source: git/Singular/LIB/poly.lib @ 73b186

///////////////////////////////////////////////////////////////////////////////
version="$Id: poly.lib,v 1.49 2008-12-12 11:26:34 Singular Exp $";
category="General purpose";
info="
LIBRARY:  poly.lib      Procedures for Manipulating Polys, Ideals, Modules
AUTHORS:  O. Bachmann, G.-M. Greuel, A. Fruehbis

PROCEDURES:
 cyclic(int);           ideal of cyclic n-roots
 katsura([i]);          katsura [i] ideal
 freerank(poly/...)     rank of coker(input) if coker is free else -1
 is_zero(poly/...);     int, =1 resp. =0 if coker(input) is 0 resp. not
 lcm(ideal);            lcm of given generators of ideal
 maxcoef(poly/...);     maximal length of coefficient occurring in poly/...
 maxdeg(poly/...);      int/intmat = degree/s of terms of maximal order
 maxdeg1(poly/...);     int = [weighted] maximal degree of input
 mindeg(poly/...);      int/intmat = degree/s of terms of minimal order
 mindeg1(poly/...);     int = [weighted] minimal degree of input
 normalize(poly/...);   normalize poly/... such that leading coefficient is 1
 rad_con(p,I);          check radical containment of poly p in ideal I
 content(f);            content of polynomial/vector f
 numerator(n);          numerator of number n
 denominator(n)         denominator of number n
 mod2id(M,iv);          conversion of a module M to an ideal
 id2mod(i,iv);          conversion inverse to mod2id
 substitute(I,...)      substitute in I variables by polynomials
 subrInterred(i1,i2,iv);interred w.r.t. a subset of variables
 newtonDiag(f);         Newton diagram of a polynomial
 hilbPoly(I);           Hilbert polynomial of basering/I
          (parameters in square brackets [] are optional)

";

LIB "general.lib";
LIB "ring.lib";
///////////////////////////////////////////////////////////////////////////////
static proc bino(int a, int b)
{
//computes binomial var(1)+a over b
   int i;
   if(b==0){return(1);}
   poly p=(var(1)+a)/b;
   for(i=1;i<=b-1;i++)
   {
      p=p*(var(1)+a-i)/i;
   }
   return(p);
}

proc hilbPoly(ideal I)
"USAGE: hilbPoly(I); I a homogeneous ideal
RETURN: the Hilbert polynomial of basering/I as an intvec v=v_0,...,v_r
        such that the Hilbert polynomial is (v_0+v_1*t+...v_r*t^r)/r!
EXAMPLE: example hilbPoly; shows an example
"
{
   def R=basering;
   if(!attrib(I,"isSB")){I=std(I);}
   intvec v=hilb(I,2);
   int s=dim(I);
   intvec hp;
   if(s==0){return(hp);}
   int d=size(v)-2;
   ring S=0,t,dp;
   poly p=v[1+d]*bino(s-1-d,s-1);
   int i;
   for(i=1;i<=d;i++)
   {
      p=p+v[d-i+1]*bino(s-1-d+i,s-1);
   }
   int n=1;
   for(i=2;i<=s-1;i++){n=n*i;}
   p=n*p;
   for(i=1;i<=s;i++)
   {
      hp[i]=int(leadcoef(p[s-i+1]));
   }
   setring R;
   return(hp);
}
example
{ "EXAMPLE:"; echo = 2;
   ring r = 0,(b,c,t,h),dp;
   ideal I=
   bct-t2h+2th2+h3,
   bt3-ct3-t4+b2th+c2th-2bt2h+2ct2h+2t3h-bch2-2bth2+2cth2+2th3,
   b2c2+bt2h-ct2h-t3h+b2h2+2bch2+c2h2-2bth2+2cth2+t2h2-2bh3+2ch3+2th3+3h4,
   c2t3+ct4-c3th-2c2t2h-2ct3h-t4h+bc2h2-2c2th2-bt2h2+4t3h2+2bth3-2cth3-t2h3
   +bh4-6th4-2h5;
   hilbPoly(I);
}

///////////////////////////////////////////////////////////////////////////////
proc substitute (I,list #)
"USAGE:  - case 1: typeof(#[1])==poly:
           substitute (I,v,f[,v1,f1,v2,f2,...]); I object of basering which
           can be mapped, v,v1,v2,.. ring variables, f,f1,f2,... poly
@*       - case 2: typeof(#[1])==ideal:
           substitute (I,v,f); I object of basering which can be mapped,
           v ideal of ring variables, f ideal
RETURN:  object of same type as I,
@*       - case 1: ring variable v,v1,v2,... substituted by polynomials
           f,f1,f2,..., in this order
@*       - case 2: ring variables in v substituted by polynomials in f:
           v[i] is substituted by f[i], i=1,...,i=min(size(v),ncols(f))
NOTE:    this procedure extends the built-in command subst via maps
EXAMPLE: example substitute; shows an example
"
{
  def bas = basering;
  ideal m = maxideal(1);
  int i,ii;
  if(typeof(#[1])=="poly")
  {
    poly v = #[1];
    poly f = #[2];
    map phi = bas,m;
    def J = I;
    for (ii=1; ii<=size(#) - 1; ii=ii+2)
    {
      m = maxideal(1);
      i=rvar(#[ii]);
      m[i] = #[ii+1];
      phi = bas,m;
      J = phi(J);
    }
    return(J);
  }
  if(typeof(#[1])=="ideal")
  {
    ideal v = #[1];
    ideal f = #[2];
    int mi = ncols(v);
    if(ncols(f)<mi)
    {
      mi = ncols(f);
    }
    def J = I;
    for (ii=1; ii<=mi; ii++)
    {
      m[rvar(v[ii])]=f[ii];
    }
    map phi = bas,m;
    J = phi(I);
    return(J);
  }
}
example
{ "EXAMPLE:"; echo = 2;
   ring r = 0,(b,c,t),dp;
   ideal I = -bc+4b2c2t,bc2t-5b2c;
   substitute(I,c,b+c,t,0,b,b-1);
   ideal v = c,t,b;
   ideal f = b+c,0,b-1;
   substitute(I,v,f);
}
///////////////////////////////////////////////////////////////////////////////
proc cyclic (int n)
"USAGE:   cyclic(n);  n integer
RETURN:  ideal of cyclic n-roots from 1-st n variables of basering
EXAMPLE: example cyclic; shows examples
"
{
//----------------------------- procedure body --------------------------------
   ideal m = maxideal(1);
   m = m[1..n],m[1..n];
   int i,j;
   ideal s; poly t;
   for ( j=0; j<=n-2; j++ )
   {
      t=0;
      for( i=1;i<=n;i++ ) { t=t+product(m,i..i+j); }
      s=s+t;
   }
   s=s,product(m,1..n)-1;
   return (s);
}
//-------------------------------- examples -----------------------------------
example
{ "EXAMPLE:"; echo = 2;
   ring r=0,(u,v,w,x,y,z),lp;
   cyclic(nvars(basering));
   homog(cyclic(5),z);
}
///////////////////////////////////////////////////////////////////////////////

proc katsura
"USAGE: katsura([n]); n integer
RETURN: katsura(n) : n-th katsura ideal of
         (1) newly created and set ring (32003, x(0..n), dp), if
             nvars(basering) < n
         (2) basering, if nvars(basering) >= n
        katsura()  : katsura ideal of basering
EXAMPLE: example katsura; shows examples
"
{
  int n;
  if ( size(#) == 1 && typeof(#[1]) == "int")
  {
    n = #[1] - 1;
    while (1)
    {
      if (defined(basering))
      {
        if (nvars(basering) >= #[1]) {break;}
      }
      ring katsura_ring = 32003, x(0..#[1]), dp;
      keepring katsura_ring;
      break;
    }
  }
  else
  {
    n = nvars(basering) -1;
  }

  ideal s;
  int i, j;
  poly p;

  p = -1;
  for (i = -n; i <= n; i++)
  {
    p = p + kat_var(i, n);
  }
  s[1] = p;

  for (i = 0; i < n; i++)
  {
    p = -1 * kat_var(i,n);
    for (j = -n; j <= n; j++)
    {
      p = p + kat_var(j,n) * kat_var(i-j, n);
    }
    s = s,p;
  }
  return (s);
}
//-------------------------------- examples -----------------------------------
example
{
  "EXAMPLE:"; echo = 2;
  ring r; basering;
  katsura();
  katsura(4); basering;
}

proc kat_var(int i, int n)
{
  poly p;
  if (i < 0)  { i = -i;}
  if (i <= n) { p = var(i+1); }
  return (p);
}
///////////////////////////////////////////////////////////////////////////////

proc freerank
"USAGE:   freerank(M[,any]);  M=poly/ideal/vector/module/matrix
COMPUTE: rank of module presented by M in case it is free.
         By definition this is vdim(coker(M)/m*coker(M)) if coker(M)
         is free, where m = maximal ideal of the variables of the
         basering and M is considered as matrix.
         (the 0-module is free of rank 0)
RETURN:  rank of coker(M) if coker(M) is free and -1 else;
         in case of a second argument return a list:
                L[1] = rank of coker(M) or -1
                L[2] = minbase(M)
NOTE:    freerank(syz(M)); computes the rank of M if M is free (and -1 else)
EXAMPLE: example freerank; shows examples
"
{
  int rk;
  def M = simplify(#[1],10);
  resolution mre = res(M,2);
  intmat B = betti(mre);
  if ( ncols(B)>1 ) { rk = -1; }
  else { rk = sum(B[1..nrows(B),1]); }
  if (size(#) == 2) { list L=rk,mre[1]; return(L);}
  return(rk);
}
example
{"EXAMPLE";   echo=2;
  ring r;
  ideal i=x;
  module M=[x,0,1],[-x,0,-1];
  freerank(M);          // should be 2, coker(M) is not free
  freerank(syz (M),"");
  // [1] should be 1, coker(syz(M))=M is free of rank 1
  // [2] should be gen(2)+gen(1) (minimal relation of M)
  freerank(i);
  freerank(syz(i));     // should be 1, coker(syz(i))=i is free of rank 1
}
///////////////////////////////////////////////////////////////////////////////

proc is_zero
"USAGE:   is_zero(M[,any]); M=poly/ideal/vector/module/matrix
RETURN:  integer, 1 if coker(M)=0 resp. 0 if coker(M)!=0, where M is
         considered as matrix.
         If a second argument is given, return a list:
                L[1] = 1 if coker(M)=0 resp. 0 if coker(M)!=0
                L[2] = dim(M)
EXAMPLE: example is_zero; shows examples
"
{
  int d=dim(std(#[1]));
  int a = ( d==-1 );
  if( size(#) >1 ) { return(list(a,d)); }
  return(a);
}
example
{ "EXAMPLE:";   echo=2;
  ring r;
  module m = [x],[y],[1,z];
  is_zero(m,1);
  qring q = std(ideal(x2+y3+z2));
  ideal j = x2+y3+z2-37;
  is_zero(j);
}
///////////////////////////////////////////////////////////////////////////////

proc maxcoef (f)
"USAGE:   maxcoef(f);  f  poly/ideal/vector/module/matrix
RETURN:  maximal length of coefficient of f of type int (by measuring the
         length of the string of each coefficient)
EXAMPLE: example maxcoef; shows examples
"
{
//----------------------------- procedure body --------------------------------
   int max,s,ii,jj; string t;
   ideal i = ideal(matrix(f));
   i = simplify(i,6);            // delete 0's and keep first of equal elements
   poly m = var(1); matrix C;
   for (ii=2;ii<=nvars(basering);ii++) { m = m*var(ii); }
   for (ii=1; ii<=size(i); ii++)
   {
      C = coef(i[ii],m);
      for (jj=1; jj<=ncols(C); jj++)
      {
         t = string(C[2,jj]);  s = size(t);
         if ( t[1] == "-" ) { s = s - 1; }
         if ( s > max ) { max = s; }
      }
   }
   return(max);
}
//-------------------------------- examples -----------------------------------
example
{ "EXAMPLE:"; echo = 2;
   ring r= 0,(x,y,z),ds;
   poly g = 345x2-1234567890y+7/4z;
   maxcoef(g);
   ideal i = g,10/1234567890;
   maxcoef(i);
   // since i[2]=1/123456789
}
///////////////////////////////////////////////////////////////////////////////

proc maxdeg (id)
"USAGE:   maxdeg(id);  id  poly/ideal/vector/module/matrix
RETURN:  int/intmat, each component equals maximal degree of monomials in the
         corresponding component of id, independent of ring ordering
         (maxdeg of each var is 1).
         Of type int if id is of type poly, of type intmat else
NOTE:    proc maxdeg1 returns an integer, the absolute maximum; moreover, it has
         an option for computing weighted degrees
SEE ALSO: maxdeg1
EXAMPLE: example maxdeg; shows examples
"
{
   //-------- subprocedure to find maximal degree of given component ----------
   proc findmaxdeg
   {
      poly c = #[1];
      if (c==0) { return(-1); }
   //--- guess upper 'o' and lower 'u' bound, in case of negative weights -----
      int d = (deg(c)>=0)*deg(c)-(deg(c)<0)*deg(c);
      int i = d;
      while ( c-jet(c,i) != 0 ) { i = 2*(i+1); }
      int o = i-1;
      int u = (d != i)*((i /  2)-1);
   //----------------------- "quick search" for maxdeg ------------------------
      while ( (c-jet(c,i)==0)*(c-jet(c,i-1)!=0) == 0)
      {
         i = (o+1+u) /  2;
         if (c-jet(c,i)!=0) { u = i+1; }
         else { o = i-1; }
      }
      return(i);
   }
//------------------------------ main program ---------------------------------
   matrix M = matrix(id);
   int r,c = nrows(M), ncols(M); int i,j;
   intmat m[r][c];
   for (i=r; i>0; i--)
   {
      for (j=c; j>0; j--) { m[i,j] = findmaxdeg(M[i,j]); }
   }
   if (typeof(id)=="poly") { return(m[1,1]); }
   return(m);
}
//-------------------------------- examples -----------------------------------
example
{ "EXAMPLE:"; echo = 2;
   ring r = 0,(x,y,z),wp(1,2,3);
   poly f = x+y2+z3;
   deg(f);             //deg; returns weighted degree (in case of 1 block)!
   maxdeg(f);
   matrix m[2][2]=f+x10,1,0,f^2;
   maxdeg(m);
}
///////////////////////////////////////////////////////////////////////////////

proc maxdeg1 (id,list #)
"USAGE:   maxdeg1(id[,v]);  id=poly/ideal/vector/module/matrix, v=intvec
RETURN:  integer, maximal [weighted] degree of monomials of id independent of
         ring ordering, maxdeg1 of i-th variable is v[i] (default: v=1..1).
NOTE:    This proc returns one integer while maxdeg returns, in general,
         a matrix of integers. For one polynomial and if no intvec v is given
         maxdeg is faster
EXAMPLE: example maxdeg1; shows examples
"
{
   //-------- subprocedure to find maximal degree of given component ----------
   proc findmaxdeg
   {
      poly c = #[1];
      if (c==0) { return(-1); }
      intvec v = #[2];
   //--- guess upper 'o' and lower 'u' bound, in case of negative weights -----
      int d = (deg(c)>=0)*deg(c)-(deg(c)<0)*deg(c);
      int i = d;
      if ( c == jet(c,-1,v))      //case: maxdeg is negative
      {
         i = -d;
         while ( c == jet(c,i,v) ) { i = 2*(i-1); }
         int o = (d != -i)*((i /  2)+2) - 1;
         int u = i+1;
         int e = -1;
      }
      else                        //case: maxdeg is nonnegative
      {
         while ( c != jet(c,i,v) ) { i = 2*(i+1); }
         int o = i-1;
         int u = (d != i)*((i /  2)-1);
         int e = 1;
      }
   //----------------------- "quick search" for maxdeg ------------------------
      while ( ( c==jet(c,i,v) )*( c!=jet(c,i-1,v) ) == 0 )
      {
         i = (o+e+u) /  2;
         if ( c!=jet(c,i,v) ) { u = i+1; }
         else { o = i-1; }
      }
      return(i);
   }
//------------------------------ main program ---------------------------------
   ideal M = simplify(ideal(matrix(id)),8);   //delete scalar multiples from id
   int c = ncols(M);
   int i,n;
   if( size(#)==0 )
   {
      int m = maxdeg(M[c]);
      for (i=c-1; i>0; i--)
      {
          n = maxdeg(M[i]);
          m = (m>=n)*m + (m<n)*n;             //let m be the maximum of m and n
      }
   }
   else
   {
      intvec v=#[1];                          //weight vector for the variables
      int m = findmaxdeg(M[c],v);
      for (i=c-1; i>0; i--)
      {
         n = findmaxdeg(M[i],v);
         if( n>m ) { m=n; }
      }
   }
   return(m);
}
//-------------------------------- examples -----------------------------------
example
{ "EXAMPLE:"; echo = 2;
   ring r = 0,(x,y,z),wp(1,2,3);
   poly f = x+y2+z3;
   deg(f);            //deg returns weighted degree (in case of 1 block)!
   maxdeg1(f);
   intvec v = ringweights(r);
   maxdeg1(f,v);                        //weighted maximal degree
   matrix m[2][2]=f+x10,1,0,f^2;
   maxdeg1(m,v);                        //absolute weighted maximal degree
}
///////////////////////////////////////////////////////////////////////////////

proc mindeg (id)
"USAGE:   mindeg(id);  id  poly/ideal/vector/module/matrix
RETURN:  minimal degree/s of monomials of id, independent of ring ordering
         (mindeg of each variable is 1) of type int if id of type poly, else
         of type intmat.
NOTE:    proc mindeg1 returns one integer, the absolute minimum; moreover it
         has an option for computing weighted degrees.
EXAMPLE: example mindeg; shows examples
"
{
   //--------- subprocedure to find minimal degree of given component ---------
   proc findmindeg
   {
      poly c = #[1];
      if (c==0) { return(-1); }
   //--- guess upper 'o' and lower 'u' bound, in case of negative weights -----
      int d = (ord(c)>=0)*ord(c)-(ord(c)<0)*ord(c);
      int i = d;
      while ( jet(c,i) == 0 ) { i = 2*(i+1); }
      int o = i-1;
      int u = (d != i)*((i /  2)-1);
   //----------------------- "quick search" for mindeg ------------------------
      while ( (jet(c,u)==0)*(jet(c,o)!=0) )
      {
         i = (o+u) / 2;
         if (jet(c,i)==0) { u = i+1; }
         else { o = i-1; }
      }
      if (jet(c,u)!=0) { return(u); }
      else { return(o+1); }
   }
//------------------------------ main program ---------------------------------
   matrix M = matrix(id);
   int r,c = nrows(M), ncols(M); int i,j;
   intmat m[r][c];
   for (i=r; i>0; i--)
   {
      for (j=c; j>0; j--) { m[i,j] = findmindeg(M[i,j]); }
   }
   if (typeof(id)=="poly") { return(m[1,1]); }
   return(m);
}
//-------------------------------- examples -----------------------------------
example
{ "EXAMPLE:"; echo = 2;
   ring r = 0,(x,y,z),ls;
   poly f = x5+y2+z3;
   ord(f);                  // ord returns weighted order of leading term!
   mindeg(f);               // computes minimal degree
   matrix m[2][2]=x10,1,0,f^2;
   mindeg(m);               // computes matrix of minimum degrees
}
///////////////////////////////////////////////////////////////////////////////

proc mindeg1 (id, list #)
"USAGE:   mindeg1(id[,v]);  id=poly/ideal/vector/module/matrix, v=intvec
RETURN:  integer, minimal [weighted] degree of monomials of id independent of
         ring ordering, mindeg1 of i-th variable is v[i] (default v=1..1).
NOTE:    This proc returns one integer while mindeg returns, in general,
         a matrix of integers. For one polynomial and if no intvec v is given
         mindeg is faster.
EXAMPLE: example mindeg1; shows examples
"
{
   //--------- subprocedure to find minimal degree of given component ---------
   proc findmindeg
   {
      poly c = #[1];
      intvec v = #[2];
      if (c==0) { return(-1); }
   //--- guess upper 'o' and lower 'u' bound, in case of negative weights -----
      int d = (ord(c)>=0)*ord(c)-(ord(c)<0)*ord(c);
      int i = d;
      if ( jet(c,-1,v) !=0 )      //case: mindeg is negative
      {
         i = -d;
         while ( jet(c,i,v) != 0 ) { i = 2*(i-1); }
         int o = (d != -i)*((i /  2)+2) - 1;
         int u = i+1;
         int e = -1; i=u;
      }
      else                        //case: inded is nonnegative
      {
         while ( jet(c,i,v) == 0 ) { i = 2*(i+1); }
         int o = i-1;
         int u = (d != i)*((i /  2)-1);
         int e = 1; i=u;
      }
   //----------------------- "quick search" for mindeg ------------------------
      while ( (jet(c,i-1,v)==0)*(jet(c,i,v)!=0) == 0 )
      {
         i = (o+e+u) /  2;
         if (jet(c,i,v)==0) { u = i+1; }
         else { o = i-1; }
      }
      return(i);
   }
//------------------------------ main program ---------------------------------
   ideal M = simplify(ideal(matrix(id)),8);   //delete scalar multiples from id
   int c = ncols(M);
   int i,n;
   if( size(#)==0 )
   {
      int m = mindeg(M[c]);
      for (i=c-1; i>0; i--)
      {
          n = mindeg(M[i]);
          m = (m<=n)*m + (m>n)*n;             //let m be the maximum of m and n
      }
   }
   else
   {
      intvec v=#[1];                          //weight vector for the variables
      int m = findmindeg(M[c],v);
      for (i=c-1; i>0; i--)
      {
         n = findmindeg(M[i],v);
         m = (m<=n)*m + (m>n)*n;              //let m be the maximum of m and n
      }
   }
   return(m);
}
//-------------------------------- examples -----------------------------------
example
{ "EXAMPLE:"; echo = 2;
   ring r = 0,(x,y,z),ls;
   poly f = x5+y2+z3;
   ord(f);                  // ord returns weighted order of leading term!
   intvec v = 1,-3,2;
   mindeg1(f,v);            // computes minimal weighted degree
   matrix m[2][2]=x10,1,0,f^2;
   mindeg1(m,1..3);         // computes absolute minimum of weighted degrees
}
///////////////////////////////////////////////////////////////////////////////

proc normalize (id)
"USAGE:   normalize(id);  id=poly/vector/ideal/module
RETURN:  object of same type,s
         each element is normalized to make its leading coefficient equal to 1
EXAMPLE: example normalize; shows an example
"
{
   return(simplify(id,1));
}
//-------------------------------- examples -----------------------------------
example
{ "EXAMPLE:"; echo = 2;
   ring r = 0,(x,y,z),ls;
   poly f = 2x5+3y2+4z3;
   normalize(f);
   module m=[9xy,0,3z3],[4z,6y,2x];
   normalize(m);
   ring s = 0,(x,y,z),(c,ls);
   module m=[9xy,0,3z3],[4z,6y,2x];
   normalize(m);
}
///////////////////////////////////////////////////////////////////////////////

///////////////////////////////////////////////////////////////////////////////
// Input: <ideal>=<f1,f2,...,fm> and <polynomial> g
// Question: Does g lie in the radical of <ideal>?
// Solution: Compute a standard basis G for <f1,f2,...,fm,gz-1> where z is a
//           new variable. Then g is contained in the radical of <ideal> <=>
//           1 is generator in G.
///////////////////////////////////////////////////////////////////////////////
proc rad_con (poly g,ideal I)
"USAGE:   rad_con(g,I); g polynomial, I ideal
RETURN:  1 (TRUE) (type int) if g is contained in the radical of I
@*       0 (FALSE) (type int) otherwise
EXAMPLE: example rad_con; shows an example
"
{ def br=basering;
  int n=nvars(br);
  int dB=degBound;
  degBound=0;
  string mp=string(minpoly);
  if (attrib(br,"global")==1)
  {
    execute("ring R=("+charstr(br)+"),(@x(1..n),@z),dp;");
  }
  else
  {
    execute("ring R=("+charstr(br)+"),(@z,@x(1..n)),(dp(1),"+ordstr(br)+");");
  }
  execute("minpoly=number("+mp+");");
  ideal irrel=@x(1..n);
  map f=br,irrel;
  poly p=f(g);
  ideal J=f(I),ideal(p*@z-1);
  J=std(J);
  degBound=dB;
  if (J[1]==1)
  {
     setring br;
     return(1);
  }
  else
  {
     setring br;
     return(0);
  }
}
example
{ "EXAMPLE:"; echo=2;
   ring R=0,(x,y,z),dp;
   ideal I=x2+y2,z2;
   poly f=x4+y4;
   rad_con(f,I);
   ideal J=x2+y2,z2,x4+y4;
   poly g=z;
   rad_con(g,I);
}
///////////////////////////////////////////////////////////////////////////////

proc lcm (id, list #)
"USAGE:   lcm(p[,q]); p int/intvec q a list of integers or
          p poly/ideal q a list of polynomials
RETURN:  the least common multiple of p and q:
@*         - of type int if p is an int/intvec
@*         - of type poly if p is a poly/ideal
EXAMPLE: example lcm; shows an example
"
{
  int k,j;
//------------------------------ integer case --------------------------------
  if( typeof(id) == "int" or typeof(id) == "intvec" )
  {
     intvec i = id;
     if (size(#)!=0)
     {
        for (j = 1; j<=size(#); j++)
        {
           if (typeof(#[j]) !="int" and typeof(#[j]) !="intvec")
           { ERROR("// ** list element must be an integer");}
           else
           { i = i,#[j]; }
        }
     }
     int p,q;
     if( i == 0 )
     {
        return(0);
     }
     for(j=1;j<=size(i);j++)
     {
       if( i[j] != 0 )
       {
         p=i[j];
         break;
       }
     }
     for (k=j+1;k<=size(i);k++)
     {
        if( i[k] !=0)
        {
           q=gcd(p,i[k]);
           p=p/q;
           p=p*i[k];
        }
     }
     if(p <0 )
     {return(-p);}
     return(p);
   }

//----------------------------- polynomial case ------------------------------
  if( typeof(id) == "poly" or typeof(id) == "ideal" )
  {
     ideal i = id;
     if (size(#)!=0)
     {
        for (j = 1; j<=size(#); j++)
        {
           if (typeof(#[j]) !="poly" and typeof(#[j]) !="ideal"
              and typeof(#[j]) !="int" and typeof(#[j]) !="intvec")
           { ERROR("// ** list element must be a polynomial");}
           else
           { i = i,#[j]; }
        }
     }
     poly p,q;
     i=simplify(i,10);
     if(size(i) == 0)
     {
        return(0);
     }
     for(j=1;j<=size(i);j++)
     {
       if(maxdeg(i[j])!= 0)
       {
         p=i[j];
         break;
       }
     }
     if(deg(p)==-1)
     {
       return(1);
     }
     for (k=j+1;k<=size(i);k++)
     {
        if(maxdeg(i[k])!=0)
        {
           q=gcd(p,i[k]);
           if(maxdeg(q)==0)
           {
                 p=p*i[k];
           }
           else
           {
              p=p/q;
              p=p*i[k];
           }
        }
     }
     return(p);
   }
}
example
{ "EXAMPLE:"; echo = 2;
   ring  r = 0,(x,y,z),lp;
   poly  p = (x+y)*(y+z);
   poly  q = (z4+2)*(y+z);
   lcm(p,q);
   ideal i=p,q,y+z;
   lcm(i,p);
   lcm(2,3,6);
   lcm(2..6);
}

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

proc content(f)
"USAGE:   content(f); f polynomial/vector
RETURN:  number, the content (greatest common factor of coefficients)
         of the polynomial/vector f
EXAMPLE: example content; shows an example
"
{
  if (f==0) { return(number(1)); }
  return(leadcoef(f)/leadcoef(cleardenom(f)));
}
example
{ "EXAMPLE:"; echo = 2;
   ring r=0,(x,y,z),(c,lp);
   content(3x2+18xy-27xyz);
   vector v=[3x2+18xy-27xyz,15x2+12y4,3];
   content(v);
}
///////////////////////////////////////////////////////////////////////////////

proc numerator(number n)
"USAGE:    numerator(n); n number
RETURN:   number, the numerator of n
SEE ALSO: denominator, content, cleardenom
EXAMPLE:  example numerator; shows an example
"
{
  poly p = cleardenom(n+var(1));
  return (number(coeffs(p,var(1))[1,1]));
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,x, dp;
  number n = 3/2;
  numerator(n);
}
///////////////////////////////////////////////////////////////////////////////

proc denominator(number n)
"USAGE:    denominator(n); n number
RETURN:   number, the denominator of n
SEE ALSO: denominator, content, cleardenom
EXAMPLE:  example denominator; shows an example
"
{
  poly p = cleardenom(n+var(1));
  return (number(coeffs(p,var(1))[2,1]));
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,x, dp;
  number n = 3/2;
  denominator(n);
}
////////////////////////////////////////////////////////////////////////

////////////////////////////////////////////////////////////////////////
// The idea of the procedures mod2id, id2mod and subrInterred is, to
// perform standard basis computation or interreduction of a submodule
// of a free module with generators gen(1),...,gen(n) over a ring R
// in a ring R[t1,...,tn]/<ti*tj> with gen(i) maped to ti
////////////////////////////////////////////////////////////////////////

proc mod2id(matrix M,intvec vpos)
"USAGE:   mod2id(M,vpos); M matrix, vpos intvec
ASSUME:  vpos is an integer vector such that gen(i) corresponds
         to var(vpos[i]).
         The basering contains variables var(vpos[i]) which do not occur
         in M.
RETURN:  ideal I in which each gen(i) from the module is replaced by
         var(vpos[i]) and all monomials var(vpos[i])*var(vpos[j]) have
         been added to the generating set of I.
NOTE:    This procedure should be used in the following situation:
         one wants to pass to a ring with new variables, say e(1),..,e(s),
         which correspond to the components gen(1),..,gen(s) of the
         module M such that e(i)*e(j)=0 for all i,j.
         The new ring should already exist and be the current ring
EXAMPLE: example mod2id; shows an example"
{
  int p = printlevel-voice+3;  // p=printlevel+1 (default: p=1)
//----------------------------------------------------------------------
// define the ideal generated by the var(vpos[i]) and set up the matrix
// for the multiplication
//----------------------------------------------------------------------
  ideal vars=var(vpos[1]);
  for(int i=2;i<=size(vpos);i++)
  {
    vars=vars,var(vpos[i]);
  }
  matrix varm[1][size(vpos)]=vars;
  if (size(vpos) > nrows(M))
  {
    matrix Mt[size(vpos)][ncols(M)];
    Mt[1..nrows(M),1..ncols(M)]=M;
    kill M;
    matrix M=Mt;
  }
//----------------------------------------------------------------------
// define the desired ideal
//----------------------------------------------------------------------
  ideal ret=vars^2,varm*M;
  return(ret);
}
example
{ "EXAMPLE:"; echo=2;
  ring r=0,(e(1),e(2),x,y,z),(dp(2),ds(3));
  module mo=x*gen(1)+y*gen(2);
  intvec iv=2,1;
  mod2id(mo,iv);
}
////////////////////////////////////////////////////////////////////////

proc id2mod(ideal i,intvec vpos)
"USAGE:   id2mod(I,vpos); I ideal, vpos intvec
RETURN:  module corresponding to the ideal by replacing var(vpos[i]) by
         gen(i) and omitting all generators var(vpos[i])*var(vpos[j])
NOTE:    * This procedure only makes sense if the ideal contains
           all var(vpos[i])*var(vpos[j]) as monomial generators and
           all other generators of I are linear combinations of the
           var(vpos[i]) over the ring in the other variables.
         * This is the inverse procedure to mod2id and should be applied
           only to ideals created by mod2id using the same intvec vpos
           (possibly after a standard basis computation)
EXAMPLE: example id2mod; shows an example"
{
  int p = printlevel-voice+3;  // p=printlevel+1 (default: p=1)
//---------------------------------------------------------------------
// Initialization
//---------------------------------------------------------------------
  int n=size(i);
  int v=size(vpos);
  matrix tempmat;
  matrix mm[v][n];
//---------------------------------------------------------------------
// Conversion
//---------------------------------------------------------------------
  for(int j=1;j<=v;j++)
  {
    tempmat=coeffs(i,var(vpos[j]));
    mm[j,1..n]=tempmat[2,1..n];
  }
  for(j=1;j<=v;j++)
  {
    mm=subst(mm,var(vpos[j]),0);
  }
  module ret=simplify(mm,10);
  return(ret);
}
example
{ "EXAMPLE:"; echo=2;
  ring r=0,(e(1),e(2),x,y,z),(dp(2),ds(3));
  ideal i=e(2)^2,e(1)*e(2),e(1)^2,e(1)*y+e(2)*x;
  intvec iv=2,1;
  id2mod(i,iv);
}
///////////////////////////////////////////////////////////////////////

proc subrInterred(ideal mon, ideal sm, intvec iv)
"USAGE:   subrInterred(mon,sm,iv);
         sm:   ideal in a ring r with n + s variables,
               e.g. x_1,..,x_n and t_1,..,t_s
         mon:  ideal with monomial generators (not divisible by
               any of the t_i) such that sm is contained in the module
               k[t_1,..,t_s]*mon[1]+..+k[t_1,..,t_s]*mon[size(mon)]
         iv:   intvec listing the variables which are supposed to be used
               as x_i
RETURN:  list l:
          l[1]=the monomials from mon in the order used
          l[2]=their coefficients after interreduction
          l[3]=l[1]*l[2]
PURPOSE:  Do interred only w.r.t. a subset of variables.
         The procedure returns an interreduced system of generators of
         sm considered as a k[t_1,..,t_s]-submodule of the free module
         k[t_1,..,t_s]*mon[1]+..+k[t_1,..,t_s]*mon[size(mon)]).
EXAMPLE: example subrInterred; shows an example
"
{
  int p = printlevel-voice+3;  // p=printlevel+1 (default: p=1)
//-----------------------------------------------------------------------
// check that mon is really generated by monomials
// and sort its generators with respect to the monomial ordering
//-----------------------------------------------------------------------
  int err;
  for(int i=1;i<=ncols(mon);i++)
  {
    if ( size(mon[i]) > 1 )
    {
      err=1;
    }
  }
  if (err==1)
  {
    ERROR("mon has to be generated by monomials");
  }
  intvec sv=sortvec(mon);
  ideal mons;
  for(i=1;i<=size(sv);i++)
  {
    mons[i]=mon[sv[i]];
  }
  ideal itemp=maxideal(1);
  for(i=1;i<=size(iv);i++)
  {
    itemp=subst(itemp,var(iv[i]),0);
  }
  itemp=simplify(itemp,10);
  def r=basering;
  string tempstr="ring rtemp=" + charstr(basering) + ",(" + string(itemp)
                               + "),(C,dp);";
//-----------------------------------------------------------------------
// express m in terms of the generators of mon and check whether m
// can be considered as a submodule of k[t_1,..,t_n]*mon
//-----------------------------------------------------------------------
  module motemp;
  motemp[ncols(sm)]=0;
  poly ptemp;
  int j;
  for(i=1;i<=ncols(mons);i++)
  {
    for(j=1;j<=ncols(sm);j++)
    {
      ptemp=sm[j]/mons[i];
      motemp[j]=motemp[j]+ptemp*gen(i);
    }
  }
  for(i=1;i<=size(iv);i++)
  {
    motemp=subst(motemp,var(iv[i]),0);
  }
  matrix monmat[1][ncols(mons)]=mons;
  ideal dummy=monmat*motemp;
  for(i=1;i<=size(sm);i++)
  {
    if(sm[i]-dummy[i]!=0)
    {
      ERROR("the second argument is not a submodule of the assumed structure");
    }
  }
//----------------------------------------------------------------------
// do the interreduction and convert back
//----------------------------------------------------------------------
  execute(tempstr);
  def motemp=imap(r,motemp);
  intvec save=option(get);
  option(redSB);
  motemp=interred(motemp);
  option(set,save);
  setring r;
  kill motemp;
  def motemp=imap(rtemp,motemp);
  //list ret=monmat,motemp,monmat*motemp;
  module motemp2=motemp;
  for(i=1;i<=ncols(motemp2);i++)
  {
    motemp2[i]=cleardenom(motemp2[i]);
  }
  module motemp3=monmat*motemp;
  for(i=1;i<=ncols(motemp3);i++)
  {
    motemp3[i]=cleardenom(motemp3[i]);
  }
  list ret=monmat,motemp2,matrix(motemp3);
  return(ret);
}
example
{ "EXAMPLE:"; echo=2;
  ring r=0,(x,y,z),dp;
  ideal i=x^2+x*y^2,x*y+x^2*y,z;
  ideal j=x^2+x*y^2,x*y,z;
  ideal mon=x^2,z,x*y;
  intvec iv=1,3;
  subrInterred(mon,i,iv);
  subrInterred(mon,j,iv);
}

///////////////////////////////////////////////////////////////////////////////
// moved here from nctools.lib
// This procedure calculates the Newton diagram of the polynomial f
// The output is a intmat M, each row of M is the exp of a monomial in f
////////////////////////////////////////////////////////////////////////
proc newtonDiag(poly f)
"USAGE:  newtonDiag(f); f a poly
RETURN:  intmat
PURPOSE: compute the Newton diagram of f
NOTE:    each row is the exponent of a monomial of f
EXAMPLE: example newtonDiag; shows examples
"{
  int n = nvars(basering);
  intvec N=0;
  if ( f != 0 )
  {
    while ( f != 0 )
    {
      N = N, leadexp(f);
      f = f-lead(f);
    }
  }
  else
  {
    N=N, leadexp(f);
  }
  N = N[2..size(N)]; // Deletes the zero added in the definition of T
  intmat M=intmat(N,(size(N)/n),n); // Conversion from vector to matrix
  return (M);
}
example
{
  "EXAMPLE:";echo=2;
  ring r = 0,(x,y,z),lp;
  poly f = x2y+3xz-5y+3;
  newtonDiag(f);
}

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