6701y4u-12741x3zu-1179x2yzu+1671xy2zu-93/10y3zu-77/78x2z2u-35/101xyz2u-41/89y2z2u+1556xz3u+13/121yz3u+1288z4u-53/89x3u2-3/109x2yu2-12018xy2u2-5219y3u2+86/49x2zu2-1/84xyzu2+86/113y2zu2-85/123xz2u2+94/85yz2u2-10590z3u2-99/38x2u3+2696xyu3+107/13y2u3-144xzu3+8/69yzu3-107/33z2u3-117/53xu4+1924yu4+75/7zu4+98/113u5+13120x4v-7308x3yv+49/85x2y2v+55/64xy3v-4634y4v+9344x3zv-126/121x2yzv+23/81xy2zv+5086y3zv-4215x2z2v+2442xyz2v-9336y2z2v-43/44xz3v+114/109yz3v+10963z4v-2/111x3uv-5/126x2yuv-41/56xy2uv+47/79y3uv+74x2zuv-15011xyzuv+11187y2zuv-104/115xz2uv-94/67yz2uv+6674z3uv-5/78x2u2v-123/79xyu2v+7/36y2u2v+79/97xzu2v-55/62yzu2v-121/3z2u2v+109/4xu3v+9102yu3v-9526zu3v+8388u4v-47/98x3v2+14967x2yv2+5139xy2v2+83/16y3v2-15926x2zv2+8971xyzv2-15679y2zv2+8686xz2v2+13783yz2v2-57/103z3v2-73/100x2uv2-106/119xyuv2-98/113y2uv2+12096xzuv2+14898yzuv2+115/61z2uv2-125/119xu2v2+6048yu2v2+1/102zu2v2+31/125u3v2+7/27x2v3+64/123xyv3+2569y2v3+113/96xzv3-7216yzv3+2236z2v3-7127xuv3-38/17yuv3+94/73zuv3+708u2v3+87/88xv4-15428yv4-41/64zv4-55/86uv4-41/89v5,109/21x3yz-8388x2y2z-17/10xy3z-14404y4z+12672x3z2+73/52x2yz2+13291xy2z2+118/57y3z2+6159x2z3+4294xyz3-35/92y2z3-55/116xz4+83/64yz4-40/81z5-21/88x3yu-23/58x2y2u+83/118xy3u+12766y4u-45/101x3zu-38/71x2yzu+13/14xy2zu+50y3zu-2123x2z2u-61/102xyz2u+7/111y2z2u+48/121xz3u-15030yz3u-16/21z4u-34/121x3u2+34/111x2yu2+7/48xy2u2+1018y3u2-90/71x2zu2+10585xyzu2-117/56y2zu2+15739xz2u2-37/123yz2u2+4841z3u2-3527x2u3+107/77xyu3+14476y2u3-53/14xzu3-44/41yzu3+83/99z2u3+57/83xu4+1763yu4+3291zu4-2214u5+4710x3yv+4019x2y2v+106/103xy3v-33/67y4v-65/74x3zv-19/36x2yzv+83/97xy2zv+3597y3zv+76/25x2z2v-2159xyz2v+73/67y2z2v-68/55xz3v+36/29yz3v+96/55z4v-7809x3uv-101/51x2yuv+107/99xy2uv+3452y3uv-32/63x2zuv-61/125xyzuv+700y2zuv+14/45xz2uv-10/27yz2uv-64/109z3uv-12507x2u2v-3286xyu2v-8787y2u2v+11244xzu2v-7142yzu2v+51/52z2u2v-11/15xu3v+14/15yu3v-71/78zu3v+12840u4v+1676x3v2-7/22x2yv2+13461xy2v2+91/17y3v2-211x2zv2+2520xyzv2+37/91y2zv2+57/43xz2v2+79/61yz2v2+126/5z3v2+15390x2uv2+2521xyuv2-9728y2uv2-31/100xzuv2-67/120yzuv2-2/89z2uv2+3/40xu2v2-123/5yu2v2+95/44zu2v2+55/82u3v2+61/73x2v3+4656xyv3-59/61y2v3+53/5xzv3+81/19yzv3+7838z2v3-30xuv3-24/7yuv3-88/67zuv3-97/109u2v3+1/32xv4-7809yv4+115/33zv4+95/9uv4-51/31v5,10723x3yu-16/79x2y2u-62/49xy3u+5204y4u+49/3x3zu-91/71x2yzu-77/3xy2zu+88/43y3zu+13949x2z2u+9045xyz2u-109/16y2z2u+89/62xz3u+15674yz3u+10313z4u+12767x3u2-3679x2yu2+115/94xy2u2-95/72y3u2+83/126x2zu2+3070xyzu2+44/91y2zu2+105/53xz2u2-12618yz2u2-13345z3u2+61/47x2u3+108/25xyu3-3/17y2u3-76/71xzu3-36yzu3+96/47z2u3-7291xu4+7566yu4+1445zu4-48/79u5-4731x3yv+35/38x2y2v+26/47xy3v-6365y4v-15522x3zv+6045x2yzv+121/111xy2zv-15013y3zv+4541x2z2v+12683xyz2v+21/61y2z2v+96/29xz3v+98/57yz3v+57/112z4v+1027x3uv+220x2yuv+63/122xy2uv-6828y3uv+21/92x2zuv-13076xyzuv+4327y2zuv+65/107xz2uv-109/120yz2uv+46/87z3uv-13/102x2u2v+20/61xyu2v-41/47y2u2v-7345xzu2v+125/42yzu2v+124/49z2u2v-118/115xu3v+53/36yu3v-15/56zu3v+107/76u4v-13479x3v2+104/73x2yv2+5583xy2v2+12198y3v2-10858x2zv2+244xyzv2+15876y2zv2-38/123xz2v2+93/122yz2v2+82/15z3v2-35x2uv2-90/103xyuv2-16/73y2uv2+5638xzuv2-39/35yzuv2-82/39z2uv2+39/106xu2v2-3530yu2v2+7014zu2v2-9/16u3v2+105/46x2v3+1/9xyv3-10361y2v3+3039xzv3-5956yzv3+750z2v3-2053xuv3+27/112yuv3+4/121zuv3-13/108u2v3-70/33xv4-115/108yv4-59/61zv4-10442uv4-17/39v5,-80/61x3y2+68/57x2y3-35/101xy4+19/75y5+635x3yz+287x2y2z-31/49xy3z+89/93y4z-20/41x3z2-10544x2yz2+14618xy2z2-92/49y3z2+33/86x2z3-1/121xyz3-9325y2z3+2/67xz4-49/106yz4+15003z5+67/49x3yu-15884x2y2u-11/9xy3u+9349y4u+63/20x3zu+79/25x2yzu-23/77xy2zu+99/20y3zu-8233x2z2u-5/84xyz2u+71/97y2z2u+124/43xz3u-64/101yz3u-13567z4u-59/84x3u2+55/94x2yu2+7962xy2u2+36/95y3u2+5/8x2zu2+104/27xyzu2+59/12y2zu2+8231xz2u2-138yz2u2-8432z3u2+25/37x2u3+19/97xyu3+55/97y2u3+12659xzu3+27/119yzu3-101/6z2u3+69/103xu4-111/124yu4+96/107zu4-97/5u5+11725x3yv+16/105x2y2v-79/16xy3v-43/107y4v+17/54x3zv-12757x2yzv+7/124xy2zv-93/73y3zv+69/22x2z2v+73/40xyz2v+15350y2z2v+108/65xz3v+190yz3v+4395z4v+14431x3uv+8x2yuv+6264xy2uv+38/41y3uv-15601x2zuv-6559xyzuv-4074y2zuv+8619xz2uv+83/28yz2uv+15701z3uv-35/37x2u2v+75/62xyu2v+9293y2u2v-32/123xzu2v-64/15yzu2v+101/106z2u2v-13/111xu3v+89/37yu3v+49/36zu3v-78/125u4v+32/87x3v2-15591x2yv2-637xy2v2+31/61y3v2+39/41x2zv2-15874xyzv2-12988y2zv2+37/109xz2v2-18/119yz2v2-7257z3v2+97/16x2uv2+21/29xyuv2+81/95y2uv2-109/34xzuv2-35/27yzuv2+13439z2uv2-55/81xu2v2-23/90yu2v2-79/37zu2v2-60/103u3v2+41/24x2v3-95/54xyv3-104/25y2v3-27/97xzv3+43/52yzv3+82/65z2v3+15/82xuv3+1/44yuv3+65/119zuv3-5/32u2v3-1039xv4+39/31yv4-2847zv4-35/123uv4+59/20v5,80/61x4y-116/11x3y2-79/103x2y3-10762xy4-67/101y5+83/38x4z+61/10x3yz+31/8x2y2z-107/103xy3z-85/113y4z-3/97x3z2+6296x2yz2+47/86xy2z2+15041y3z2+14353x2z3+1990xyz3+15780y2z3-92/43xz4+64/47yz4-108/7z5+88/49x4u+98/57x3yu-41/44x2y2u+92/119xy3u+43/71y4u+14335x3zu+58/7x2yzu-79/39xy2zu-3487y3zu+56/83x2z2u-75/73xyz2u-7173y2z2u-99/64xz3u-51/65yz3u+11138z4u+8299x3u2+47/116x2yu2+98/95xy2u2+117/53y3u2-123/46x2zu2+125/74xyzu2-82/19y2zu2-4508xz2u2+49/29yz2u2-3337z3u2-109/7x2u3+103/41xyu3+28/13y2u3-6790xzu3-8289yzu3+59/41z2u3+15888xu4-9252yu4+10/99zu4+1485u5-101/23x4v-6626x3yv+19/25x2y2v-73/49xy3v-17/14y4v-33/46x3zv+109/58x2yzv+1/44xy2zv+8/65y3zv+5700x2z2v+16/69xyz2v+18/37y2z2v+3905xz3v+84/95yz3v+83/65z4v-700x3uv-8/7x2yuv+3184xy2uv-8373y3uv+106/115x2zuv+68/125xyzuv-2829y2zuv-38/41xz2uv+101/19yz2uv+66/97z3uv-47/7x2u2v+21/97xyu2v+9/74y2u2v+57/55xzu2v-107/89yzu2v-14157z2u2v-1693xu3v+9/10yu3v-53/114zu3v-10049u4v+9871x3v2+14318x2yv2-8447xy2v2+10730y3v2-98/87x2zv2-3822xyzv2+38/39y2zv2+111/59xz2v2-4603yz2v2+35/69z3v2-161x2uv2-73/48xyuv2-58/79y2uv2+71/63xzuv2+1832yzuv2+7891z2uv2-13900xu2v2+113/31yu2v2-7626zu2v2-1223u3v2-33/94x2v3-49/20xyv3+20/113y2v3-47/15xzv3+109/32yzv3-49/95z2v3+92/17xuv3-16/99yuv3-10817zuv3+3754u2v3-94/9xv4-67/79yv4+12659zv4+101/117uv4+113/17v5,-3756x3yz-24/35x2y2z-29/112xy3z+7944y4z-8968x3z2+62/115x2yz2-38/101xy2z2+28/39y3z2+61/49x2z3-88/119xyz3-119/87y2z3-7/123xz4-4935yz4-1239z5+8x3yu-68/7x2y2u+11910xy3u+5043y4u+7293x3zu-121/47x2yzu+8/77xy2zu-53/109y3zu+15439x2z2u-6359xyz2u-81/109y2z2u-3423xz3u+12246yz3u-1382z4u-2355x3u2+12015x2yu2-79/38xy2u2-14798y3u2+623x2zu2+47/113xyzu2+11457y2zu2+5127xz2u2-9/16yz2u2-70/41z3u2+107/112x2u3-73/49xyu3-38/15y2u3+8748xzu3-5150yzu3-94/67z2u3+10449xu4-993yu4-53/6zu4+46/79u5+59/46x3yv+7/44x2y2v+11979xy3v+78/61y4v+69/109x3zv-424x2yzv+3267xy2zv-82/17y3zv+15/92x2z2v-106/121xyz2v-32/37y2z2v+15746xz3v+97/30yz3v-7/29z4v-9484x3uv+64/45x2yuv-75xy2uv+14513y3uv-11/103x2zuv-82/3xyzuv+8740y2zuv-67/101xz2uv+584yz2uv+15/121z3uv+11669x2u2v+51/112xyu2v+124/101y2u2v+106/17xzu2v+3108yzu2v-67/118z2u2v+6866xu3v+5705yu3v-68/77zu3v-89/98u4v+14035x3v2-1896x2yv2+107/99xy2v2+46/49y3v2+109/29x2zv2-14831xyzv2-13871y2zv2+61/62xz2v2-35/9yz2v2+61/40z3v2+101/115x2uv2-14723xyuv2+13811y2uv2+66/13xzuv2-9031yzuv2-118/77z2uv2-86/121xu2v2-10616yu2v2-60/107zu2v2+123/85u3v2-92/47x2v3+23/98xyv3-5986y2v3-8636xzv3+63/89yzv3+1899z2v3+117/19xuv3-65/41yuv3+2817zuv3-1578u2v3-35/82xv4+2155yv4+7844zv4-40/67uv4-16/111v5,-4394x3y2-8801x2y3+12722xy4+15341y5+85/73x3yz+3786x2y2z-156xy3z-11841y4z-93/47x3z2+79/20x2yz2+79/99xy2z2+10343y3z2-631x2z3+7178xyz3-4119y2z3+15660xz4+14318yz4+1/69z5-9/85x3yu-59/33x2y2u-4250xy3u-9233y4u-103/75x3zu-72/53x2yzu+50/7xy2zu+6476y3zu-37/114x2z2u-5/7xyz2u-13/93y2z2u+7262xz3u+40/93yz3u-5837z4u-13/116x3u2+12622x2yu2+2615xy2u2-235y3u2+99/115x2zu2+101/9xyzu2+83/28y2zu2+14490xz2u2+119/96yz2u2-11229z3u2+41/86x2u3+14636xyu3-88/45y2u3-9311xzu3-8967yzu3-71/105z2u3+56/53xu4+14093yu4+115zu4+118/67u5-95/54x3yv-11922x2y2v-101/117xy3v-15245y4v-3849x3zv+100x2yzv+101/64xy2zv+82/5y3zv-53/52x2z2v-26/67xyz2v+17/72y2z2v+33/41xz3v-91/82yz3v-29/12z4v+73/126x3uv+109/73x2yuv-28/55xy2uv-51/29y3uv+17/35x2zuv+108/67xyzuv+81/2y2zuv+95/101xz2uv-57/8yz2uv-7142z3uv-8230x2u2v-91/107xyu2v-51/11y2u2v+67/125xzu2v-59/17yzu2v+23/121z2u2v+14520xu3v+2632yu3v+10602zu3v+19/75u4v-98/115x3v2-13270x2yv2+23/28xy2v2-97/24y3v2+18/37x2zv2+2394xyzv2+9634y2zv2-12683xz2v2-4723yz2v2+93/11z3v2+76/15x2uv2-7066xyuv2-3496y2uv2-31/33xzuv2+23/114yzuv2+75/4z2uv2+6637xu2v2+5/31yu2v2-220zu2v2-15145u3v2-16/57x2v3+994xyv3+13/111y2v3-35/111xzv3-1460yzv3-13818z2v3-12748xuv3+6/19yuv3+3481zuv3+4768u2v3+31/83xv4-21/34yv4-117/95zv4+7890uv4-5747v5,4394x4y+25/118x3y2+27/122x2y3-1/20xy4-21/58y5+7367x4z+11281x3yz+2518x2y2z-55/92xy3z+7/17y4z-6716x3z2+116/11x2yz2+67/35xy2z2+14506y3z2-70/17x2z3-92/125xyz3-11235y2z3+28/79xz4+6770yz4+13311z5-12323x4u-36x3yu-74/35x2y2u+28/11xy3u-14140y4u+29/76x3zu+10357x2yzu-13657xy2zu-65/9y3zu+8/49x2z2u-186xyz2u+7854y2z2u+20/111xz3u+53/20yz3u-4420z4u-49x3u2+106/73x2yu2-57/112xy2u2-1767y3u2-67/36x2zu2+97/31xyzu2-4/15y2zu2-27/119xz2u2-6837yz2u2+81/88z3u2-803x2u3-14237xyu3+116/45y2u3+74/53xzu3-44/23yzu3+7957z2u3+2551xu4-5262yu4-3014zu4+115/98u5+46/39x4v+71/2x3yv+13841x2y2v-12/7xy3v+7491y4v+19/67x3zv-21/82x2yzv+29/92xy2zv+124/123y3zv+4436x2z2v+4318xyz2v-14/111y2z2v+105/118xz3v+12883yz3v-105/22z4v-124/109x3uv-107/103x2yuv+7/4xy2uv-114/121y3uv-74/69x2zuv-5025xyzuv-114/97y2zuv-5872xz2uv+24/95yz2uv-49/88z3uv-21/29x2u2v+121/52xyu2v+721y2u2v+119/67xzu2v+85/48yzu2v+50/67z2u2v+24/53xu3v+32/115yu3v-2809zu3v+2966u4v+5211x3v2-190x2yv2+2551xy2v2+2895y3v2+124/63x2zv2-6160xyzv2+71/122y2zv2+8635xz2v2-9/28yz2v2+193z3v2+9/119x2uv2-110/101xyuv2-11713y2uv2-6363xzuv2+61/93yzuv2-10139z2uv2+89/96xu2v2+94/103yu2v2+61/93zu2v2-61/78u3v2-67/22x2v3+15674xyv3-121/5y2v3+121/76xzv3+15523yzv3-93/88z2v3+8667xuv3-11344yuv3-97/108zuv3-14490u2v3+10228xv4-18/107yv4-54/37zv4-2193uv4+14370v5,-103/79x3yz-16/41x2y2z+114/83xy3z-104/85y4z+14/37x3z2-3023x2yz2+723xy2z2-7/55y3z2-82/85x2z3-33/83xyz3+109/119y2z3-13119xz4-4627yz4-35/37z5-11662x3yu+121/57x2y2u-7212xy3u+59/83y4u+2149x3zu-30/13x2yzu+125/119xy2zu+79/23y3zu-3/88x2z2u+7168xyz2u-10717y2z2u+112/33xz3u-10509yz3u+121/56z4u+125/88x3u2-115/121x2yu2+49/4xy2u2-27/88y3u2-2867x2zu2-43/85xyzu2-4734y2zu2-37/114xz2u2-2/43yz2u2+106/95z3u2-108/35x2u3+4531xyu3-19/44y2u3-73/27xzu3+59/117yzu3+25/9z2u3-65/89xu4+81/20yu4+5365zu4+1733u5-2363x3yv-10326x2y2v-5/24xy3v-17/106y4v-14412x3zv+6361x2yzv+10187xy2zv-76/101y3zv+4991x2z2v+14252xyz2v-107/18y2z2v+4857xz3v-13162yz3v+1088z4v-3/64x3uv-14/113x2yuv+96/41xy2uv-76/103y3uv+8924x2zuv-11/30xyzuv-4493y2zuv+3070xz2uv+10/41yz2uv-35/51z3uv+4318x2u2v+15309xyu2v+107/101y2u2v-1748xzu2v+11082yzu2v+29/90z2u2v-19/8xu3v+107/64yu3v+3039zu3v-21/59u4v+14/87x3v2+16/59x2yv2+77/9xy2v2+3679y3v2+53/113x2zv2+97/80xyzv2-899y2zv2+10016xz2v2-10049yz2v2+29/10z3v2+43/7x2uv2-5919xyuv2-7/34y2uv2+42/85xzuv2+102/23yzuv2-8300z2uv2+125/2xu2v2-1700yu2v2-107/21zu2v2+12942u3v2-3054x2v3-87/53xyv3+4745y2v3+52/83xzv3-11747yzv3-119/64z2v3-5987xuv3+2016yuv3-2889zuv3+4535u2v3-1588xv4-27/5yv4+8541zv4-15581uv4-3/119v5,-x3y2-7492x2y3+118/35xy4-43/5y5-2479x3yz-8522x2y2z-105/26xy3z-103/18y4z+5868x3z2-10040x2yz2-50/81xy2z2+3378y3z2-71/44x2z3-12323xyz3-25/44y2z3+12979xz4-15146yz4+5962z5+8852x3yu-39/77x2y2u-9554xy3u-14590y4u+61/84x3zu-2295x2yzu-5969xy2zu+14241y3zu-29/74x2z2u+107/17xyz2u+107y2z2u-14/99xz3u+109/93yz3u-73/41z4u-9170x3u2+8877x2yu2-109/118xy2u2+65/36y3u2+122/93x2zu2-15020xyzu2+33/50y2zu2-11/46xz2u2-91/116yz2u2-47/10z3u2-34/57x2u3-90/79xyu3+12760y2u3+81/71xzu3+1436yzu3+14839z2u3+6598xu4-12418yu4-47/59zu4-1357u5+6/91x3yv-46/21x2y2v+100/53xy3v-69/125y4v+98x3zv-63/23x2yzv+96/121xy2zv+12/11y3zv-37/120x2z2v-37/106xyz2v-25/122y2z2v-13657xz3v+79/47yz3v+15519z4v+45/46x3uv+73/100x2yuv-5361xy2uv-39/101y3uv+1161x2zuv-105/22xyzuv+79/34y2zuv+96/31xz2uv-20/51yz2uv-2683z3uv-122/125x2u2v-66/109xyu2v-6357y2u2v-3021xzu2v+86/33yzu2v-105/8z2u2v+95/22xu3v-9046yu3v+69/83zu3v-3166u4v+91/115x3v2+9661x2yv2-8757xy2v2+79/23y3v2+125/107x2zv2+6133xyzv2+8975y2zv2-38/103xz2v2-120/43yz2v2+59/124z3v2-9252x2uv2+27/64xyuv2+99/80y2uv2+125/28xzuv2-60/109yzuv2-73/107z2uv2-15162xu2v2+91/71yu2v2-89/122zu2v2-1/9u3v2+40/117x2v3-71/41xyv3+10502y2v3-5618xzv3-995yzv3-117/112z2v3-15185xuv3-4470yuv3-13/4zuv3-14631u2v3-5804xv4-15527yv4-47/87zv4+5084uv4-64/11v5,x4y+46/47x3y2+13/47x2y3-13553xy4-5/59y5-15/76x4z+370x3yz+15/56x2y2z-8199xy3z+17/37y4z+32/47x3z2-26/29x2yz2-1/109xy2z2+68/77y3z2-88/123x2z3-1397xyz3-90y2z3+117/80xz4+14919yz4+11248z5+1742x4u-6060x3yu+97/38x2y2u+54/53xy3u-26/45y4u-315x3zu+9/76x2yzu+113/14xy2zu-28/79y3zu-109/14x2z2u+53/54xyz2u-213y2z2u-18/11xz3u-12129yz3u+6321z4u-13361x3u2+109/81x2yu2+37/84xy2u2-19/69y3u2+82/31x2zu2-20/41xyzu2-13135y2zu2-52/99xz2u2-12942yz2u2-8237z3u2-12377x2u3-41/43xyu3+13143y2u3-6701xzu3-98/53yzu3+97/43z2u3+1570xu4-41/5yu4-7/86zu4+306u5-7920x4v+76/53x3yv-91/73x2y2v-49/64xy3v+12976y4v-1787x3zv-95/123x2yzv-10033xy2zv-83/100y3zv-17/112x2z2v+1060xyz2v-121/54y2z2v+25/77xz3v+23/73yz3v-3576z4v+41/98x3uv+15265x2yuv+9014xy2uv-86/33y3uv-83/51x2zuv-15747xyzuv+11/16y2zuv-12110xz2uv-15554yz2uv-7654z3uv+68/53x2u2v+1673xyu2v-123/64y2u2v-3/19xzu2v+29/34yzu2v-5457z2u2v+51/58xu3v-4242yu3v-70/67zu3v+7159u4v+8620x3v2-9381x2yv2+52/79xy2v2-73/113y3v2+83/22x2zv2-1445xyzv2+6524y2zv2-122xz2v2-107/85yz2v2-1240z3v2+1710x2uv2-101/59xyuv2-2950y2uv2-12343xzuv2-79/27yzuv2-5/26z2uv2+89/68xu2v2-14985yu2v2+8382zu2v2-5109u3v2-12322x2v3-97/106xyv3-86/23y2v3+115/11xzv3-45/2yzv3-79/110z2v3+7472xuv3-17/71yuv3+1659zuv3-4468u2v3-85/47xv4+11/73yv4-1700zv4-32/51uv4+7855v5,62/49x4-125/36x3y+2x2y2+56/39xy3+7307y4-42/43x3z+83/42x2yz-11648xy2z-43/106y3z-49/109x2z2-95/28xyz2-194y2z2+71/95xz3-83/59yz3-66/95z4-1/51x3u-82/21x2yu+121/103xy2u-11106y3u-55/91x2zu+55/119xyzu-123/79y2zu-20/11xz2u+6943yz2u+69z3u+5908x2u2-56/89xyu2-13119y2u2+32/115xzu2+11412yzu2+63/50z2u2+73/85xu3-12527yu3+1991zu3+26/33u4+79/91x3v-10/27x2yv+20/117xy2v-49/102y3v+98/33x2zv+115/76xyzv+12412y2zv-47/48xz2v+121/113yz2v+105/26z3v-10954x2uv+10730xyuv+15721y2uv-101/63xzuv-32/19yzuv+67/71z2uv+77/45xu2v+27/98yu2v-670zu2v-13396u3v-55/119x2v2-7890xyv2-101/7y2v2-9563xzv2+3/95yzv2+4857z2v2+9397xuv2+3/116yuv2-12257zuv2-79/49u2v2+11697xv3+26/19yv3+4885zv3-53/22uv3-8355v4;TestGRRes(Name, I); kill R, Name, @p; "";

  string Name = "k3.d10.g9.quart2"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = x3yz2+31/15x2y2z2-7231xy3z2+99/37y4z2+28/95x3z3+97/32x2yz3+13247xy2z3+12717y3z3-113/31x2z4-61/30xyz4-6844y2z4+104/3xz5-13849yz5+43/39z6+13061x3yzu-8463x2y2zu+94/69xy3zu-8/61y4zu-13297x3z2u+7217x2yz2u-7830xy2z2u-75/14y3z2u+2839x2z3u-14657xyz3u-52/7y2z3u-6/89xz4u-6169yz4u+44/7z5u-98/33x3yu2+41/30x2y2u2-65/98xy3u2+122/13y4u2+9906x3zu2-11587x2yzu2+17/53xy2zu2+6504y3zu2+49/106x2z2u2+11480xyz2u2+97/71y2z2u2+12560xz3u2-114/83yz3u2-13761z4u2-67/112x3u3-18/49x2yu3+21/67xy2u3-44/43y3u3+123/116x2zu3+4459xyzu3-13841y2zu3-805xz2u3-1382yz2u3-5293z3u3+133x2u4-122/79xyu4+9724y2u4+61/24xzu4+113/119yzu4-19/108z2u4+15893xu5+57/22yu5+4600zu5-618u6-27/53x3yzv+44/103x2y2zv-142xy3zv+19/84y4zv+105/8x3z2v+10532x2yz2v-75/74xy2z2v-70/19y3z2v+31/80x2z3v-481xyz3v+47/30y2z3v+14318xz4v+51/28yz4v-15/113z5v-46/17x3yuv-99/100x2y2uv-106/5xy3uv+14384y4uv+7/100x3zuv-15/64x2yzuv-6976xy2zuv+12051y3zuv-67/42x2z2uv-2627xyz2uv-49/104y2z2uv+77/16xz3uv+15766yz3uv+85/117z4uv-107/101x3u2v-6699x2yu2v+2443xy2u2v-27/28y3u2v+11945x2zu2v-14467xyzu2v-4873y2zu2v-63/124xz2u2v-8270yz2u2v+11900z3u2v+47/14x2u3v+53/8xyu3v-10/51y2u3v-87/119xzu3v+114/73yzu3v+86/57z2u3v+52/63xu4v-11587yu4v+1/18zu4v-121/109u5v+116/11x3yv2+19/108x2y2v2-31/3xy3v2-43/9y4v2-81/100x3zv2-7728x2yzv2-1037xy2zv2+24/101y3zv2-61/103x2z2v2-8/51xyz2v2+117/109y2z2v2+98/23xz3v2+1646yz3v2-3356z4v2+105/59x3uv2+117/31x2yuv2+519xy2uv2+12633y3uv2+25/6x2zuv2-963xyzuv2-49/23y2zuv2-116/25xz2uv2+14146yz2uv2+11480z3uv2-95/8x2u2v2-10928xyu2v2-51/23y2u2v2-12770xzu2v2-92/91yzu2v2+3872z2u2v2+3183xu3v2+6871yu3v2+90/37zu3v2+10019u4v2-69/88x3v3-1398x2yv3-97/72xy2v3-46/97y3v3+107/14x2zv3-20/89xyzv3-11367y2zv3+120/29xz2v3-86/81yz2v3+107/69z3v3-39/17x2uv3+83/11xyuv3+169y2uv3-11/71xzuv3-22/17yzuv3-14862z2uv3-13009xu2v3-101/12yu2v3+10617zu2v3+2567u3v3-23/85x2v4+27/50xyv4+113/51y2v4+97/16xzv4+4438yzv4-11857z2v4+14580xuv4-6426yuv4+9421zuv4-10585u2v4+14670xv5+1807yv5+10298zv5-116/53uv5+7869v6,x3yzu+31/15x2y2zu-7231xy3zu+99/37y4zu+28/95x3z2u+97/32x2yz2u+13247xy2z2u+12717y3z2u-113/31x2z3u-61/30xyz3u-6844y2z3u+104/3xz4u-13849yz4u+43/39z5u-124/43x3yu2-90/13x2y2u2-13244xy3u2-78/73y4u2+118/43x3zu2+37/67x2yzu2-10426xy2zu2+2412y3zu2-32/113x2z2u2+35/104xyz2u2+3952y2z2u2+9028xz3u2-1990yz3u2-59/109z4u2+15499x3u3+116/23x2yu3+95/58xy2u3+8/47y3u3+59/109x2zu3-29xyzu3+12412y2zu3+20/81xz2u3+2200yz2u3-13809z3u3+3889x2u4-8136xyu4+8922y2u4-4/121xzu4+82/113yzu4-65/23z2u4+101/53xu5+103/113yu5-99/118zu5-9524u6-2749x3yzv-7814x2y2zv+73/113xy3zv+9937y4zv-59/62x3z2v-23/12x2yz2v-10245xy2z2v+7130y3z2v-4427x2z3v+6656xyz3v+3448y2z3v-46/79xz4v+1611yz4v+8453z5v+12013x3yuv+49/17x2y2uv-4/115xy3uv-121/91y4uv-63/29x3zuv+64/7x2yzuv-8785xy2zuv-87/14y3zuv+36/121x2z2uv+9525xyz2uv+4215y2z2uv-17/13xz3uv-117/125yz3uv+101/122z4uv+42/37x3u2v-8747x2yu2v-105/79xy2u2v+10799y3u2v-58/49x2zu2v-8/75xyzu2v-67/49y2zu2v-38/11xz2u2v+53/27yz2u2v+113/52z3u2v+18/59x2u3v+71/106xyu3v+47/2y2u3v-4594xzu3v+95/4yzu3v-121/46z2u3v-55/62xu4v-101/72yu4v+40/53zu4v+15227u5v-15553x3yv2+29/94x2y2v2-4076xy3v2-7133y4v2+27/125x3zv2+33/29x2yzv2-63/95xy2zv2+9166y3zv2-480x2z2v2+9941xyz2v2+107/46y2z2v2+13018xz3v2+53/98yz3v2+92/35z4v2+17/30x3uv2+77/95x2yuv2+11/67xy2uv2+8262y3uv2+65/11x2zuv2+2567xyzuv2-33/94y2zuv2+85/92xz2uv2+103/25yz2uv2-27/100z3uv2+13210x2u2v2-109/90xyu2v2+141y2u2v2-124/51xzu2v2-3/109yzu2v2-4910z2u2v2+205xu3v2+14357yu3v2+85/57zu3v2-109/28u4v2-68/39x3v3+10545x2yv3-2176xy2v3-8743y3v3+15111x2zv3+25/119xyzv3+8/103y2zv3-6046xz2v3+8658yz2v3+106/5z3v3-31/126x2uv3-7762xyuv3+2315y2uv3+124/67xzuv3-77/104yzuv3+95/71z2uv3+69/119xu2v3+13069yu2v3-8620zu2v3+105/41u3v3-15772x2v4-11212xyv4-61/36y2v4+38/125xzv4-15860yzv4+8/63z2v4+7519xuv4-94/41yuv4+45/32zuv4+9417u2v4-71/35xv5-6287yv5+6481zv5+106/99uv5+3/41v6,-8036x3yu2+7966x2y2u2-151xy3u2-14/111y4u2-111/76x3zu2-102/11x2yzu2+7956xy2zu2-7397y3zu2-113/16x2z2u2-8049xyz2u2+7230y2z2u2+3978xz3u2-36/113yz3u2-8147z4u2-107/83x3u3+78/97x2yu3+12700xy2u3+11/72y3u3+88/31x2zu3-63/40xyzu3+101/35y2zu3-220xz2u3+3/103yz2u3-49/45z3u3-21/113x2u4+104/123xyu4+98/47y2u4-56/61xzu4-87/50yzu4+5913z2u4-120/17xu5+64/11yu5-109/80zu5+10371u6-118/25x3yzv+58/99x2y2zv-5/64xy3zv+7/46y4zv-49/103x3z2v-77/106x2yz2v-44/7xy2z2v-7559y3z2v-17/35x2z3v+948xyz3v-15043y2z3v-3576xz4v-2/109yz4v+74/11z5v+6436x3yuv+7316x2y2uv+29/5xy3uv-1326y4uv+34/49x3zuv-122/27x2yzuv-632xy2zuv+46/49y3zuv-13463x2z2uv-808xyz2uv-17/32y2z2uv-13149xz3uv-117/88yz3uv-45/79z4uv-65/94x3u2v+6/67x2yu2v+34/39xy2u2v-14026y3u2v+42/107x2zu2v-3287xyzu2v-70/43y2zu2v+29/104xz2u2v-47/18yz2u2v-11038z3u2v+6262x2u3v-5255xyu3v-7/10y2u3v+7065xzu3v+5608yzu3v+4675z2u3v-73/90xu4v-15822yu4v-71/63zu4v+110/97u5v-69/5x3yv2+4315x2y2v2-124/45xy3v2-79/16y4v2-10739x3zv2-93/46x2yzv2+12499xy2zv2-73/86y3zv2+6367x2z2v2-12876xyz2v2-306y2z2v2-89xz3v2-70/51yz3v2+13120z4v2+61/57x3uv2+14782x2yuv2-91/9xy2uv2-2625y3uv2+14747x2zuv2-5899xyzuv2-12944y2zuv2-47/14xz2uv2-4551yz2uv2-99/101z3uv2-12618x2u2v2+1507xyu2v2-11951y2u2v2+68/49xzu2v2+49/39yzu2v2-56/103z2u2v2-31/85xu3v2-32/49yu3v2-65/14zu3v2+15/7u4v2+5749x3v3-3667x2yv3-107/29xy2v3+11301y3v3+95/18x2zv3-121/74xyzv3+75/26y2zv3+101/98xz2v3-111/76yz2v3-11335z3v3-15923x2uv3-36/83xyuv3-4134y2uv3-87/118xzuv3-41/11yzuv3+104/61z2uv3+12583xu2v3-50/23yu2v3-31/44zu2v3-29/23u3v3+108/107x2v4-8216xyv4-5009y2v4+101/26xzv4-9779yzv4+71/74z2v4-3358xuv4+83/84yuv4-34/39zuv4+44/47u2v4-112/83xv5+113/74yv5+82/79zv5-115/99uv5+12/109v6,-x4y-31/15x3y2+7231x2y3-99/37xy4-28/95x4z-53/107x3yz-4623x2y2z+5300xy3z-41/111y4z+12205x3z2+113/120x2yz2+54/49xy2z2-85/63y3z2+104/89x2z3-52/121xyz3-22/49y2z3+14367xz4+71/93yz4+55/56z5-5/81x4u-67/81x3yu-83/13x2y2u+98/55xy3u+15289y4u-94/111x3zu+40/29x2yzu-16/59xy2zu-107/14y3zu+2965x2z2u-459xyz2u-2/47y2z2u+35/22xz3u+119/39yz3u-12180z4u-13679x3u2+1534x2yu2+11305xy2u2-62/9y3u2-68/39x2zu2+11/90xyzu2-36/101y2zu2-2896xz2u2-15114yz2u2-49/114z3u2+19/16x2u3-11401xyu3-109/3y2u3+67/80xzu3+53/92yzu3+2894z2u3+119/74xu4+407yu4-65/53zu4+95/94u5-9309x4v+21/40x3yv+1436x2y2v+2194xy3v+6994y4v-116/81x3zv+13/2x2yzv-12/13xy2zv-23/84y3zv-61/83x2z2v+2023xyz2v+19/40y2z2v+43/26xz3v-59/113yz3v-47/53z4v+15580x3uv+21x2yuv+113/97xy2uv-15419y3uv-15243x2zuv+5128xyzuv-34/47y2zuv+13206xz2uv-4833yz2uv+107/91z3uv-1693x2u2v+54/53xyu2v-86/67y2u2v+98/9xzu2v+86/17yzu2v+64/89z2u2v+25/113xu3v+7884yu3v+14089zu3v-12027u4v-9471x3v2-36/85x2yv2-21/13xy2v2+15888y3v2+76/109x2zv2+4547xyzv2+115/12y2zv2-11/107xz2v2+6764yz2v2-8321z3v2+84/101x2uv2-202xyuv2+3251y2uv2+91/4xzuv2+7124yzuv2-53/81z2uv2+47/84xu2v2-8833yu2v2+117/14zu2v2-3/113u3v2+126/97x2v3-78/115xyv3+68/63y2v3-34/109xzv3+5913yzv3+6226z2v3-2365xuv3+91/120yuv3+14120zuv3-69/8u2v3+71/12xv4-13094yv4-7262zv4-33uv4+5367v5,-9533x4y-318x3y2+8/49x2y3+83/29xy4+13129y5+221x4z+115/48x3yz+12508x2y2z+97/52xy3z+11479y4z+8941x3z2+104/109x2yz2+9191xy2z2+103/64y3z2+10584x2z3-7728xyz3+3979y2z3+15/82xz4+5409yz4-1326z5+3756x4u-57/62x3yu+63/47x2y2u-14600xy3u+159y4u-11/4x3zu-113/57x2yzu-26/125xy2zu-32/87y3zu-10/21x2z2u+12927xyz2u-73/62y2z2u+115/99xz3u-13/3yz3u-126/25z4u-3969x3u2-122/57x2yu2-5003xy2u2-100/117y3u2-71/30x2zu2+7356xyzu2-2211y2zu2+31/40xz2u2-6722yz2u2-139z3u2+4426x2u3+1/115xyu3-72/85y2u3+15260xzu3+7938yzu3+4/115z2u3-33/89xu4+31/108yu4-50/83zu4+14/107u5+24/95x4v-113/17x3yv+81/14x2y2v-9957xy3v-10075y4v-122/113x3zv+65/118x2yzv-96/29xy2zv-19/41y3zv+113/35x2z2v+121/31xyz2v-9/68y2z2v+91/45xz3v-23/116yz3v-67/99z4v-5355x3uv-3112x2yuv-12824xy2uv-58/123y3uv-13/22x2zuv-19/85xyzuv-121/24y2zuv-14093xz2uv+99/95yz2uv+89/50z3uv+13096x2u2v-109/120xyu2v+121/61y2u2v+80/41xzu2v-39yzu2v-8/99z2u2v+5/17xu3v+112/69yu3v+14346zu3v-7173u4v+125/13x3v2+43/53x2yv2-78/103xy2v2-109/111y3v2+33/13x2zv2-15333xyzv2+87/49y2zv2-7212xz2v2+7729yz2v2-86/123z3v2-119/103x2uv2-71/122xyuv2-81/113y2uv2+6133xzuv2+55/72yzuv2+69/31z2uv2+12828xu2v2+94/15yu2v2-7588zu2v2+21/41u3v2-8712x2v3+74/9xyv3-11/87y2v3+1446xzv3-3/95yzv3-87/55z2v3-717xuv3-110/97yuv3-13/113zuv3-95/81u2v3-37/68xv4+5112yv4-56/11zv4-6/115uv4+7910v5,25/42x4y-42/79x3y2-59/21x2y3+2736xy4-107/115x4z-203x3yz+47/101x2y2z+7686xy3z-63/64y4z+103/57x3z2-12082x2yz2+11/102xy2z2-83/43y3z2+13/49x2z3-2685xyz3+123/44y2z3+31/12xz4+126/83yz4+14745z5+83/37x4u+7362x3yu-14615x2y2u-14109xy3u+49/47y4u+1929x3zu+83/71x2yzu-13640xy2zu-97/58y3zu-11141x2z2u-61/49xyz2u-3745y2z2u-74/21xz3u+3493yz3u-7540z4u-103/118x3u2-43/32x2yu2-9200xy2u2-23/65y3u2+15895x2zu2-13924xyzu2-14291y2zu2-11039xz2u2-31/37yz2u2-101/93z3u2-39/83x2u3-4536xyu3-78/47y2u3+75/44xzu3-24/121yzu3-81/113z2u3-81/89xu4+15825yu4-4111zu4+5850u5-12534x4v-69/94x3yv-10076x2y2v+3952xy3v+25/12y4v+21/34x3zv+11002x2yzv-54xy2zv+20/23y3zv+4991x2z2v+549xyz2v+2687y2z2v-110/9xz3v+11359yz3v+49/24z4v+62/107x3uv-27/41x2yuv-17/52xy2uv-10972y3uv+12/103x2zuv-318xyzuv-77/40y2zuv-114/53xz2uv+17/28yz2uv-8084z3uv+85/36x2u2v+7/100xyu2v-5772y2u2v-89/114xzu2v-40/121yzu2v+3340z2u2v+36/113xu3v-38/93yu3v+2519zu3v-7084u4v+8136x3v2-55/23x2yv2+27/7xy2v2+74/39y3v2+63/16x2zv2-8661xyzv2+2/91y2zv2+3773xz2v2-75/122yz2v2+447z3v2-59/109x2uv2-119/9xyuv2-67/49y2uv2-11334xzuv2-10482yzuv2-60/91z2uv2+94/65xu2v2-108/17yu2v2-69/70zu2v2-23/20u3v2+8/115x2v3+29/41xyv3+8/15y2v3-95/6xzv3-9714yzv3+2550z2v3-121/80xuv3+67/18yuv3+43/5zuv3+23/124u2v3-12509xv4-104/79yv4-73/21zv4-1238uv4+9038v5,94/107x4y+47/14x3y2-6362x2y3-20/59xy4-43/120y5-3028x4z-15141x3yz-2028x2y2z+84/115xy3z-3024y4z+2811x3z2+47/45x2yz2+121/101xy2z2-100/57y3z2+8/115x2z3+1/101xyz3-13/112y2z3+3618xz4+88/67yz4-52/63z5+102/97x4u-12/89x3yu-102x2y2u-3846xy3u-61/86y4u+85/54x3zu+78/29x2yzu-13381xy2zu-49/95y3zu-77/2x2z2u-5784xyz2u+1557y2z2u-9163xz3u-114/121yz3u-57/103z4u+36/31x3u2-9062x2yu2-23/111xy2u2+7362y3u2-7671x2zu2+14945xyzu2+7901y2zu2+51/5xz2u2-109/48yz2u2+7696z3u2+11280x2u3-44/57xyu3-13736y2u3-13458xzu3-14723yzu3-707z2u3+899xu4-10381yu4+99/25zu4-7788u5-237x4v+45/43x3yv-7666x2y2v-4/109xy3v+4303y4v-13107x3zv-108/91x2yzv-7707xy2zv-73/47y3zv+61/118x2z2v-11/65xyz2v+2970y2z2v-104/37xz3v-15408yz3v-64/55z4v+47/113x3uv+2185x2yuv+7941xy2uv-61/37y3uv+6482x2zuv-11/70xyzuv+83/110y2zuv-109/83xz2uv-86/95yz2uv-7583z3uv+83/45x2u2v+89/38xyu2v-2/11y2u2v+3577xzu2v+124/125yzu2v-1151z2u2v+109/85xu3v+70/13yu3v+37/104zu3v-210u4v+51/29x3v2-104/111x2yv2+105/58xy2v2-13459y3v2-80/79x2zv2-3006xyzv2-115/16y2zv2+8208xz2v2+35/38yz2v2+49/27z3v2-1647x2uv2+10482xyuv2-34/93y2uv2+97/18xzuv2+101/20yzuv2+1711z2uv2+91/36xu2v2-96/23yu2v2+7006zu2v2+86/31u3v2-10734x2v3-43/18xyv3-4597y2v3-11174xzv3-7334yzv3+7/96z2v3+4/97xuv3-5/82yuv3-15600zuv3-69/94u2v3-71/25xv4+21/97yv4+117/23zv4-6557uv4-67/83v5,8164x4y+19/73x3y2-1592x2y3-28/87xy4-63/103x4z+11/42x3yz-52/67x2y2z-13766xy3z+11378y4z+10/37x3z2+115/41x2yz2+11/100xy2z2-49/40y3z2+86/111x2z3+124/5xyz3-25/79y2z3-14525xz4+11380yz4-53/42z5-12169x4u-14/51x3yu+68/33x2y2u-3/62xy3u-31/22y4u-74/93x3zu+12924x2yzu-103/123xy2zu-74/97y3zu-2789x2z2u-95/32xyz2u+45/13y2z2u+40/71xz3u+49/110yz3u+34/75z4u+9829x3u2-59/92x2yu2+106/65xy2u2+123/86y3u2+7133x2zu2-73/46xyzu2-7/29y2zu2-937xz2u2-65/67yz2u2-88/111z3u2-61/119x2u3+975xyu3-54/7y2u3-37/33xzu3+61/59yzu3+51/115z2u3+117/43xu4+8506yu4+13941zu4-14945u5-115/63x4v-14237x3yv-74/87x2y2v+104/47xy3v-95/104y4v+11535x3zv-119/75x2yzv-44xy2zv+11299y3zv-21/113x2z2v-2852xyz2v+95/77y2z2v-75/19xz3v-4864yz3v-79/88z4v+139x3uv-10068x2yuv+2049xy2uv+7515y3uv+97/56x2zuv+109/113xyzuv+7778y2zuv-71/11xz2uv-80/19yz2uv+55/59z3uv-69/98x2u2v-15679xyu2v+114/11y2u2v+69/65xzu2v+879yzu2v+45/104z2u2v+47/97xu3v-1373yu3v+15885zu3v+11121u4v-5042x3v2+4/25x2yv2-8607xy2v2-25/33y3v2+93/55x2zv2+68xyzv2-4167y2zv2+14180xz2v2-115/47yz2v2-81/67z3v2-12099x2uv2+34/107xyuv2+122/59y2uv2+775xzuv2-91yzuv2-85/96z2uv2-59/95xu2v2+174yu2v2+11/16zu2v2+66/37u3v2-121/36x2v3+6070xyv3-83/52y2v3-121/59xzv3-55/12yzv3+8088z2v3-20/29xuv3+76/125yuv3-10858zuv3+1833u2v3-103/50xv4+76/93yv4-119/18zv4+37/114uv4+51/7v5,85/56x4y-7839x3y2+12/37x2y3+6558xy4-8191x4z+115/7x3yz+81/23x2y2z-4121xy3z-1131y4z-23/37x3z2-71/32x2yz2+30/97xy2z2+5070y3z2-49/123x2z3+103/88xyz3-45/19y2z3+5132xz4+7277yz4+1896z5-103/75x4u-12020x3yu+12337x2y2u+6248xy3u+14290y4u-87/44x3zu-5364x2yzu-11801xy2zu-59/37y3zu+34/109x2z2u-14482xyz2u-10338y2z2u+118/73xz3u+7/8yz3u+158z4u+10590x3u2-5182x2yu2+83/62xy2u2+11557y3u2-92/119x2zu2-37/94xyzu2+5383y2zu2-365xz2u2+7/62yz2u2-7965z3u2-10/43x2u3+119/101xyu3-113/83y2u3-121/41xzu3+61/104yzu3+37/60z2u3-74/95xu4-113/66yu4-205zu4+4787u5-94/93x4v+14871x3yv-14723x2y2v+10730xy3v+112/17y4v-35/19x3zv-3487x2yzv-65/43xy2zv-7445y3zv-79/124x2z2v+7423xyz2v+91/2y2z2v+91/34xz3v-6970yz3v-50/113z4v+75/43x3uv-127x2yuv+11978xy2uv+48/113y3uv+113/62x2zuv-8941xyzuv-101/112y2zuv-5737xz2uv-31/123yz2uv+9490z3uv+19/92x2u2v-107/73xyu2v-23/121y2u2v+38/65xzu2v-672yzu2v+13/77z2u2v+46/119xu3v-103/18yu3v+107/59zu3v-52/21u4v-94/87x3v2-74/31x2yv2-9/22xy2v2-2896y3v2+113/3x2zv2-5386xyzv2-11391y2zv2+42/97xz2v2+77/64yz2v2-1610z3v2-102/43x2uv2+124/39xyuv2+14829y2uv2+88/113xzuv2-10411yzuv2-51/43z2uv2-36/121xu2v2+9487yu2v2-5589zu2v2+4335u3v2-5/91x2v3+6084xyv3-56/39y2v3-84/101xzv3-81/85yzv3-6521z2v3-2432xuv3+14317yuv3-43/82zuv3+121/8u2v3+14783xv4-92/45yv4+112/27zv4-8410uv4+31/105v5,-6691x4y-10158x3y2-5372x2y3+4132xy4+106/9y5+15600x4z-803x3yz+43/29x2y2z+9/91xy3z-92/61y4z+4807x3z2-12562x2yz2+14234xy2z2-91/17y3z2-91/30x2z3-10615xyz3-4206y2z3-29/45xz4-11/86yz4-115/9z5+125/112x4u+52/59x3yu+92/49x2y2u+121/85xy3u-51/14y4u-73/48x3zu-1/110x2yzu+12/65xy2zu+15045y3zu+12826x2z2u-123/89xyz2u+9465y2z2u-67/31xz3u-5080yz3u-7944z4u-107/72x3u2+1473x2yu2+7965xy2u2+15753y3u2-95/98x2zu2-9827xyzu2-25/53y2zu2-83/54xz2u2-13217yz2u2-117/110z3u2+230x2u3-12120xyu3+11/36y2u3-2071xzu3+109/59yzu3+6909z2u3-15/64xu4+45/82yu4-3091zu4-15711u5+5957x4v-45/86x3yv+26/29x2y2v-40/57xy3v+25/43y4v+126/37x3zv-38/33x2yzv+65/109xy2zv-33/68y3zv-7287x2z2v-4842xyz2v+35/118y2z2v+6157xz3v-97/89yz3v-91/50z4v-70/27x3uv+32/9x2yuv+78/125xy2uv+38/7y3uv-3214x2zuv-68/101xyzuv+87/55y2zuv-69/98xz2uv+5805yz2uv+41/102z3uv-43/54x2u2v-42/73xyu2v-13/49y2u2v+11864xzu2v+121/37yzu2v-100/109z2u2v-12609xu3v-9114yu3v-8746zu3v+11659u4v+3799x3v2-9581x2yv2+60/91xy2v2+2029y3v2+12075x2zv2+210xyzv2-1/22y2zv2+17/58xz2v2+1212yz2v2+118/27z3v2-3571x2uv2-3139xyuv2-23/100y2uv2-1240xzuv2+71/49yzuv2-21/103z2uv2-110/71xu2v2-40/77yu2v2-103/29zu2v2+10737u3v2+2828x2v3+14/39xyv3+7564y2v3+113/50xzv3+38/79yzv3+59/66z2v3+2726xuv3+91/94yuv3-15730zuv3-13408u2v3-97/42xv4+54/29yv4-33/73zv4+4823uv4+57/71v5,-14556x3yz-9751x2y2z-45/28xy3z+85/23y4z+5623x3z2+5369x2yz2-19/60xy2z2-36/5y3z2-95/36x2z3+5862xyz3-5/93y2z3+2949xz4+11357yz4-5679z5-52/45x3yu+4448x2y2u-9/22xy3u+2427y4u+3296x3zu+16/39x2yzu+53/57xy2zu+15/41y3zu+9473x2z2u+37xyz2u-58/69y2z2u-23/56xz3u-13/90yz3u-54/29z4u-41/67x3u2+10258x2yu2+23/44xy2u2-12952y3u2+2124x2zu2-1677xyzu2+12911y2zu2+22/45xz2u2+17/84yz2u2+5910z3u2+4782x2u3+119/39xyu3-17/84y2u3-120/91xzu3+35/59yzu3+17/77z2u3-4467xu4-77/4yu4-26/53zu4-3580u5-11977x3yv-118/77x2y2v+6040xy3v+9724y4v-47/5x3zv+59/101x2yzv+1212xy2zv-7/121y3zv+93/53x2z2v-56/23xyz2v-4470y2z2v+110/111xz3v-41/99yz3v-81/10z4v-71/24x3uv+26/115x2yuv+59/39xy2uv-10029y3uv+11748x2zuv+5749xyzuv+6887y2zuv+38/3xz2uv-116/61yz2uv-55/118z3uv+105/22x2u2v+70/87xyu2v-28/13y2u2v-109/123xzu2v-102/47yzu2v-52/71z2u2v+101/95xu3v+51/16yu3v+15/97zu3v-78/125u4v+35/46x3v2-9526x2yv2+10781xy2v2-119/44y3v2-23/10x2zv2+59/29xyzv2-15144y2zv2+29/120xz2v2-53/126yz2v2-93/85z3v2+53/8x2uv2-487xyuv2-12143y2uv2+13825xzuv2+55/6yzuv2-4250z2uv2+4237xu2v2-109/9yu2v2+67/53zu2v2+82/33u3v2+8660x2v3+15046xyv3-79/84y2v3-10310xzv3+110yzv3-7636z2v3+57/92xuv3-22/119yuv3-95/103zuv3+5138u2v3+123/49xv4-7587yv4+30/41zv4-124/121uv4+54/71v5,-29/60x4y-108/77x3y2-109/37x2y3-3619xy4+109/6x4z-37/67x3yz+53/45x2y2z+5291xy3z-2927y4z+34/5x3z2+87/17x2yz2+100/89xy2z2-114/29y3z2-4057x2z3-1/42xyz3-14/61y2z3-398xz4-122/73yz4+66/37z5+99/37x4u-5691x3yu-8778x2y2u+17/115xy3u+51/113y4u-71/101x3zu+85/91x2yzu-92/9xy2zu-3442y3zu+109/26x2z2u+50/37xyz2u+77/94y2z2u+16/35xz3u+9985yz3u+5/102z4u-5932x3u2+89/125x2yu2-895xy2u2-12455y3u2-630x2zu2-64/47xyzu2+25/9y2zu2+7906xz2u2+6827yz2u2+9808z3u2-113/118x2u3+79/8xyu3+9484y2u3+62/39xzu3+6/85yzu3-23/49z2u3-93/115xu4-11/93yu4-15177zu4-13/2u5-7623x4v-103/73x3yv-96/115x2y2v+39/76xy3v+80/79y4v+43/68x3zv+45/97x2yzv+101/87xy2zv+4632y3zv-918x2z2v+8248xyz2v-4276y2z2v+8853xz3v-39/61yz3v-121/87z4v+9968x3uv+473x2yuv+117/56xy2uv-19/21y3uv+121/119x2zuv+3/98xyzuv-65/42y2zuv-3723xz2uv+7/34yz2uv-112/87z3uv+103x2u2v+25/41xyu2v-14459y2u2v-56/41xzu2v-59/81yzu2v-109/102z2u2v-87/16xu3v-13011yu3v+49/123zu3v+106/89u4v-61/51x3v2+14107x2yv2+8035xy2v2-8853y3v2+5723x2zv2+123/53xyzv2-9727y2zv2-102/83xz2v2+1111yz2v2-15745z3v2+83/118x2uv2-57/35xyuv2-48/73y2uv2-28/37xzuv2-27/97yzuv2-27/58z2uv2+71/93xu2v2+117/8yu2v2+12344zu2v2-2497u3v2-118/71x2v3-11/19xyv3+21/104y2v3+32/113xzv3+15544yzv3+31/18z2v3+5909xuv3-67/58yuv3+27/35zuv3+115/9u2v3+79/13xv4+6722yv4-37/114zv4-71/124uv4+4657v5,-77/61x4y-88/101x3y2+93/88x2y3-11/70xy4+9806y5+7896x4z-4699x3yz+55/122x2y2z-63/122xy3z-125/74y4z+47/45x3z2+101/17x2yz2+92/47xy2z2+69/82y3z2+12402x2z3+113/98xyz3-101/33y2z3-15376xz4+47/71yz4-73/10z5+65/74x4u-14409x3yu-14478x2y2u+13593xy3u+102/97y4u+39/62x3zu-34/125x2yzu-83/9xy2zu+45/113y3zu+14484x2z2u-15293xyz2u-26/55y2z2u-958xz3u+67/35yz3u-93/19z4u+25/16x3u2+107/52x2yu2-4599xy2u2-86/51y3u2-9885x2zu2-77/47xyzu2+33/65y2zu2+90/109xz2u2-61/26yz2u2+6198z3u2-38/37x2u3-13935xyu3-142y2u3-64/5xzu3-7228yzu3+1251z2u3+1556xu4+117/121yu4-92/35zu4+99/92u5+13493x4v+12654x3yv+32/101x2y2v-11118xy3v+43/51y4v-575x3zv+103/21x2yzv+85/24xy2zv+1788y3zv+85/3x2z2v-64/25xyz2v+57/35y2z2v+37/120xz3v-69/110yz3v+48/49z4v+55/114x3uv-6439x2yuv+31/51xy2uv-90/49y3uv-45/104x2zuv-12018xyzuv+6/119y2zuv+40/63xz2uv+20/91yz2uv+50/43z3uv+1/26x2u2v-109/47xyu2v+99/7y2u2v+72/83xzu2v+61/118yzu2v+3530z2u2v+6146xu3v+117yu3v-9921zu3v-8708u4v-10/47x3v2-15294x2yv2-7336xy2v2+1/66y3v2-3057x2zv2+74/123xyzv2+146y2zv2-103/34xz2v2-117/76yz2v2+8472z3v2-7/92x2uv2+10033xyuv2+43/53y2uv2+4694xzuv2-49/2yzuv2-71/73z2uv2-125/17xu2v2-9817yu2v2+7218zu2v2+6897u3v2-19/90x2v3+11899xyv3-11779y2v3-5456xzv3+17/42yzv3+15340z2v3+12/7xuv3+9580yuv3-502zuv3-14069u2v3-4371xv4+14452yv4-9423zv4-117/122uv4+1126v5,49/108x4-39/4x3y-67/21x2y2-8/69xy3-9779y4+57/14x3z-11145x2yz+6928xy2z-7824y3z+1/79x2z2+5173xyz2-62/15y2z2-123/112xz3+88/79yz3+1/125z4+57/23x3u-11856x2yu-7444xy2u+115/8y3u-11133x2zu+71/73xyzu-7941y2zu+69/65xz2u+22/75yz2u+65/121z3u+9471x2u2+9167xyu2+51/59y2u2+12835xzu2+15047yzu2+11102z2u2-10059xu3+19/28yu3+65/21zu3-39/28u4-3/73x3v+94/61x2yv+8778xy2v-12922y3v-8711x2zv-37/97xyzv+14270y2zv+4487xz2v-59/112yz2v-14183z3v+15553x2uv+3579xyuv+114/91y2uv-4/97xzuv+13/85yzuv-89/15z2uv+58/75xu2v-34/7yu2v-90/61zu2v+90/101u3v-14673x2v2+90/19xyv2-45/37y2v2+23/49xzv2-71/11yzv2+119/8z2v2+89/10xuv2+109/91yuv2+36/49zuv2-7/31u2v2-40/113xv3-21/121yv3+9910zv3+33/14uv3-23/79v4,93/70x4-43/125x3y+9582x2y2+7565xy3-11511y4-3/79x3z-36/107x2yz-2038xy2z+879y3z-4700x2z2+103/14xyz2+102/79y2z2-67/68xz3-44/25yz3+105/79z4-29/24x3u-74/83x2yu+67/43xy2u+49/12y3u-115/11x2zu+23/67xyzu-61/27y2zu+12257xz2u+14068yz2u+23/15z3u+607x2u2+73/8xyu2+14237y2u2-13/33xzu2+110/71yzu2+41/101z2u2+5708xu3+88/67yu3+1460zu3-2472u4-1629x3v-51/70x2yv-88/73xy2v-36/97y3v+38/11x2zv+15899xyzv+54/19y2zv+9460xz2v-5150yz2v+3462z3v+5522x2uv-19/123xyuv+14871y2uv+53/5xzuv-7535yzuv-13430z2uv+107/47xu2v-8307yu2v-55/79zu2v-11945u3v-16/83x2v2+115/48xyv2+12389y2v2+11545xzv2-25/26yzv2-3755z2v2+4724xuv2-31/21yuv2+7872zuv2+89/45u2v2+87/47xv3+7625yv3+13494zv3-15376uv3-25/126v4;TestGRRes(Name, I); kill R, Name, @p; "";

  string Name = "rat.d10.g9.quart2"; int @p=31991; ring R = (@p),(x,y,z,u,v), dp;ideal I = x3yu2-48/11x2y2u2-8356xy3u2+35/121y4u2+31/66x3zu2-54/83x2yzu2-61/18xy2zu2+11526y3zu2+7372x2z2u2-91/60xyz2u2-95/97y2z2u2-45/71xz3u2+71/115yz3u2+25/54z4u2-61/102x3u3-12668x2yu3+6653xy2u3+41/54y3u3+87/50x2zu3-5004xyzu3+13924y2zu3+2310xz2u3-93/14yz2u3-2/93z3u3-97/125x2u4-58/11xyu4+46/73y2u4-4417xzu4+60/101yzu4+56/75z2u4-113/118xu5+115/4yu5-40zu5-8554u6-54/83x3yuv-9770x2y2uv-590xy3uv+15/49y4uv+94/69x3zuv+121/105x2yzuv+95/88xy2zuv+3186y3zuv+11/6x2z2uv-44/81xyz2uv+637y2z2uv+109/121xz3uv-33yz3uv-94/115z4uv-49/95x3u2v-11/109x2yu2v+45/113xy2u2v+97/84y3u2v+5257x2zu2v+99/49xyzu2v+12584y2zu2v-4294xz2u2v+1137yz2u2v-58/69z3u2v-4749x2u3v+120/97xyu3v-31/103y2u3v+62/97xzu3v-107/74yzu3v+53/59z2u3v+91/33xu4v+1291yu4v+23/34zu4v+58/77u5v+16/17x3yv2-750x2y2v2+86/89xy3v2+123/46y4v2+53/123x3zv2-61/99x2yzv2+12389xy2zv2+10419y3zv2+43/11x2z2v2-146xyz2v2-116/51y2z2v2+13/62xz3v2-5524yz3v2-111/118z4v2-56/55x3uv2-3038x2yuv2+14/27xy2uv2-43/64y3uv2+3385x2zuv2+25/11xyzuv2+92/41y2zuv2+28/113xz2uv2-2049yz2uv2+89/37z3uv2-13094x2u2v2-2774xyu2v2+15474y2u2v2-15791xzu2v2-71/116yzu2v2+77/41z2u2v2-83/68xu3v2-33/106yu3v2+71/37zu3v2-41/17u4v2+12052x3v3+1906x2yv3+13825xy2v3+80/7y3v3-125/96x2zv3-9661xyzv3+85/116y2zv3-72/91xz2v3+13/112yz2v3-126/97z3v3-1637x2uv3+34/103xyuv3+3844y2uv3+77/10xzuv3+6359yzuv3-11185z2uv3-124/121xu2v3+66/91yu2v3-14636zu2v3-1051u3v3+9/64x2v4-12924xyv4-119/41y2v4+74/23xzv4+1622yzv4+73/37z2v4-60/101xuv4+111/22yuv4-45/124zuv4+59/37u2v4-66/37xv5-71/99yv5+12409zv5-113/64uv5-5267v6,-x4y-22/79x3y2-125/42x2y3-116/7xy4+98/111y5-31/66x4z-118/75x3yz+110/93x2y2z-43/92xy3z-788y4z-7372x3z2-2701x2yz2-67/124xy2z2-117/62y3z2+45/71x2z3-8396xyz3-10343y2z3-25/54xz4+30/59yz4+61/102x4u+11736x3yu+12726x2y2u+41/118xy3u-15832y4u-87/50x3zu-130x2yzu+41/8xy2zu-10300y3zu-2310x2z2u-101/5xyz2u+6205y2z2u+2/93xz3u+8679yz3u+97/125x3u2-43/37x2yu2-39/80xy2u2+12139y3u2+4417x2zu2+4294xyzu2+11/58y2zu2-56/75xz2u2+8338yz2u2+113/118x2u3-10190xyu3-37/16y2u3+40xzu3+74/23yzu3+8554xu4+115/22yu4-39/79x4v+61/72x3yv+8048x2y2v-9201xy3v+16/121y4v+113/93x3zv+109/75x2yzv+12700xy2zv-10607y3zv+50/11x2z2v+1223xyz2v-103/79y2z2v-123/58xz3v+31/26yz3v-15/122z4v+122/25x3uv-99/17x2yuv+1723xy2uv-38/121y3uv+11016x2zuv-25/102xyzuv-14970y2zuv-61/6xz2uv-14981yz2uv+15900z3uv+3268x2u2v-75/19xyu2v-1436y2u2v-1764xzu2v-57/41yzu2v+12741z2u2v-14615xu3v+119/61yu3v-115/119zu3v+10501u4v-8502x3v2-51/76x2yv2-6281xy2v2+17/49y3v2-106/7x2zv2+63/101xyzv2-27/95y2zv2-1606xz2v2+9245yz2v2+1912z3v2+11155x2uv2+223xyuv2-13/18y2uv2+110/43xzuv2+76/81yzuv2-6291z2uv2+1400xu2v2-95/23yu2v2-9701zu2v2+106/105u3v2+72/47x2v3-13118xyv3+14409y2v3+37/86xzv3+44/69yzv3-325z2v3+113/71xuv3+16/81yuv3+6/19zuv3-119/39u2v3-89/9xv4+72/53yv4+112/55zv4-8587uv4-6604v5,-x3y2+48/11x2y3+8356xy4-35/121y5-12750x3yz+100/111x2y2z+45/74xy3z+99/74y4z-6/7x3z2-47/67x2yz2+11465xy2z2-11865y3z2+7776x2z3+124/45xyz3-98/115y2z3+117/85xz4-59/120yz4-8748z5+61/102x3yu+12668x2y2u-6653xy3u-41/54y4u+13408x3zu-2185x2yzu-1240xy2zu+1161y3zu+44/27x2z2u-11164xyz2u-13388y2z2u-107/13xz3u+90/71yz3u+4204z4u+97/125x2yu2+58/11xy2u2-46/73y3u2+55/48x2zu2+121/31xyzu2+126/61y2zu2-55/69xz2u2+5988yz2u2+3755z3u2+113/118xyu3-115/4y2u3+3390xzu3-5762yzu3+30/61z2u3+8554yu4-14317zu4+99/116x3yv-113/119x2y2v+50/23xy3v-37/79y4v-8668x3zv+14049x2yzv+111/35xy2zv+61/28y3zv-10171x2z2v+68/21xyz2v+2023y2z2v-9/109xz3v+8520yz3v-2683z4v-13547x3uv+28/65x2yuv-5988xy2uv+61/111y3uv+12314x2zuv+29/44xyzuv+6141y2zuv+11280xz2uv+79/22yz2uv-38/111z3uv+19/51x2u2v+5093xyu2v-10291y2u2v-5009xzu2v-111/49yzu2v+3813z2u2v-61/37xu3v+15914yu3v-3218zu3v-12915u4v-118/101x3v2-7/57x2yv2+13128xy2v2+11606y3v2+42/101x2zv2-54/17xyzv2-43/49y2zv2-119/110xz2v2+9742yz2v2-43/4z3v2-55/8x2uv2-29/88xyuv2+12042y2uv2+101/37xzuv2-57/62yzuv2+106/97z2uv2+38/83xu2v2+8152yu2v2-5492zu2v2-47/79u3v2+15112x2v3+69/44xyv3-6/71y2v3+113/54xzv3-13210yzv3-707z2v3-119/8xuv3+3845yuv3-19/20zuv3+4852u2v3+15761xv4-12372yv4+74/69zv4-2100uv4-12833v5,-x3yz+48/11x2y2z+8356xy3z-35/121y4z-31/66x3z2+54/83x2yz2+61/18xy2z2-11526y3z2-7372x2z3+91/60xyz3+95/97y2z3+45/71xz4-71/115yz4-25/54z5+15/52x3yu+6039x2y2u+74/99xy3u-17/40y4u+29/50x3zu-7775x2yzu+6368xy2zu+14170y3zu+52/41x2z2u+7003xyz2u-5787y2z2u-101/37xz3u-23/28yz3u-20/63z4u+41/77x3u2+8650x2yu2-15922xy2u2-16/83y3u2+7278x2zu2+31/30xyzu2-2/107y2zu2+35/122xz2u2+85/58yz2u2-757z3u2+2/101x2u3+86/17xyu3+95/59y2u3+123/22xzu3-6869yzu3-9311z2u3-105/97xu4+5699yu4+15925zu4+13528u5-154x3yv+4187x2y2v+56/107xy3v-15932y4v-5137x3zv-37/56x2yzv+9401xy2zv+92/123y3zv-79/97x2z2v+9201xyz2v+19/53y2z2v+107/20xz3v+17/77yz3v-15306z4v+3215x3uv-79/117x2yuv-9/76xy2uv-6352y3uv+93/13x2zuv-65/89xyzuv-115/4y2zuv-34/57xz2uv+39/107yz2uv+31/9z3uv+107/48x2u2v+2632xyu2v+29/96y2u2v-125/89xzu2v+29/113yzu2v+3940z2u2v-116/111xu3v+6145yu3v-105/62zu3v+101/17u4v-9281x3v2-49/107x2yv2-12154xy2v2+4/19y3v2-114/71x2zv2-15/118xyzv2+4372y2zv2+45/121xz2v2+46/111yz2v2+6614z3v2+17x2uv2+10806xyuv2-10617y2uv2-25/111xzuv2-116/27yzuv2-7/58z2uv2-686xu2v2+3/13yu2v2-17/49zu2v2-40/107u3v2+47/90x2v3-83/43xyv3-6326y2v3+49/64xzv3+113/76yzv3-122/73z2v3+10232xuv3-116/109yuv3-1990zuv3+70/51u2v3-118/19xv4-27/55yv4+21/19zv4-23/57uv4-11721v5,-3399x4y+1849x3y2-3/29x2y3+28/87xy4+10/29y5-9788x4z-49/73x3yz+13829x2y2z+118/73xy3z+13129y4z-618x3z2+92/13x2yz2+101/117xy2z2-162y3z2+24/5x2z3-29/74xyz3+2687y2z3-74/39xz4+2/57yz4+68/73x4u-13787x3yu-11659x2y2u+14729xy3u+92/53y4u+15/71x3zu-62/15x2yzu+21/85xy2zu+4938y3zu-120/37x2z2u-77/102xyz2u-4785y2z2u-83/70xz3u-12128yz3u-13592z4u-123/20x3u2+2607x2yu2+40/19xy2u2+6361y3u2-3091x2zu2+89/113xyzu2+149y2zu2-2890xz2u2-8374yz2u2+11886z3u2-49/43x2u3-9854xyu3-6943y2u3+10743xzu3-122/45yzu3-13902z2u3-103/19xu4-48/59yu4+27/86zu4+46/35u5-117/17x4v-15/7x3yv+8409x2y2v-83/28xy3v+86/35y4v+37/45x3zv+4/3x2yzv+35/38xy2zv+4015y3zv-49/111x2z2v-1260xyz2v-25/33y2z2v+116/19xz3v+93/8yz3v+5755z4v-25/89x3uv-11669x2yuv-64/107xy2uv+2993y3uv+7767x2zuv-17/95xyzuv-103/80y2zuv-14576xz2uv+80/47yz2uv+25/107z3uv+103/2x2u2v+125/117xyu2v-2/89y2u2v-5298xzu2v-50/27yzu2v-71/53z2u2v+2652xu3v+15761yu3v+2124zu3v+11/82u4v+100/63x3v2+4180x2yv2+11/39xy2v2-1221y3v2+108/125x2zv2+97/126xyzv2-7698y2zv2+13984xz2v2+1342yz2v2-84/121z3v2-26/73x2uv2-14/15xyuv2-22/37y2uv2-71/82xzuv2+12430yzuv2+103/52z2uv2-13095xu2v2+10114yu2v2-8/73zu2v2-33/97u3v2+83/105x2v3+22/45xyv3-7961y2v3-9654xzv3-54/55yzv3-3/71z2v3-10148xuv3-117/98yuv3+101/102zuv3-606u2v3+97/43xv4-68/21yv4+63/16zv4+42/17uv4+5834v5,-3399x3y2-32/113x2y3+14/99xy4+15001y5-121/115x3yz+4604x2y2z+7/2xy3z+9532y4z-3267x3z2+97/118x2yz2-14238xy2z2-80/21y3z2-12332x2z3-19/69xyz3+116/15y2z3-103/32xz4+15340yz4+10509z5+112/109x3yu-97x2y2u-40/11xy3u+90/29y4u-95/106x3zu-114/67x2yzu+113/48xy2zu+12080y3zu-44x2z2u+18/17xyz2u-4814y2z2u-103/100xz3u-96/61yz3u-205z4u-87/82x3u2-97/108x2yu2+3230xy2u2+104/83y3u2+41/86x2zu2+116/49xyzu2-59/110y2zu2+14/59xz2u2-6962yz2u2-2185z3u2+59/91x2u3+2497xyu3+3/37y2u3-13010xzu3+6/83yzu3-11448z2u3+13/72xu4-69/62yu4-2869zu4+23/73u5-20/43x3yv+5074x2y2v+28/125xy3v-2706y4v+13010x3zv-17/109x2yzv+21/4xy2zv+59/93y3zv-2406x2z2v+117/11xyz2v-14978y2z2v+70/89xz3v-33/7yz3v-13676z4v-13690x3uv+9825x2yuv-117/107xy2uv+12760y3uv-93/98x2zuv-113/64xyzuv+113/103y2zuv-9748xz2uv+11016yz2uv-10729z3uv+90/13x2u2v-13/47xyu2v-11/39y2u2v+20/69xzu2v+5531yzu2v+125/49z2u2v-11025xu3v-9621yu3v+113/109zu3v+4710u4v-107/7x3v2+110/119x2yv2-10025xy2v2-6644y3v2-5041x2zv2+5/96xyzv2+11472y2zv2-5128xz2v2+2927yz2v2+121/18z3v2-125/89x2uv2+12936xyuv2-71/47y2uv2+34/47xzuv2-75/103yzuv2-2654z2uv2-2350xu2v2-7707yu2v2+47/72zu2v2-952u3v2-21/67x2v3+58/37xyv3-8757y2v3+3615xzv3+44/123yzv3-13027z2v3-9/10xuv3+75/43yuv3+115/18zuv3+8071u2v3-26/3xv4-67/65yv4+14186zv4-41/122uv4+33/28v5,-3399x3yz-32/113x2y2z+14/99xy3z+15001y4z-9788x3z2+37/96x2yz2+7743xy2z2+31/55y3z2-618x2z3-8171xyz3+82/109y2z3+24/5xz4+88/85yz4-74/39z5-13165x3yu+3407x2y2u-12509xy3u-23/45y4u-11774x3zu-10/67x2yzu+69/79xy2zu-10/123y3zu-7636x2z2u+83/32xyz2u+51/112y2z2u+19/8xz3u+9309yz3u-44/49z4u+4089x3u2-374x2yu2-919xy2u2+98/107y3u2+2776x2zu2+85/26xyzu2+31/13y2zu2-103/82xz2u2+35/76yz2u2+59/45z3u2+2950x2u3+27/44xyu3+88/71y2u3+7/114xzu3-72/77yzu3+12917z2u3-34/67xu4-85/82yu4-55/84zu4+4690u5+11/42x3yv-19/125x2y2v-8288xy3v+9199y4v-12929x3zv+13357x2yzv-4903xy2zv-584y3zv-10/33x2z2v+59/113xyz2v+103/92y2z2v+101/69xz3v+8708yz3v-8/7z4v+13560x3uv-43/49x2yuv-121/98xy2uv+75/79y3uv-39x2zuv-88/69xyzuv-89/78y2zuv+110/67xz2uv+61/4yz2uv-98/45z3uv+82/7x2u2v-85/41xyu2v+6548y2u2v+9367xzu2v-59/81yzu2v-14408z2u2v+2363xu3v-80/11yu3v-50/17zu3v-14799u4v-53/21x3v2+9437x2yv2-117/80xy2v2+81/85y3v2-8/45x2zv2-6428xyzv2+15126y2zv2+68/89xz2v2+7/122yz2v2+9639z3v2+113/4x2uv2-8678xyuv2-104/45y2uv2-79/90xzuv2+39/101yzuv2-7234z2uv2-28/43xu2v2+1251yu2v2-97/56zu2v2+17/41u3v2+107/24x2v3+2747xyv3+9933y2v3-4199xzv3+53/83yzv3+6364z2v3-5456xuv3+618yuv3-123/55zuv3+2375u2v3+63/76xv4-115/106yv4-8811zv4-31/75uv4+10/109v5,13/89x4y+77/31x3y2+36/83x2y3-11411xy4+6936y5-12223x4z+7400x3yz+33/118x2y2z-12146xy3z+108/79y4z+82/99x3z2-9877x2yz2-79/70xy2z2-19/123y3z2-1491x2z3+7953xyz3-43/126y2z3+60/17xz4+98/57yz4-13317x4u-77/27x3yu-6811x2y2u-69/61xy3u+6144y4u+5404x3zu+121/120x2yzu-91/23xy2zu-71/106y3zu+1435x2z2u-120/13xyz2u-12019y2z2u-68/7xz3u-113/82yz3u+11526z4u-8706x3u2-89/53x2yu2-14804xy2u2+120/107y3u2+71/94x2zu2-1/70xyzu2+1532y2zu2+4470xz2u2+13/60yz2u2-115/102z3u2-82/21x2u3+27/121xyu3-4439y2u3-101/47xzu3-3186yzu3-106/101z2u3-10169xu4+19/58yu4-96/73zu4-7959u5-10526x4v-107/92x3yv+47/6x2y2v-23/43xy3v-69/62y4v+59/65x3zv-28/95x2yzv+5479xy2zv-39/77y3zv+11/69x2z2v-11713xyz2v+43/79y2z2v-15602xz3v+16/73yz3v-13952z4v+61/82x3uv-2219x2yuv-91/106xy2uv+5/37y3uv-148x2zuv+31/51xyzuv+18/101y2zuv+97/68xz2uv-73/32yz2uv+47/2z3uv+2/41x2u2v-13009xyu2v-7/60y2u2v+15779xzu2v+72/7yzu2v-11/73z2u2v-119/44xu3v-9067yu3v+3249zu3v+61/51u4v+12525x3v2-118/9x2yv2-3270xy2v2-4/25y3v2-5075x2zv2+77/40xyzv2-89/65y2zv2+17/58xz2v2-15609yz2v2+95/54z3v2-75/79x2uv2-4907xyuv2+12418y2uv2-57/17xzuv2-8746yzuv2+13/95z2uv2-124/67xu2v2+16/13yu2v2+28/23zu2v2-10847u3v2-645x2v3+106/75xyv3+6/115y2v3-8495xzv3+58/35yzv3-9398z2v3-101/72xuv3-71/20yuv3-124/65zuv3-8971u2v3+27/28xv4+12/29yv4-4276zv4+10858uv4+29/12v5,13/89x3y2+12068x2y3-15543xy4-77/79y5+6626x3yz+64/53x2y2z-6/23xy3z-47/125y4z+14403x3z2-43/78x2yz2-31/115xy2z2+94/59y3z2-118/117x2z3-11229xyz3+2268y2z3-116/85xz4+25/58yz4+3085z5+59/27x3yu+67/82x2y2u+11/6xy3u+103/47y4u-63/80x3zu-81/47x2yzu+7760xy2zu-115/56y3zu-10/17x2z2u+101/5xyz2u+15634y2z2u+1/107xz3u-9282yz3u+43/62z4u+62/55x3u2+100/113x2yu2-9205xy2u2-46/13y3u2+43/96x2zu2+10159xyzu2+692y2zu2+859xz2u2-19/74yz2u2+123/47z3u2-9/20x2u3-11391xyu3-2375y2u3+109/24xzu3-57/53yzu3-925z2u3-82/45xu4+97/34yu4+13/82zu4-108/29u5+63/10x3yv+38/17x2y2v-19/115xy3v+3150y4v+22/69x3zv+26/57x2yzv+110/27xy2zv+87/77y3zv+85/18x2z2v+39/47xyz2v-48/17y2z2v-7/27xz3v-13/100yz3v-11662z4v-17/8x3uv+37/11x2yuv+29/11xy2uv-109/88y3uv-2817x2zuv-61/44xyzuv+10/31y2zuv+10010xz2uv+51/86yz2uv-97/83z3uv-89/96x2u2v+4030xyu2v-58/77y2u2v-114/43xzu2v-37/10yzu2v-2011z2u2v+14483xu3v-109/101yu3v+121/102zu3v-79/92u4v+15113x3v2+10781x2yv2-14259xy2v2-113/48y3v2-7/94x2zv2-17/74xyzv2-5/117y2zv2-59/75xz2v2+13188yz2v2+103/43z3v2+4/125x2uv2-52/59xyuv2+85/92y2uv2-1/46xzuv2-9106yzuv2-83/11z2uv2-23/94xu2v2+6742yu2v2-35/107zu2v2-14596u3v2-117/43x2v3+1026xyv3+90/19y2v3+14671xzv3-101/100yzv3+6962z2v3+61/68xuv3+108/37yuv3-4157zuv3-3974u2v3+15677xv4+8661yv4+8459zv4-16/23uv4-37/119v5,13/89x3yz+12068x2y2z-15543xy3z-77/79y4z-12223x3z2-13941x2yz2+115/84xy2z2+13/98y3z2+82/99x2z3+7751xyz3+122/17y2z3-1491xz4+1327yz4+60/17z5+15363x3yu+9780x2y2u+19/117xy3u-1924y4u-14600x3zu+46/41x2yzu-5466xy2zu-73/12y3zu+10838x2z2u-8302xyz2u-89/113y2z2u+53/69xz3u-9224yz3u+47/33z4u-7399x3u2+89/77x2yu2+9312xy2u2-41/80y3u2-732x2zu2-6781xyzu2-8608y2zu2-9270xz2u2-117/58yz2u2-115/68z3u2-48/31x2u3-9067xyu3+97/107y2u3+73/57xzu3-2719yzu3-110/59z2u3-37/86xu4-15796yu4-61/4zu4-115/72u5+6161x3yv+4134x2y2v+677xy3v-8375y4v+1150x3zv+1551x2yzv+4157xy2zv+112/87y3zv+8171x2z2v+6040xyz2v+15651y2z2v-7/66xz3v-47/61yz3v+77/64z4v+14848x3uv+48/119x2yuv-9534xy2uv-117/95y3uv+5/4x2zuv+122xyzuv+90/31y2zuv-41/26xz2uv+31/30yz2uv-10428z3uv-9896x2u2v-71/21xyu2v-55/38y2u2v-29/22xzu2v-11092yzu2v+39/122z2u2v+93/73xu3v+22/49yu3v-21/106zu3v+56u4v+8565x3v2-1695x2yv2+2/17xy2v2+1/78y3v2-113/71x2zv2-41/100xyzv2+55/14y2zv2+15286xz2v2+17/53yz2v2+126/71z3v2-79/87x2uv2+109/97xyuv2-28/31y2uv2-6533xzuv2+22/5yzuv2-10449z2uv2+10830xu2v2-15516yu2v2+28/57zu2v2-81/22u3v2+4198x2v3+5667xyv3-7133y2v3-8408xzv3+11066yzv3-26/125z2v3-808xuv3+95/54yuv3-64/17zuv3-5267u2v3-15333xv4+42/89yv4+63/85zv4+119/113uv4-2011v5,5583x4y+1725x3y2-8652x2y3-91/25xy4-8495x4z-13731x3yz+9298x2y2z-41/111xy3z-15503y4z-13805x3z2+3962x2yz2-2/63xy2z2+3314y3z2+2522x2z3-10/87xyz3-408y2z3+7/16xz4+69/22yz4-7254z5-59/21x4u+115/7x3yu-1718x2y2u+7851xy3u+2632y4u-82/3x3zu+37/86x2yzu+101/113xy2zu+6747y3zu-109/113x2z2u+7399xyz2u+24/103y2z2u+89/9xz3u-14630yz3u+15066z4u-12561x3u2+113/115x2yu2+87/97xy2u2-126/67y3u2-48/7x2zu2+123/103xyzu2-11/107y2zu2-2747xz2u2+8158yz2u2-3/107z3u2+41/6x2u3+12767xyu3+3873y2u3+74/83xzu3-55/119yzu3-24/83z2u3+55xu4-7/95yu4+57/44zu4+2/101u5-6928x4v-121/57x3yv+111/104x2y2v+946xy3v-29y4v+3057x3zv-14/25x2yzv+43/31xy2zv-105/2y3zv+2336x2z2v+61/77xyz2v-7880y2z2v+5/58xz3v+10593yz3v+7094z4v+63/59x3uv-5/69x2yuv-11/81xy2uv-4157y3uv+73/65x2zuv-1676xyzuv-2376y2zuv-85/63xz2uv-95/2yz2uv-14903z3uv-119/110x2u2v-115/24xyu2v+125/9y2u2v+106/87xzu2v-13/12yzu2v-4/19z2u2v+7838xu3v-43/111yu3v+7/113zu3v-12500u4v+7743x3v2-2023x2yv2-85/83xy2v2+49/41y3v2+20/87x2zv2+3932xyzv2-77/6y2zv2+47/90xz2v2-15580yz2v2+39/4z3v2-61/8x2uv2+2518xyuv2+29/98y2uv2+11057xzuv2-18/107yzuv2+708z2uv2+14720xu2v2-3175yu2v2-113/59zu2v2-14735u3v2+7/69x2v3-4029xyv3+54/91y2v3+12372xzv3+67/2yzv3+8856z2v3-2178xuv3+995yuv3+64/95zuv3+4039u2v3-37/44xv4+23/17yv4-3035zv4-103/124uv4+69/64v5,-5583x3y2-1725x2y3+8652xy4+91/25y5+6201x3yz-73/49x2y2z-3844xy3z+10548y4z-11057x3z2-105/122x2yz2+31/53xy2z2+79/89y3z2-24/101x2z3+107/119xyz3-126y2z3+8164xz4+2/77yz4-51/8z5-14941x3yu-106x2y2u+8695xy3u+125/62y4u+4328x3zu+29/117x2yzu-6249xy2zu-2791y3zu+67/49x2z2u-38/29xyz2u+122/41y2z2u+10603xz3u-3029yz3u+5578z4u+14754x3u2-108/79x2yu2+4408xy2u2-12401y3u2-1426x2zu2-1741xyzu2-83/86y2zu2+79/95xz2u2+122/121yz2u2+81/2z3u2-1172x2u3-41/68xyu3-70/3y2u3+24/107xzu3+120/79yzu3+18/119z2u3-65/122xu4+1018yu4+22/107zu4+15189u5+5/8x3yv-12060x2y2v+3/62xy3v-227y4v+60/41x3zv-123/115x2yzv+110/123xy2zv+12864y3zv-86/121x2z2v-69/94xyz2v+14/79y2z2v+118/45xz3v+10842yz3v-37/58z4v+100/69x3uv-47/65x2yuv-7/67xy2uv-93/100y3uv-6262x2zuv-4/75xyzuv+2082y2zuv-9117xz2uv+12450yz2uv-84/67z3uv+123/26x2u2v-51/89xyu2v+19/74y2u2v-104/77xzu2v+318yzu2v+12402z2u2v+95/8xu3v-81/26yu3v-4486zu3v+3872u4v+72/91x3v2-83/63x2yv2+93/92xy2v2-15924y3v2-53/62x2zv2+6046xyzv2+1408y2zv2+60/107xz2v2-1150yz2v2-126/19z3v2-7429x2uv2+2554xyuv2+3602y2uv2+10738xzuv2-57/64yzuv2+86/69z2uv2+8172xu2v2+91/113yu2v2+92/65zu2v2+118/37u3v2+47/83x2v3+12750xyv3+10851y2v3+4216xzv3+6/101yzv3-108z2v3+2920xuv3-101/102yuv3-157zuv3+7742u2v3-7234xv4-2/111yv4+59/33zv4-93/91uv4+24/19v5,1592x4y+75/121x3y2+40/19x2y3-2651xy4+9934x4z+245x3yz+11665x2y2z+30/41xy3z+1823y4z+89/88x3z2-105/46x2yz2+79/58xy2z2-4191y3z2-76/61x2z3-21/32xyz3-9516y2z3-14896xz4-85/77yz4+51/109z5+61/30x4u-10/101x3yu+11796x2y2u+76/101xy3u+123/88y4u-5932x3zu-11857x2yzu+7128xy2zu-45/79y3zu+119/18x2z2u+9/74xyz2u+7042y2z2u-1114xz3u-11/82yz3u-1466z4u-6/85x3u2+27/106x2yu2+14246xy2u2-6216y3u2+47/6x2zu2-45/59xyzu2+89/41y2zu2+41/80xz2u2-7583yz2u2-75/113z3u2-14808x2u3-10873xyu3-90/67y2u3-11081xzu3-7369yzu3-7131z2u3-1402xu4-15386yu4-108/73zu4-5039u5+120/113x4v+10617x3yv-50/87x2y2v-2395xy3v-20/69y4v-8587x3zv+12960x2yzv-41/50xy2zv-13844y3zv-65/32x2z2v-77/122xyz2v-85/66y2z2v+13/100xz3v-20/51yz3v-13676z4v+76/97x3uv+1046x2yuv-8059xy2uv-117/59y3uv-29/105x2zuv+7287xyzuv-107/119y2zuv-35/118xz2uv+79/86yz2uv-2211z3uv+5448x2u2v+62/35xyu2v-2275y2u2v+29/121xzu2v-1674yzu2v-56/43z2u2v-3377xu3v-43/110yu3v+23/10zu3v-24/61u4v+121/53x3v2-4745x2yv2-57/64xy2v2+9554y3v2-12741x2zv2+10449xyzv2+37/108y2zv2+8621xz2v2-11/57yz2v2+1566z3v2+125/49x2uv2-121/118xyuv2+109/84y2uv2-335xzuv2+10167yzuv2-59/109z2uv2-103/119xu2v2+43/13yu2v2-73/87zu2v2+2037u3v2+13002x2v3+83/48xyv3-10713y2v3+1026xzv3-105/64yzv3-37/6z2v3+14779xuv3-6448yuv3+19/69zuv3-1/110u2v3+10010xv4+79/12yv4+12/19zv4-35/61uv4-11/57v5,-1592x3y2-75/121x2y3-40/19xy4+2651y5+39/121x3yz+122/77x2y2z-114/31xy3z+1544y4z+2/3x3z2-10271x2yz2-8373xy2z2+56/61y3z2+55/48x2z3-116xyz3-25/7y2z3-108/113xz4-34/53yz4+5548z5-122x3yu-9690x2y2u+43/87xy3u-5/19y4u+97/54x3zu-17/19x2yzu+4355xy2zu+12/5y3zu-1/100x2z2u+12754xyz2u+13600y2z2u+17/45xz3u-12091yz3u+5145z4u-63/64x3u2-84/31x2yu2-97/41xy2u2+7/13y3u2-79/62x2zu2-80/103xyzu2-69/14y2zu2+119/4xz2u2-35/87yz2u2-13840z3u2+14101x2u3+7952xyu3-1857y2u3-9861xzu3+3180yzu3+75/107z2u3-250xu4-15134yu4+4717zu4-2/41u5+22/27x3yv-8983x2y2v+10520xy3v-113/2y4v+10/73x3zv-1986x2yzv-110/13xy2zv+1550y3zv+32/111x2z2v-111/35xyz2v+101/98y2z2v+8045xz3v-2/89yz3v+2924z4v-79/11x3uv-15178x2yuv+10874xy2uv+54/11y3uv-8950x2zuv+70/53xyzuv-2403y2zuv-8249xz2uv+6935yz2uv+20/89z3uv+885x2u2v-76/71xyu2v-4/17y2u2v-31/52xzu2v-4/99yzu2v+10333z2u2v-93/104xu3v+82/101yu3v-71/37zu3v+9397u4v-15/112x3v2-6614x2yv2+119/2xy2v2+88/119y3v2+306x2zv2+2790xyzv2+10992y2zv2-115/74xz2v2-14711yz2v2+11612z3v2-1788x2uv2-75/97xyuv2+79/30y2uv2+99/59xzuv2-11439yzuv2-121/113z2uv2+108/37xu2v2+37/36yu2v2-3/65zu2v2-55/42u3v2+13/100x2v3-209xyv3-1272y2v3-117/68xzv3+63/94yzv3+32/59z2v3+1013xuv3-3463yuv3+6946zuv3-37/86u2v3+67/117xv4+85/28yv4-3024zv4-82/9uv4-32/65v5,-35/52x4y-12140x3y2+23/83x2y3+69/5xy4-80/79y5+120/43x4z-11865x3yz-3487x2y2z+53/59xy3z+53/102y4z-14083x3z2-14430x2yz2-2442xy2z2-33/104y3z2-91/38x2z3+4/87xyz3-26/57y2z3+4097xz4-9/122yz4+6364z5+9634x4u-97/95x3yu-46/99x2y2u+3847xy3u+121/106y4u+12765x3zu-5292x2yzu+1607xy2zu-67/121y3zu-12/35x2z2u+4/55xyz2u-17/27y2z2u+91/122xz3u-23/31yz3u+65/49z4u+73/46x3u2-124/27x2yu2-9933xy2u2+46/75y3u2+53/114x2zu2+3503xyzu2-14147y2zu2-11283xz2u2+11889yz2u2+99/104z3u2+3117x2u3+12624xyu3-10060y2u3+2193xzu3-80/47yzu3-77/13z2u3+11/31xu4-47/90yu4+49/48zu4-2/105u5-92/61x4v+7443x3yv+35/76x2y2v+114/67xy3v-73/126y4v+97/107x3zv+9464x2yzv+10869xy2zv+15718y3zv-37/33x2z2v+124/13xyz2v-11/26y2z2v-61/40xz3v+91/100yz3v-18/103z4v+60/29x3uv+21/125x2yuv-11117xy2uv+11748y3uv-16/117x2zuv+18/103xyzuv-1711y2zuv+1872xz2uv-109/123yz2uv-18/113z3uv-26/103x2u2v+14140xyu2v+11065y2u2v+8686xzu2v-5/111yzu2v+30/101z2u2v-10501xu3v-36/113yu3v-73/74zu3v+12753u4v-43/52x3v2-76/15x2yv2-5793xy2v2+18/13y3v2+1/79x2zv2+84/23xyzv2-172y2zv2+86/77xz2v2+15/37yz2v2+11835z3v2-6482x2uv2+94/113xyuv2+10727y2uv2-102/41xzuv2+15914yzuv2-12973z2uv2-9038xu2v2-13107yu2v2+1533zu2v2+12549u3v2-13528x2v3+903xyv3+23/114y2v3-123/64xzv3-81/5yzv3+111/103z2v3+4734xuv3-33/20yuv3-7954zuv3-2478u2v3+15518xv4-6723yv4-14/31zv4-3482uv4+10919v5,-3/94x4y-12936x3y2+2/11x2y3+32/23xy4-15921y5+61/93x4z+82/111x3yz-93/2x2y2z-6659xy3z-97/90y4z+402x3z2-14586x2yz2-121/39xy2z2+68/7y3z2+1212x2z3-2980xyz3+49/52y2z3-72/89xz4+92/47yz4+8478z5+2733x4u-103/89x3yu+1166x2y2u-7/53xy3u-106/23y4u+677x3zu+907x2yzu+7891xy2zu-9014y3zu+76/47x2z2u+49/116xyz2u-49/78y2z2u+12261xz3u+118/105yz3u-126/13z4u-8812x3u2-97/120x2yu2-9534xy2u2+92/5y3u2-54/71x2zu2+94/103xyzu2+2256y2zu2+4182xz2u2-5798yz2u2-31/115z3u2-73/98x2u3+15822xyu3+1004y2u3-578xzu3+9494yzu3-6779z2u3+14506xu4+10/121yu4+58/27zu4-2817u5-19/119x4v+7128x3yv+75/64x2y2v-65/109xy3v+5129y4v-53/55x3zv+54/125x2yzv-3009xy2zv+6144y3zv+15601x2z2v+123/55xyz2v-58/77y2z2v-56/61xz3v+121/10yz3v-103/86z4v-93/25x3uv+94/123x2yuv-25/107xy2uv+14807y3uv+65/7x2zuv+87/44xyzuv+6605y2zuv+23/99xz2uv-413yz2uv-17/15z3uv-79/46x2u2v+15240xyu2v-42/67y2u2v+8932xzu2v-5888yzu2v-4204z2u2v+7002xu3v-36/97yu3v-1634zu3v+61/102u4v-14/33x3v2-6520x2yv2+9004xy2v2-67/36y3v2-7/8x2zv2-24/11xyzv2-9373y2zv2+1556xz2v2-79/74yz2v2-6691z3v2+108x2uv2-76/61xyuv2+220y2uv2-1191xzuv2-4/9yzuv2+4546z2uv2+12205xu2v2+9/22yu2v2+64/93zu2v2-44/125u3v2+292x2v3+41/74xyv3+16/79y2v3-15892xzv3+5733yzv3+6796z2v3-42/55xuv3+71/79yuv3-19/104zuv3-38/15u2v3+6436xv4+28/15yv4+87/55zv4+2270uv4-30/41v5,-117/4x3y+97/122x2y2-3618xy3+6566y4+97/113x3z-12634x2yz+9865xy2z-1764y3z+114/31x2z2+5006xyz2+7/44y2z2-15040xz3+8/125yz3+11134z4-12980x3u-79/41x2yu-79/98xy2u+89/65y3u-1217x2zu+89/87xyzu+83/66y2zu+115/11xz2u+123/107yz2u+10920z3u-86/73x2u2-11/94xyu2-14054y2u2+6752xzu2-123/124yzu2+12129z2u2-13310xu3-52/63yu3+12847zu3-1545u4-11064x3v+11499x2yv-37/64xy2v+50/103y3v+123/94x2zv-126xyzv-111/44y2zv+95/14xz2v+113/83yz2v-77/103z3v+41/64x2uv+91/90xyuv-4932y2uv+103/31xzuv+62/63yzuv+1161z2uv-99/106xu2v-3181yu2v-11741zu2v-33/8u3v-3/118x2v2-9369xyv2+527y2v2-113/39xzv2-88/49yzv2-113/101z2v2+95/68xuv2-5930yuv2-20/43zuv2+7/41u2v2+109/93xv3-107/61yv3-8352zv3-5255uv3+12021v4,-2159x4-94/3x3y-4602x2y2+1609xy3+10721y4+28/9x3z-99/35x2yz+1/110xy2z+113/114y3z-118/75x2z2-103/93xyz2-68/67y2z2+13687xz3-1531yz3+61/107z4+6076x3u+9004x2yu+2211xy2u+110/53y3u+47/102x2zu+8495xyzu-9238y2zu+57/121xz2u-8543yz2u+8/19z3u-13527x2u2-13293xyu2+1138y2u2+26/115xzu2+78/53yzu2-12556z2u2+7299xu3+70/19yu3-14687zu3+13559u4+113/9x3v-85/126x2yv-83/3xy2v-3/46y3v+1814x2zv+28/79xyzv+103/51y2zv+78/31xz2v-14387yz2v+1/88z3v+116/75x2uv-101/59xyuv-70/3y2uv+109/71xzuv+13/88yzuv-147z2uv-113/76xu2v-9661yu2v+13855zu2v-6162u3v-1857x2v2-8208xyv2-4634y2v2-6178xzv2-7352yzv2-8247z2v2-113/15xuv2+99/40yuv2+21/97zuv2+11/37u2v2-6605xv3+8964yv3+35/121zv3+8543uv3-6008v4;TestGRRes(Name, I); kill R, Name, @p; "";

//  if( defined(assumeLevel0) ){ assumeLevel = assumeLevel0; } else { kill assumeLevel; } // restore the state of aL
}

/////////////////////////////////////////////////////////

proc grzero()
"USAGE:  grzero()
RETURN:  graded object representing S(0)^1
PURPOSE: compute presentation of S(0)^1
EXAMPLE: example grzero; shows an example
"
{
 return ( grobj(freemodule(0), intvec(0:0), intvec(0:0)) );
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;


  grview( grobj(freemodule(0), intvec(0:0), intvec(0:0)) );
  grview( grobj(freemodule(0), intvec(0:0)) );

  grview( grzero() );

//  def M = grpower( grshift( grzero(), 3), 2 ); grview(M);
}

proc grtwists(intvec v)
"USAGE:  grtwists(v), intvec v
RETURN:  graded object representing S(v[1]) + ... + S(v[size(v)])
PURPOSE: compute presentation of S(v[1]) + ... + S(v[size(v)])
EXAMPLE: example grtwists; shows an example
"
{
  matrix m[size(v)][0];
  return( grobj(m, -v) ); // will set the rank as well
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  grview( grtwists ( intvec(-4, 1, 6 )) );

  grview( grtwists ( intvec(0:0) ) );
}

proc grtwist(int a, int d)
"USAGE:  grtwist(a,d), int a, d
RETURN:  graded object representing S(d)^a
PURPOSE: compute presentation of S(d)^a
EXAMPLE: example grtwist; shows an example
"
{
  ASSUME(0, a > 0);
  def Z = grtwists( intvec(d:a) ); // will set the rank as well
//  ASSUME(2, grisequal(Z, grpower( grshift(grzero(), d), a ) )); // optional check
  return(Z);
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

//  grview(grpower( grshift(grzero(), 10), 5 ) );

  grview( grtwist (5, 10) );
}

proc grpower(def A, int p)
"USAGE:  grpower(A, p), graded object A, int p > 0
RETURN:  graded direct power A^p
PURPOSE: compute the graded direct power A^p
NOTE:    the power p must be positive
EXAMPLE: example grpower; shows an example
"
{
  if(p==0){ ERROR("Sorry, we don't know what is A^0!?!?"); } // grzero!?

  ASSUME(0, p > 0);
  ASSUME(1, grtest(A) );

  if(p==1){ return(A); }

  def N = grsum(A,A);

  if(p==2){ return(N); }

  // TODO: replace recursion with a loop!
  // see http://en.wikipedia.org/wiki/Exponentiation_by_squaring
  if((p%2)==0)
    { return ( grpower(N, p div 2) ); }
  else
    { return ( grsum( A, grpower(N, (p-1) div 2) )); }
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module A = grobj( module([x+y, x, 0], [0, x+y, y]), intvec(1,1,1) );
  grview(A);

  module B = grobj( module([x,y]), intvec(2,2) );
  grview(B);

  module D = grsum( grpower(A,2), grpower(B,2) );

  print(D);
  homog(D);
  grview(D);
}


proc grsum(A,B)
"USAGE:  grsum(A, B), graded objects A and B
RETURN:  graded direct sum of input objects
PURPOSE: compute the graded direct sum of A and B
EXAMPLE: example grsum; shows an example
"
{
  ASSUME(1, grtest(A) );
  ASSUME(1, grtest(B) );

  intvec a = grrange(A);
  intvec b = grrange(B);
  intvec c = a,b;

  if( (ncols(B)>0) && (size(B)>0) )
  {
    int r = nrows(A);
    module T = align(module(B), r); //  T;  print(T);  nrows(T); // BUG!!!!
    module S = module(A), T;
  }
  else { def S = A; }

  intvec da = grdeg(A);
  intvec db = grdeg(B);
  intvec dc = da, db;


  def SS = grobj(S, c, dc);

  ASSUME(0, size( grrange(SS) ) == (size(a) + size(b)) );
  ASSUME(0, size( grdeg(SS) ) == (size(da) + size(db)) );
  ASSUME(0, ncols( SS ) == size( grdeg(SS) ) );
  ASSUME(0, nrows( SS ) == size( grrange(SS) ) );

  return(SS);
}
example
{ "EXAMPLE:"; echo = 2;

//  if( defined(assumeLevel) ){ int assumeLevel0 = assumeLevel; } else { int assumeLevel; export(assumeLevel); }; assumeLevel = 5;

  ring r=32003,(x,y,z),dp;

  module A = grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0,0,1) );
  grview(A);

  module B = grobj( module([0,x,y]), intvec(15,1,1) );
  grview(B);

  module C = grsum(A,B);

  print(C);
  homog(C);
  grview(C);

  module D = grsum(
     grsum(grpower(A,2), grtwist(1,1)),
     grsum(grtwist(1,2), grpower(B,2))
     );

  print(D);
  homog(D);
  grview(D);

  module F = grobj( module([x,y,0]), intvec(1,1,5) );
  grview(F);

  module T = grsum( F, grsum( grtwist(1, 10), B ) );
  grview(T);

//  if( defined(assumeLevel0) ){ assumeLevel = assumeLevel0; } else { kill assumeLevel; } // restore the state of aL
}

proc grshift( def M, int d)
"USAGE:  grshift(A, d), graded objects A, int d
RETURN:  shifted graded object
PURPOSE: shift the grading on A by d: A(i) -> A(i+d)
EXAMPLE: example grshift; shows an example
"
{
  ASSUME(1, grtest(M) );



  intvec a = grrange(M);
  intvec t = grdeg(M);

  if( size(a) == 0 && size(t) == 0 )
  {
    "!! Warning: shifting '0 <- 0' leaves it as it unchanged!";
    return (M);
  }

  a = a - intvec(d:size(a));
  t = t - intvec(d:size(t));

  return (grobj(M, a, t));
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module A = grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0,0,1) );

  grview(A);

  module S = grshift( A,  6);

  grview(S);

  grview( grshift( grzero(), 100 ) ); // does nothing...
}


proc grisequal (def A, def B)
"USAGE:  grisequal(A, B), graded objects A and B
RETURN:  1 if A == B as graded objects, 0 otherwise
PURPOSE: test the equality of two graded object
NOTE: A and B should be literarly the same at the moment. TODO?
EXAMPLE: example grisequal; shows an example
"
{
  ASSUME(1, grtest(A) );
  ASSUME(1, grtest(B) );

  int ra = nrows(A);
  int rb = nrows(B);

  intvec wa = grrange(A);
  intvec wb = grrange(B);

  if( (ra != rb) || (ncols(A) != ncols(B)) ){ return (0); } // TODO: ???

  intvec da = grdeg(A);
  intvec db = grdeg(B);

  return ( (da == db) && (wa == wb) &&
           (size(module(matrix(A) - matrix(B))) == 0)    ); // TODO: ???
}
example
{ "EXAMPLE:"; echo = 2;
//  TODO
}

proc grobj(def A, intvec w, list #)
"USAGE:  grobj(M, w[, d]), matrix/ideal/module M, intvec w, d
RETURN:  graded object with matrix presentation M, row weighting w [and total graded degrees d of columns]
PURPOSE: create a valid graded object with a given matrix presentation, weighting [and total graded degrees (in case of zero columns)]
EXAMPLE: example grobj; shows an example
"
{
  ASSUME(0, size(w) >= nrows(A) );

  module M = module(A);

  attrib( M, "rank", size(w) );
  attrib( M, "isHomog", w );

  intvec @ww = 0:0;

  if( size(#) > 0 )
  {
    ASSUME(0, typeof(#[1]) == "intvec" );

    @ww = intvec( #[1] );

    if( size(@ww) != ncols(M) )
    {
      if( (size(M) == 0) && (ncols(M) <= 1) && (size(w) == 0) && (size(@ww) > 0) )
      {
        matrix m[size(w)][size(@ww)]; M = module(m); attrib( M, "isHomog", w );
      }
    }

    ASSUME(0, size(@ww) == ncols(M) );
  }
  else
  {
    if( size(M) == ncols(M) ) /* no zero cols? */
    {
      @ww = grdeg(M); // let us compute them all :)
    } else
    {
      if( (ncols(M) == 1) && (size(M) == 0) )
      {
        M = freemodule(0);
      }

      attrib( M, "rank", size(w) );
      attrib( M, "isHomog", w );

//      ASSUME(0, /* PROBLEM WITH ZERO COLUMNS / THEIR DEGREES! */ (ncols(M) == 0) );
    }
  }

//  type(@ww);  type(M);

  ASSUME(0, size(@ww) == ncols(M) ); // ?!

  attrib(M, "degHomog", @ww);

  ASSUME(0, grtest(M) );
  return (M);
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  def A = grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0,0,1) );
  grview(A);

  def F = grobj( module([x,y,0]), intvec(1,1,5) );
  grview(F);

  int d = 666; // zero can have any degree...
  def Z = grobj( module([x,0], [0,0,0], [0, y]), intvec(1,2,3), intvec(2, d, 3) );
  grview(Z);

  print(Z);
  attrib(Z);
  grrange(Z); // module weights
  attrib(Z, "degHomog"); // total degrees

  // Zero object:
  matrix z[3][0];  grview( grobj( z, intvec(1,2,3) ) );
  grview( grobj( freemodule(0), intvec(1,2,3) ) );

  matrix z1[0][3]; grview( grobj( z1, 0:0, intvec(1,2,3) ) );
  grview( grobj( freemodule(0), 0:0, intvec(1,2,3) ) );

  matrix z0[0][0]; grview( grobj( z0, 0:0 ) );
  grview( grobj( freemodule(0), 0:0 ) );



}

proc grtest(def N, list #)
"USAGE:  grtest(M[,b]), anyting M, optionally int b
RETURN:  1 if M is a valid graded object, 0 otherwise
PURPOSE: validate a graded object. Print an invalid object message if b is not given
NOTE: M should be an ideal or module or matrix, with weighting attribute
   'isHomog' and optionally total graded degrees attribute 'degHomog'.
   Attributes should be compatible with the presentation matrix.
EXAMPLE: example grtest; shows an example
"
{
  int b = (size(#) == 0);
  string t = typeof(N);
  if( (t != "ideal") && (t != "module") && (t != "matrix") )
  {
    if(b) { "   ? grtest: Input should be something like a matrix!"; };
    return (0);
  };

  if ( typeof(attrib(N,"isHomog")) != "intvec" )
  {
    if(b) { type(N); attrib(N); "   ? grtest: Input must be graded!";  };
    return (0);
  };

  intvec gr = grrange(N); // grading weights...
  if ( nrows(N) != size(gr) )
  {
    if(b) { "   ? grtest: Input has wrong number of rows!"; };
    return (0);
  };

  if( ncols(N) == 0 ) // zero-column matrix?
  {
    return(1);
  }

//  if( attrib(N, "rank") != size(gr) ){ return (0); } // wrong rank :(

  if ( typeof(attrib(N, "degHomog")) == "intvec" )
  {
    intvec T = attrib(N, "degHomog"); // graded degrees

    if( (ncols(N) == 1) && (size(T) == 0) && (size(N) == 0) )
    {
      //  if(b) { "Input seems to be a valid graded ZERO-arrow!"; };
      return (1);
    }

    if ( ncols(N) != size(T) )
    {
      if(b) { "   ? grtest: Input has wrong number of cols!"; };
      return (0);
    };

    int k = nvars(basering) + 1; // index of mod. column in the leadexp

    module L = lead(module(N)); vector v;

    // checking T for non-zero N[i]
    int i = size(T);

    for (; i > 0; i-- )
    {
      v = L[i];
      if( v != 0 )
      {
        if( (deg(v) + gr[ leadexp(v)[k] ]) != T[i] )
        {
          if(b) { "   ? grtest: Input has wrong total grade of " + string(i) + "-th column!"; };
          return (0);
        };  // wrong T[i]
      }
    }

    // TODO: check t on nonzero cols...
  } else
  {
    if( ncols(N) != size(N) )
    {
      if(b) { "   ? grtest: Input should have exclusively non-zero columns, please give total grades otherwise!"; };
      return (0);
    };
  }

  if( !homog(N) )
  {
    if(b) { "   ? grtest: Input should be graded homogenous!"; };
    return (0);
  };

//  if(b) { "Input seems to be a valid graded object (map)!"; };
  return (1);
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  // the following calls will fail due to tests in grtest:

 grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0,0,0) ); // enough row weights
// grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0) ); // not enough row weights
// grobj( module([x,0], [0,0,0], [0, y]), intvec(1,2,3) ); // zero column needs (otherwise optional) total degrees
 grobj( module([x,0], [0,0,0], [0, y]), intvec(1,2,3), intvec(2, 10, 3) ); // compatible total degrees (on non-zero columns)
// grobj( module([x,0], [0,0,0], [0, y]), intvec(1,2,3), intvec(2-1, 10, 3+1) ); // incompatible total degrees (on both non-zero columns)

}


static proc align( def A, int d)
"analog of align kernel command for older Singular versions
 this is static since it should not be used by @code{align}-able (newer)
 Singular releases.
 Note that this proc does not care about any attributes (of A)
"
{
  ASSUME(0, d >= 0 );

  if( d == 0 ) { return (A); }

  if( ncols(A) == 0 )
  {
    matrix B[nrows(A) + d][0];
    return (B);
  }

  module T; T[d] = 0;
  T = T, module(transpose(A));
  return( module(transpose(T)) );
}
example
{ "EXAMPLE:"; echo = 2;

  ring r;
  matrix m[2][0];
  type( align(m, 3) );

  matrix m[0][2];
  type( align(m, 3) );
}


proc grgroebner(A)
"USAGE:  grgroebner(M), graded object M
RETURN:  graded object
PURPOSE: compute graded groebner basis of M
EXAMPLE: example grgroebner; shows an example
"
{
  ASSUME(1, grtest(A));

  return ( grobj( groebner(A), grrange(A) ) );
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module A = grobj( module([x+y, x, 0, 0], [0, x+y, y, 0]), intvec(0,0,0,1) );
  grview(A);

  module B = grgroebner(A);
  grview(B);
}

proc grsyz(A)
"USAGE:  grsyz(M), graded object M
RETURN:  graded object
PURPOSE: compute graded syzygy of M
EXAMPLE: example grsyz; shows an example
"
{
  ASSUME(1, grtest(A));
  return( grobj( syz(A), grdeg(A) ) );

//   if( size(syz(A)) == 0 ) : zero syzygy? //  return( grtwists( -grdeg(A) ) );
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module A = grgroebner( grobj( module([x+y, x, 0, 3], [0, x+y, y, 2], [y, y, z, 1]), intvec(0,0,0,1) ) );
  grview(A);

  grview(grsyz(A));

  module X = grgroebner( grobj( module([x]), intvec(2) ) );
  grview(X);

  // syzygy module should be zero!
  grview(grsyz(X));


}


proc grprod(A, B)
"USAGE:  grprod(M, N), graded objects M and N
RETURN:  graded object
PURPOSE: compute graded product M * N (as composition of maps)
EXAMPLE: example grprod; shows an example
"
{
  ASSUME(1, grtest(A));
  ASSUME(1, grtest(B));

  intvec a = grdeg(A);
  intvec b = grrange(B);

  ASSUME(0, (size(a) == size(b)) && (a == b));  // == for intvec :(

  return ( grobj( A*B, grrange(A), grdeg(B) ) );
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module A = grobj( module([x+y, x, 0, 3], [0, x+y, y, 2], [y, y, z, 1]), intvec(0,0,0,1) );
  grview(A);

  A = grgroebner(A);
  grview(A);

  module B = grsyz(A);
  grview(B);
  print(B);

  module D = grprod( A, B );
  grview(D);
  print(D); // must be all zeroes due to syzygy property!
  ASSUME(0, size(D) == 0);
}




// TODO: think about a proper data structure for a graded resolution!?
proc grres(def A, int l, list #)
"USAGE:  grres(M, l[, b]), graded object M, int l, int b
RETURN:  graded resolution = list of graded objects
PURPOSE: compute graded resolution of M (of length l) and minimise it if b was given
EXAMPLE: example grres; shows an example
"
{
  ASSUME(0, l >= 0);
  ASSUME(1, grtest(A));

  intvec v = grrange(A);

  int b = (size(#) > 0);
  if(b) { list r = res(A, l, #[1]); } else { list r = res(A, l); }

  l = size(r);

  int i;

  for ( i = 1; i <= l; i++ )
  {
    if( size(r[i]) == 0 )
    {
      r[i] = grobj(freemodule(0), v); // grtwists(-v);
      i++;
      break;
    }

    r[i] = grobj(r[i], v); v = grdeg(r[i]);
  }

  i = i-1;

  return( list(r[1..i]) );
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module A = grobj( module([x+y, x, 0, 3], [0, x+y, y, 2], [y, y, z, 1]), intvec(0,0,0,1) );
  grview(A);

  module B = grgroebner(A);
  grview(B);

  "graded resolution of B: "; def C = grres(B, 0); grview(C);

  int i; int l = size(C);

  "D^2 == 0: "; for (i = 1; i < l; i++ ) { i; grview( grprod(C[i], C[i+1]) ); }
}

proc grtranspose(def M)
"USAGE:   grtranspose(M), graded object M
RETURN:  graded object
PURPOSE: graded transpose of M
NOTE:    no reordering is performend by this procedure
EXAMPLE: example grtranspose; shows an example
"
{
  ASSUME(1, grtest(M) );
  return (  grobj(transpose(M), -grdeg(M), -grrange(M))  );
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module M = grtwists( intvec(-2, 0, 4, 4) ); grview(M);

  module N = grsyz( grtranspose( M ) ); grview(N);

  module L = grtranspose(N); grview( L );

  module K = grsyz( L ); grview(K);


  // Corner cases: 0 <- 0!
  module Z = grzero(); grview(Z);
  grview( grtranspose( Z ) );


  // Corner cases: * <- 0
  matrix M1[3][0];

  module Z1 = grobj( M1, intvec(-1, 0, 1) ); grview(Z1);
  grview( grtranspose( Z1 ) );


  // Corner cases: 0 <- *
  matrix M2[0][3];

  module Z2 = grobj( M2, 0:0, intvec(-1, 0, 1) ); grview(Z2);
  grview( grtranspose( Z2 ) );

}


proc grgens(def M)
"USAGE:   grgens(M), graded object M (map)
RETURN:  graded object
PURPOSE: try compute graded generators of coker(M) and return them as columns
         of a graded map.
NOTE:    presentation of resulting generated submodule may be different to M!
EXAMPLE: example grgens; shows an example
"
{
  ASSUME(1, grtest(M) );

  module N = grtranspose( grsyz( grtranspose(M) ) );

//  ASSUME(3, grisequal( grgroebner(M), grgroebner( grpres( N ) ) ) ); // FIXME: not always true!?

  return ( N );
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module M = grtwists( intvec(-2, 0, 4, 4) ); grview(M);

  module N = grgens(M);

  grview( N ); print(N); // fine == M


  module A = grobj( module([x+y, x, 0, 3], [0, x+y, y, 2], [y, y, z, 1]), intvec(0,0,0,1) );

  A = grgroebner(A); grview(A);

  module B = grgens(A);

  grview( B ); print(B); // Ups :( != A

  grview( grgens( grzero() ) );

}


proc grpres(def M)
"USAGE:   grpres(M), graded object M (submodule gens)
RETURN:  graded module (via coker)
PURPOSE: compute graded presentation matrix of submodule generated by columns of M
EXAMPLE: example grpres; shows an example
"
{
  ASSUME(1, grtest(M) );

  def N = grsyz(M);

//  ASSUME(3, grisequal( M, grgens( N ) ) );

  return ( N );
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  def A = grgroebner( grobj( module([x+y, x, 0, 3], [0, x+y, y, 2], [y, y, z, 1]), intvec(0,0,0,1) ) );
  grview(A);

  "graded transpose: "; def B = grtranspose(A); grview( B ); print(B);

  "... syzygy: "; def C = grsyz(B); grview(C);

  "... transposed: "; def D = grtranspose(C); grview( D ); print (D);

  "... and back to presentation: "; def E = grsyz( D ); grview(E); print(E);

  def F = grgens( E ); grview(F); print(F);
  def G = grpres( F ); grview(G); print(G);


  def M = grtwists( intvec(-2, 0, 4, 4) ); grview(M);

  def N = grgens(M); grview( N ); print(N);

  def L = grpres( N ); grview( L ); print(L);
}



LIB "random.lib"; // for sparsepoly

proc grrndmat(intvec w, intvec v, list #)
"USAGE:  grrndmat(src,dst[,p,b]), intvec src, dst[, int p, b]
RETURN:  matrix of polynomials
PURPOSE: generate random matrix compatible with src and dst gradings
NOTE:    optional arguments p, b are for 'sparsepoly' (by default: 75%, 30000).
TODO:    this is experimental at the moment!
EXAMPLE: example grrndmat; shows an example
"
{
  // defaults for sparsepoly
  int p = 75;
  int b = 30000;

  if ( size(#) > 0 )
  {
    ASSUME( 0, (typeof(#[1]) == "int") || (typeof(#[1]) == "bigint") );
    p = #[1];

    if ( size(#) > 1 )
    {
      ASSUME( 0, (typeof(#[2]) == "int") || (typeof(#[2]) == "bigint") );
      b = #[2];
    }
  }

  int n = size(v); // destination: rows!
  int m = size(w); // source: cols

  matrix M[n][m];

  int r,c; intvec ww;

  for( c = m; c > 0; c-- )
  {
    ww = v - intvec(w[c]:n);
    for( r = n; r > 0; r-- )
    {
      if( ww[r] >= 0)
      {
        M[r,c] = sparsepoly(ww[r], ww[r], p, b);
      }

    }
  }

  return(M);
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  print( grrndmat( intvec(0, 1), intvec(1, 2, 3) ) );
}

// TODO: remove the following?
proc KeneshlouMatrixPresentation(intvec a)
"USAGE: KeneshlouMatrixPresentation(intvec a), intvec a.
RETURN: graded object
PURPOSE: matrix presentation for direct sum of omega^a[i](i) in form of a graded object
EXAMPLE: example KeneshlouMatrixPresentation; shows an example
"
{
  int n = size(a)-1;
  //  ring r = 32003,(x(0..n)),dp;
  ASSUME(0, nvars(basering)==(n+1));
  int i,j;

  // find first nonzero exponent a_i
  for(i=1;i<=size(a);i++)
    {
      if(a[i]!=0) {break; };
    }

  // all zeroes?
  if(i>size(a)) {return (grzero()); };

  for(i=2;i<=n;i++)
    {
      if(a[i]!=0) {break; };
    }

  module N;

  if(i>n)
    { // no middle part
      if(a[1]>0)
        {
          N=grtwist(a[1],0);

          if(a[n+1]>0)
            { N=grsum(N,grtwist(a[n+1],-1));}
        }
      else
        { N=grtwist(a[n+1],-1);}

      return (N); // grorder(N));
    }
  else // i <= n: middle part is present, a_i != 0
    { // a = a1  ... |  i:2, a_2 ..... i: n, a_n | .... i: n+1a_(n+1)
      j = i - 1;
      module I = maxideal(1); attrib(I,"isHomog", intvec(0)); list L = mres(I, 0); // TODO: use grres() instead!!!
      list kos = grorder(L);
      // make sure that graded maps  are represented by blocks corresponding to the betti diagram?

      def S = grpower(grshift(grobj( kos[j+2], attrib(kos[j+2], "isHomog")), j), a[i]);

      i++;

      for(; i <= n; i++)
        {
          if(a[i]==0) { i++; continue; }
          j = i - 1;
          S = grsum( S, grpower(grshift( grobj( kos[j+2], attrib(kos[j+2], "isHomog")), j), a[i])  );
        }

      // S is the middle (non-zero) part

      if(a[1] > 0 )
        {
          N=grsum(grtwist(a[1],0), S);
        }
      else
        { N = S;}

      if(a[n+1] > 0 )
        { N=grsum(N, grtwist(a[n+1],-1)); }


      return ((N)); //      return (grorder(N));
    }
}
example
{ "EXAMPLE:"; echo = 2;
  ring r = 32003,(x(0..4)),dp;

  def N1 = KeneshlouMatrixPresentation(intvec(2,0,0,0,0));
  grview(N1);

  def N2 = KeneshlouMatrixPresentation(intvec(0,0,0,0,3));
  grview(N2);

  def N = KeneshlouMatrixPresentation(intvec(2,0,0,0,3));
  grview(N);


  def M1 = KeneshlouMatrixPresentation(intvec(0,1,0,0,0));
  grview(M1);

  def M2 = KeneshlouMatrixPresentation(intvec(0,1,1,0,0));
  grview(M2);

  def M3 = KeneshlouMatrixPresentation(intvec(0,0,0,1,0));
  grview(M3);

  def M = KeneshlouMatrixPresentation(intvec(1,1,1,0,0));
  grview(M);
}

proc grconcat(A,B)
"USAGE: grconcat(A, B), graded objects A and B, dst(A) == dst(B) =: dst
RETURN: graded object
PURPOSE: construct src(A) + src(B) -----> dst  given by (A|B)
EXAMPLE: example grconcat; shows an example
"
{

  ASSUME(1, grtest(A));
  ASSUME(1, grtest(B));
  ASSUME(0, grrange(A)==grrange(B));

  intvec v = grrange(A);
  intvec w=grdeg(A),grdeg(B);
  return(grobj(concat(A,B),v,w));
}
example
{ "EXAMPLE:"; echo = 2;
  ring r;

  module R=grobj(module([x,y,z]),intvec(0:3));
  grview(R);

  module S=grobj(module([x,0,y],[xy,zy+x2,0]),intvec(0:3));
  grview(S);

  def Q=grconcat(R,S);
  grview(Q);
}

proc grlift(A, B)
"USAGE: grlift(M, N), graded objects M and N
RETURN: transformation matrix (graded object???)
PURPOSE: compute graded matrix which the generators of submodule Im(N) in terms of Im(M).
EXAMPLE: example grlift; shows an example
"
{
  ASSUME(1, grtest(A));
  ASSUME(1, grtest(B));
  ASSUME(0, grrange(A) == grrange(B));

//  matrix T;  module AA = liftstd(A, T); //  AA = module(A*T)
//  matrix U;
  matrix L =lift(A,B/*,U*/);  //  module(B*U) = module(matrix(A)*L)

  return(grobj(L, grdeg(A), grdeg(B)));
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;
  module P=grobj(module([xy,0,xz]),intvec(0,1,0));
  grview(P);


  module D=grobj(module([y,0,z],[x2+y2,z,0]),intvec(0,1,0));
  grview(D);

  def G=grlift(D,P);
  grview(G);

  ASSUME(0, grisequal( grprod(D, G), P) );
}

proc grrange(M)
"USAGE: grrange(M), graded object M
RETURN: intvec
PURPOSE: get weights of module units, thus describing the target of M
EXAMPLE: example grrange; shows an example
"
{
//  ASSUME(1, grtest(M)); // Leads to recursive call due to grtest...
  return( attrib(M, "isHomog") );
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module Z = grobj(freemodule(0),intvec(0:0),intvec(0:0));

  grrange(Z);
  grdeg(Z);

  grview(Z);

  module P=grobj(module([xy,0,xz]),intvec(0,1,0));

  grrange(P);
  grdeg(P);

  grview(P);
}

proc grlift0(M, N, alpha1)
"USAGE: grlift0(M, N, alpha1) TODO!
PURPOSE: generic random alpha0 : coker(M) -> coker(N) from random alpha1
NOTE: this proc can work only if some assumptions are fulfilled (due
to Wolfram)! e.g. at the end of a resolution for the source module...
"
{

  ASSUME(1, grtest(M));
  ASSUME(1, grtest(N));

  ASSUME(0, grdeg(M) == grdeg(alpha1) );
  ASSUME(0, grdeg(N) == grrange(alpha1) );
  return(
   grtranspose( grlift( grtranspose( M ),
       grtranspose( grprod( N,  alpha1 ) ) )
       ) ); // alpha0!

}
example
{ "EXAMPLE:"; echo = 2;

  ring S = 0, (x(0..3)), dp;
  list kos = grres(grobj(maxideal(1), intvec(0)), 0);
  print( betti(kos), "betti");
  grview(kos);


  // source module:
//  module M = grshift(kos[4], 2); // phi, Syz_3(K(2))
  def M = KeneshlouMatrixPresentation(intvec(0,0,1,0));
//   grview( grres(M, 0) );
  grview(M);

  // destination module:
//   module N = grshift(kos[3], 1); // psi, Syz_2(K(1))
  def N = KeneshlouMatrixPresentation(intvec(0,1,0,0));
//  grview( grres(N, 0) );
  grview(N);

  // random graded of degree 0, homomorphism of free presentations:
  // alpha1: src(M) -> src (N)
  def alpha1 = grrndmap( M, N ); // alpha1
  grview(alpha1);

  // random graded of degree 0, homomorphism of free presentations:
  // alpha0: dst(M) -> dst (N)
  def alpha0 = grlift0(M, N, alpha1);
  grview(alpha0);

}


proc grlifting(M,N)
"USAGE: grlifting(M,N), graded objects M and N
RETURN: map of chain complexes (as a list)
PURPOSE: construct a map of chain complexes between free resolutions of Img(M) and Img(N).
EXAMPLE: example grlifting; shows an example
"
{  ASSUME(1, grtest(M));
   ASSUME(1, grtest(N));

   list rM=grres(M,0,1);
   list rN=grres(N,0,1);
   int i,j,k;

  for(i=1;i<=size(rM);i++)
  {
    if(size(rM[i])==0){break;}
  }

  for(j=1;j<=size(rN);j++)
  {
    if(size(rN[j])==0){break;}
  }
  int t=min(i,j);

  ASSUME(0, t >= 2);

  list P;

  "t: ", t;

  P[1]= grrndmap( rM[1], rN[1] ); // alpha1

  if(t==2){return(P[1]);}

  for(k=2; k<=t; k++)
  {
    P[k] = grlift( grprod(P[k-1],rM[k]), rN[k] );
     grview(P[k]);

  }

  return(P);

}
example
{ "EXAMPLE:"; echo = 2;
/*
  ring r=32003,(x,y,z),dp;

  module P=grobj(module([xy,0,xz]),intvec(0,1,0));
  grview(P);

  module D=grobj(module([y,0,z],[x2+y2,z,0]),intvec(0,1,0));
  grview(D);

  def G=grlifting(D,P);
  grview(G);

  kill r;
  ring r=32003,(x,y,z),dp;

  module D=grobj(module([y,0,z],[x2+y2,z,0], [z3, xy, xy2]),intvec(0,1,0));
  D = grgroebner(D);
  grview( grres(D, 0));

  def G=grlifting(D, D);
  grview(G);
*/

  ring S = 0, (x(0..3)), dp;
  list kos = grres(grobj(maxideal(1), intvec(0)), 0);
  print( betti(kos), "betti");
  grview(kos);

//  module M = grshift(kos[4], 2); // phi, Syz_3(K(2))
  def M = KeneshlouMatrixPresentation(intvec(0,0,1,0));
  grview( grres(M, 0) );

//   module N = grshift(kos[3], 1); // psi, Syz_2(K(1))
  def N = KeneshlouMatrixPresentation(intvec(0,1,0,0));
  grview( grres(N, 0) );

  grlifting(M, N); // grview(G);


//  def G=grlifting( grgens(M), grgens(N) );  grview(G);


}

proc mappingcone(M,N)
"USAGE: mappingcone(M,N), M,N graded objects
RETURN: chain complex (as a list)
PURPOSE: construct a free resolution of the cokernel of a random map between Img(M), and Img(N).
EXAMPLE: example mappingcone; shows an example
"
{
  ASSUME(1, grtest(M));
  ASSUME(1, grtest(N));

  list P=grlifting(M,N);
  list rM=grres(M,1);
  list rN=grres(N,1);

  int i;
  list T;

  T[1]=grconcat(P[1],rN[2]);

  for(i=2;i<=size(P);i++)
  {
    intvec v=grrange(rM[i]);
    intvec w=grdeg(rN[i+1]);
    int r=size(v);
    int s=size(w);
    module zero = (0:s);

    module A=grconcat(P[i],rN[i+1]);
    module B=grobj(zero,v,w);
    module C=grconcat(-rM[i],B);
    module D=grconcat(grtranspose(C), grtranspose(A));

    T[i]=grtranspose(D);
  }
   return(T);
}
example
{ "EXAMPLE:"; echo = 2;

ring r=32003, (x(0..4)),dp;
def A=KeneshlouMatrixPresentation(intvec(0,0,0,0,3));
def M=grgens(A);
grview(M);

def B=KeneshlouMatrixPresentation(intvec(0,1,0,0,0));
def N=grgens(B);
grview(N);

def R=grlifting(M,N);
grview(R);
def T=mappingcone(M,N);
grview(T);

def U=grtranspose(T[1]);
resolution G=mres(U,0);
print(betti(G),"betti");
ideal I=groebner(flatten(G[2]));
resolution GI=mres(I,0);
print(betti(GI),"betti");
}

// correct
proc grrndmap(def S, def D, list #)
"USAGE: grrndmap(S,D), graded objects S and D
RETURN: graded object
PURPOSE: construct a random 0-deg graded homomorphism src(S) -> src(D)
EXAMPLE: example grrndmap; shows an example
"
{

ASSUME(1, grtest(S) );
ASSUME(1, grtest(D) );

// "src: "; grview(S);"dst: "; grview(D);

intvec v = -grdeg(S); // source
intvec w = -grdeg(D); // destination

return (grobj(grrndmat(v, w, #), -w, -v ) );
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module D=grobj(module([y,0,z],[x2+y2,z,0]),intvec(0,1,0));
  grview(D);

  module S=grobj(module([x,0,y],[xy,zy+x2,0]),intvec(0,0,0));
  grview(S);

  def H=grrndmap(D,S);
  grview(H);

}

proc grrndmap2(def D, def S, list #)
"USAGE: grrndmap2(D,S), graded objects S and D
RETURN: graded object
PURPOSE: construct a random 0-deg graded homomorphism between target of D and S.
EXAMPLE: example grrndmap2; shows an example
"
{
  ASSUME(1, grtest(D) );
  ASSUME(1, grtest(S) );
  intvec v = -grrange(D); // source
  intvec w = -grrange(S); // target
  return (grobj(grrndmat(v, w, #), -w, -v ) );
}
example
{ "EXAMPLE:"; echo = 2;

  ring r=32003,(x,y,z),dp;

  module D=grobj(module([y,0,z],[x2+y2,z,0]),intvec(0,1,0));
  grview(D);

  module S=grobj(module([x,0,y],[xy,zy+x2,0]),intvec(0,0,0));
  grview(S);

  def G=grrndmap2(D,S);
  grview(G);
}


//                A     f2     f3
// 0<---M<----F0<----F1<----F2<----F3<----
//              |p1   |p2
//
// 0<---N<----G0<----G1<----G2<----G3<----
//                B(g1)      g2     g3
//
proc grlifting2(A,B)
"USAGE: grlifting2(A,B), graded objects A and B (matrices defining maps)
RETURN: map of chain complexes (as a list)
PURPOSE: construct a map of chain complexes between free resolution of
M=coker(A) and N=coker(B).
EXAMPLE: example grlifting2; shows an example
"
{  ASSUME(1, grtest(A));
   ASSUME(1, grtest(B));

   list rM=grres(A,0);
   list rN=grres(B,0);
   int i,j,k;
   list P;

  // find first zero matrix in rM
  for(i=1;i<=size(rM);i++)
  {
    if(size(rM[i])==0){break;}
  }

  // find first zero matrix in rN
  for(j=1;j<=size(rN);j++)
  {
    if(size(rN[j])==0){break;}
  }

  int t=min(i,j);

  P[1]=grrndmap2(A,B);

  // A(or B)=0
  if(t==1){return(P[1])};

  for(k=2;k<=t;k++)
  {
   def E=grprod(P[k-1],rM[k-1]);
   P[k]=grlift(rN[k-1],E); // ---------->
   /* let yi=(pi)o(fi); to take grlift(gi,yi)
      we should have img(yi) is contained in
      img(gi)=ker(gi-1),i.e (gi-1)oyi=0. we have
      (gi-1)oyi=(gi-1)o(pi)o(fi)=(pi-1)o(fi-1)ofi=0
  */
  }
  return(P);
}
example
{ "EXAMPLE:"; echo = 2;

ring r;
module P=grobj(module([xy,0,xz]),intvec(0,1,0));
grview(P);

module D=grobj(module([y,0,z],[x2+y2,z,0]),intvec(0,1,0));
grview(D);

module PP = grpres(P);
grview(PP);

module DD = grpres(D);
grview(DD);


def T=grlifting2(DD,PP); T;

// def Z=grlifting2(P,D); Z; // WRONG!!!

}

/*
//              -f1 0       -f2 0
//   (p1 g1)     p2 g2       p3 g3
// G0<-----F0+G1<------F1+G2<-------F2+G3<-----
proc mappingcone2(A,B)
"USAGE: mappingcone2(A,B), graded objects A and B (matrices defining maps)
RETURN: chain complex (as a list)
PURPOSE: construct the free resolution of a cokernel of a random map between M=coker(A), and N=coker(B)
EXAMPLE: example mappingcone2;
"
{
  ASSUME(1, grtest(A));
  ASSUME(1, grtest(B));

  list P=grlifting2(A,B);
  list rM=grres(A,1);
  list rN=grres(B,1);

  int i;
  list T;

  T[1]=grconcat(P[1],rN[1]);

  for(i=2;i<=size(P);i++)
  {
    intvec v=grrange(rM[i-1]);
    intvec w=grdeg(rN[i]);
    int r=size(v);
    int s=size(w);
    matrix zero[r][s];

    module A=grconcat(P[i],rN[i]);
    module B=grobj(zero,v,w);

    module C=grconcat( grneg( rM[i-1] ) ,B);
    module D=grconcat(grtranspose(C), grtranspose(A));

    T[i]=grtranspose(D);
  }
  return(T);
}
example
{ "EXAMPLE:"; echo = 2;

ring r=32003,(x(0..4)),dp;
def I=maxideal(1);
module R=grobj(module(I), intvec(0));
resolution FR=mres(R,0);
print(betti(FR,0),"betti");
module K=grobj(module(FR[1]),intvec(-1),intvec(0:5));
grview(K);

module S=grsyz(K);
grview(S);
S;

module SS = grpres(S);

module B=grobj(module([1,0,0],[0,1,0],[0,0,1]),intvec(1,1,1),intvec(1,1,1));
// module B=grobj(module([1],[0,1],[0,0,1],[0,0,0,1], [0,0,0,0,1]),intvec(0:5));
grview(B);
B;
module BB = grpres(B);

def Z=grlifting2(SS,BB);Z;
def G=mappingcone2(SS,BB);G;
}
*/

proc grlifting3(A,B)
"TODO: grlifting4 was newer and had more documentation than this proc, but was removed... Please verify and update!
"
{
  ASSUME(1, grtest(A));
  ASSUME(1, grtest(B));


  list rM = grres(A,0,1);

  print( betti(rM), "betti");
  list rN = grres(B,0,1);
  print( betti(rN), "betti");

  int i,j,k;

  for(i=1;i<=size(rM);i++)
  {
    if(size(rM[i])==0){break;}
  }

  for(j=1;j<=size(rN);j++)
  {
    if(size(rN[j])==0){break;}
  }
  int t=min(i,j);

  list P;

  "t: ", t;
//  grview(rM[t]);  grview(rN[t]);

  P[t]= grrndmap2(rM[t],rN[t]);
  grview(P[t]);

  if(t==1){return(P)};

  for(k=t-1; k>=1; k--)
  {
     "k: ", k;
//  grview(rM[k]);  grview(rN[k]);

// def C = grtranspose(rM[k]); def T= grprod(rN[k],P[k+1]);
// def tT = grtranspose(T);

    P[k]= grlift0( rM[k], rN[k], P[k+1] ); // grtranspose(grlift(C,tT));

     grview(P[k]);

   }
   return(P);
}
example
{"EXAMPLE:"; echo = 2;

ring r=32003, x(0..4),dp;

def A=grtwist(3,1);
grview(A);

def T=KeneshlouMatrixPresentation(intvec(0,1,0,0,0));
grview(T);

def F=grlifting3(T,A);
grview(F);

def R=KeneshlouMatrixPresentation(intvec(0,0,0,2,0));
def S=KeneshlouMatrixPresentation(intvec(1,2,0,0,0));

def H=grlifting3(R, S);
// grview(H);

// 2nd module does not lie in the first:
// def H=grlifting3(S, R);


//def I=KeneshlouMatrixPresentation(intvec(2,3,0,6,2));
//def J=KeneshlouMatrixPresentation(intvec(4,0,1,2,1));
//def N=grlifting3(I,J); grview(N);
}

proc grneg(A)
"USAGE: grneg(A), graded object A
RETURN: graded object
PURPOSE: graded map defined by -A
EXAMPLE: example grneg; shows an example
"
{
  ASSUME(1, grtest(A));
  return( grobj(-A, grrange(A), grdeg(A)) );
}
example
{ "EXAMPLE:"; echo = 2;

   ring r=0,(x,y,z),dp;
   def A=grobj([x2,yz,xyz],intvec(1,1,0));
   grview(A);

   def F=grneg(A);
   grview(F);
}

//             -f1 0        -f2 0
//  (p1 g1)     p2 g2        p3 g3
//G0<-----F0+G1<------F1+G2<-------F2+G3<-----
proc mappingcone3(A,B)
"USAGE: mappingcone3(A,B), graded objects A and B (matrices defining maps)
RETURN: chain complex (as a list)
PURPOSE: construct a free resolution of the cokernel of a random map between M=coker(A), and N=coker(B)
EXAMPLE: example mappingcone3; shows an example
"
{
  ASSUME(1, grtest(A));
  ASSUME(1, grtest(B));

  list P=grlifting3(A,B);
  list rM=grres(A,0,1);
  list rN=grres(B,0,1);

  int i;
  list T;

  T[1]=grconcat(P[1],rN[1]);

  for(i=2;i<=size(P);i++)
  {
    intvec v= grrange(rM[i-1]);
    intvec w=grdeg(rN[i]);
    int r=size(v);
    int s=size(w);
    matrix zero[r][s];

//    ASSUME( 0, grtest(P[i]) );
//    ASSUME( 0, grtest(rN[i]) );

    module A=grconcat(P[i],rN[i]);
    module B=grobj(zero,v,w);

    module C=grconcat( grneg( rM[i-1] ) ,B);
    module D=grconcat(grtranspose(C), grtranspose(A));

    T[i]=grtranspose(D);

    kill A, B, C, D, v, w, r, s, zero;
  }
   return(T);
}
example
{ "EXAMPLE:"; echo = 2;

ring r=32003,x(0..4),dp;

def A=KeneshlouMatrixPresentation(intvec(0,0,0,0,3));
grview(A);

def T= KeneshlouMatrixPresentation(intvec(0,1,0,0,0));
grview(T);

def F=grlifting3(A,T); grview(F);

// BUG in the proc
def G=mappingcone3(A,T); grview(G);

/*
module W=grtranspose(G[1]);
resolution U=mres(W,0);
print(betti(U,0),"betti"); // ?
ideal P=groebner(flatten(U[2]));
resolution L=mres(P,0);
print(betti(L),"betti");
*/


def R=KeneshlouMatrixPresentation(intvec(0,0,0,2,0));
grview(R);

def S=KeneshlouMatrixPresentation(intvec(1,2,0,0,0));
grview(S);

def H=grlifting3(R,S); grview(H);

// BUG in the proc
def G=mappingcone3(R,S);


def I=KeneshlouMatrixPresentation(intvec(2,3,0,6,2));
def J=KeneshlouMatrixPresentation(intvec(4,0,1,2,1));
// def N=grlifting3(I,J);
// 2nd module does not lie in the first:
// def NN=mappingcone3(I,J); // ????????

}




// TODO: Please decide between KeneshlouMatrixPresentation and matrixpres, and replace one with the other!
proc matrixpres(intvec a)
"USAGE:  matrixpres(a), intvec a
RETURN:  graded object
PURPOSE: matrix presentation for direct sum of omega^a[i](i) in form of a graded object
EXAMPLE: example matrixpres; shows an example
"
{
  int n = size(a)-1;
  //  ring r = 32003,(x(0..n)),dp;
  ASSUME(0, nvars(basering)==(n+1));
  int i,j;

  // find first nonzero exponent a_i
  for(i=1;i<=size(a);i++)
    {
      if(a[i]!=0) {break; };
    }

  // all zeroes?
  if(i>size(a)) {return (grzero()); };
   for(i=2;i<=n;i++)
    {
      if(a[i]!=0) {break; };
    }

  module N;

  if(i>n)
    { // no middle part
      if(a[1]>0)
        {
          N=grtwist(a[1],-1);

          if(a[n+1]>0)
            { N=grsum(N,grtwist(a[n+1],0));}
        }
      else
        { N=grtwist(a[n+1],0);}

      return (N); // grorder(N));
    }

else // i <= n: middle part is present, a_i != 0
    { // a = a1  ... |  i:2, a_2 ..... i: n, a_n | .... i: n+1a_(n+1)
      module I = maxideal(1);
      attrib(I,"isHomog", intvec(0));
      list L = mres(I, 0);
      list kos = grorder(L);
      // make sure that graded maps  are represented by blocks corresponding to the betti diagram?
      int j=size(a)-i;
      def S = grpower(grshift(grobj( kos[j+2], attrib(kos[j+2], "isHomog")),j ), a[i]);

      i++;

      for(; i <= n; i++)
        {
          if(a[i]==0) { i++; continue; }
          int j=size(a)-i;
          S = grsum( S, grpower(grshift( grobj( kos[j+2], attrib(kos[j+2], "isHomog")), j), a[i])  );
        }

      // S is the middle (non-zero) part

      if(a[1] > 0 )
        {
          N=grsum(grtwist(a[1],-1), S);
        }
      else
        { N = S;}

      if(a[n+1] > 0 )
        { N=grsum(N, grtwist(a[n+1],0)); }


      return ((N)); //      return (grorder(N));
    }
}
example
{ "EXAMPLE:"; echo = 2;

ring r = 32003,(x(0..4)),dp;

def R=matrixpres(intvec(1,4,0,0,0));
grview(R);
def S=matrixpres(intvec(0,0,3,0,0));
grview(S);

def N1 = matrixpres(intvec(2,0,0,0,0));
grview(N1);

def N2 = matrixpres(intvec(0,0,0,0,3));
grview(N2);

def N = matrixpres(intvec(2,0,0,0,3));
grview(N);


def M1 = matrixpres(intvec(0,1,0,0,0));
grview(M1);

def M2 = matrixpres(intvec(0,1,1,0,0));
grview(M2);

def M3 = matrixpres(intvec(0,0,0,1,0));
grview(M3);

def M = matrixpres(intvec(1,1,1,0,0));
grview(M);
}