source: git/Singular/LIB/JMBTest.lib

//////////////////////////////////////////////////////////////////////
version="version JMBTest.lib 4.4.0.0 Nov_2023 "; // $Id$
category="Algebraic Geometry";
// summary description of the library
info="
LIBRARY:   JMBTest.lib  A library for Singular which performs JM basis test.
AUTHOR:    Michela Ceria, email: michela.ceria@unito.it

SEE ALSO:  JMSConst_lib
KEYWORDS:  J-marked schemes

OVERVIEW:
The library performs the J-marked basis test, as described in [CR], [BCLR].
Such a test is performed via the criterion explained in [BCLR],
concerning Eliahou-Kervaire polynomials (EK from now on).
We point out that all the polynomials are homogeneous
and they must be arranged by degree.
The fundamental steps are the following:@*
-construct the Vm polynomials, via the algorithm VConstructor
explained in [CR];@*
-construct the Eliahou-Kervaire polynomials defined in [BCLR];@*
-reduce the  Eliahou-Kervaire polynomials using the Vm's;@*
-if it exist an Eliahou-Kervaire polynomial such that its reduction
mod Vm is different from zero, the given one is not a J-Marked basis.

The algorithm terminates only if the ordering is ip.
Anyway, the number of reduction steps is bounded.

REFERENCES:
[CR] Francesca Cioffi, Margherita Roggero,Flat Families by Strongly
Stable Ideals and a Generalization of Groebner Bases,
J. Symbolic Comput. 46,  1070-1084, (2011).@*
[BCLR] Cristina Bertone, Francesca Cioffi, Paolo Lella,
Margherita Roggero, Upgraded methods for the effective
computation of marked schemes on a strongly stable ideal,
Journal of Symbolic Computation
(2012), http://dx.doi.org/10.1016/j.jsc.2012.07.006 @*

PROCEDURES:
  Minimus(ideal)            minimal variable in an ideal
  Maximus(ideal)               maximal variable in an ideal
  StartOrderingV(list,list)     ordering of polynomials as in [BCLR]
  TestJMark(list)           tests whether we have a J-marked basis
";
LIB "qhmoduli.lib";
LIB "monomialideal.lib";
LIB "ring.lib";
////////////////////////////////////////////////////////////////////
static proc mod_init()
{
newstruct("jmp", "poly h, poly t");
}

////////////////////////////////////////////////////////////////////
proc Terns(list G, int c)
"USAGE:    Terns(G,c); G list, c int
RETURN:   list: T
NOTE:     Input is a list of J-marked  polynomials
(arranged by degree) and an integer.
EXAMPLE:  example Terns; shows an example"
{
list T=list();
int z;
 for(int k=1; k<=size(G[c]);k=k+1)
    {
//Loop on G[c] making positions of polynomials in G[c]
      z=size(T);
      T=insert(T,list(1,c,k) ,size(T));
    }
return(T);
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0, (x,y,z), ip;
jmp r1;
r1.h=z^3;
r1.t=poly(0);
jmp r2;
r2.h=z^2*y;
r2.t=poly(0);
jmp r3;
r3.h=z*y^2 ;
r3.t=-x^2*y;
jmp r4;
r4.h=y^5;
r4.t=poly(0);
list G2F=list(list(r1,r2,r3),list(r4));
Terns(G2F, 1);
Terns(G2F, 2);
}
////////////////////////////////////////////////////////////////////
proc VConst(list G, int c)
"USAGE:    VConst(G, c); G list, c int
RETURN:   list: V
NOTES: this procedure computes the Vm polynomials following the
algorithm in [CR],but it only keeps in memory the monomials by
which the G's must be multiplied and their positions.
EXAMPLE:  example VConst; shows an example"
{
jmp f=G[1][1];
int aJ=deg(f.h);
// minimal degree of polynomials in G
list V=list();
V[1]=Terns(G,1);
// V[1]=G[1] (keeping in memory only [head, position])
//print(c-aJ+1);
int i;
int j;
int m;
list OO;
jmp p;
  for(m=2; m<=c-aJ+1; m=m+1)
  {
     //print("entro nel form");
     if(m>size(G))
        {V[m]=list();
//If we have not G[m] we insert a list()
        //print("vuota prima");
        }
     else
        {V[m]=Terns(G,m);
        //print("piena prima");
        }
     for(i=1; i<nvars(basering)+1; i=i+1)
        {
          //print("entrata fori");
          //print(i);
          for(j=1; j<=size(V[m-1]); j=j+1)
             {
                   p=G[V[m-1][j][2]][V[m-1][j][3]];
                  //print(p.h);
                  //print(p.t);
                  //print(var(i));
                  //print(Minimus(V[m-1][j][1]*p.h));
               if(var(i)<=Minimus(variables(V[m-1][j][1]*p.h)))
                  {
//Can I multiply by the current variable?
                    //print("minoremin");
                    //print("fin qui ci sono");
                     //print(V[m-1][j][1]);
                     OO=list(var(i)*V[m-1][j][1],V[m-1][j][2],V[m-1][j][3]);
                    V[m]=insert(V[m], OO ,size(V[m]));
                  }
             }
        }
  }
return (V);}
example
{ "EXAMPLE:"; echo = 2;
 ring r=0, (x,y,z), ip;
jmp r1;
r1.h=z^3;
r1.t=poly(0);
jmp r2;
r2.h=z^2*y;
r2.t=poly(0);
jmp r3;
r3.h=z*y^2 ;
r3.t=-x^2*y;
jmp r4;
r4.h=y^5;
r4.t=poly(0);
list G2F=list(list(r1,r2,r3),list(r4));
VConst(G2F,4,basering);}
////////////////////////////////////////////////////////////////////
proc Minimus(ideal L)
"USAGE:    Minimus(L); G list, c int
RETURN:   list: V
NOTES: it returns the minimal variable generating the ideal L.@*
The input must be an ideal generated by variables.
EXAMPLE:  example Minimus; shows an example"
{
poly min=L[1];
int i;
for(i=2;i<=size(L); i++)
  {
    if(L[i]<min){min=L[i];}
  }
return(min);
}
example
{ "EXAMPLE:"; echo = 2;
 ring r=0, (x,y,z), ip;
ideal I=y,x,z;
Minimus(I);
}
////////////////////////////////////////////////////////////////////
proc Maximus(ideal L)
"USAGE:    Maximus(L); G list, c int
RETURN:   list: V
NOTES: it returns the maximal variable generating the ideal L.@*
The input must be an ideal generated by variables.
EXAMPLE:  example Maximus; shows an example"
{
poly max=L[1];
int i;
for(i=2;i<=size(L); i++)
  {
    if(L[i]>max){max=L[i];}
  }
return(max);
}
example
{ "EXAMPLE:"; echo = 2;
 ring r=0, (x,y,z), ip;
ideal I=y,x,z;
Maximus(I);
}
////////////////////////////////////////////////////////////////////
proc GJmpMins(jmp P, jmp Q)
"USAGE:    GJmpMins(P,Q); P jmp, Q jmp
RETURN:   int: d
EXAMPLE:  example GJmpMins; shows an example"
{
  int d=1;
//-1=lower, 0=equal, 1=higher
//At the beginning suppose Q is higher
  if(deg(P.h)<deg(Q.h))
     {
//Compare degrees;
      d=-1;
     }
  if(deg(P.h)==deg(Q.h))
     {
      if(P.h==Q.h)
         {
          if(P.t==Q.t)
              {
//head=tail
               d=0;
              }
         }
     else
        {
//print(Minimus(variables(P.h/gcdMon(P.h,Q.h))));
//print(Minimus(variables(Q.h/gcdMon(P.h,Q.h))));
          if(Minimus(variables(P.h/gcdMon(P.h,Q.h)))<Minimus(variables(Q.h/gcdMon(P.h,Q.h))))
             {
             d=-1;
            //print("Per Indice");
             }
        }
     }
 return(d);
}
example
{ "EXAMPLE:"; echo = 2;
 ring r=0, (x,y,z), ip;
jmp p1;
p1.h=poly(1);
p1.t=poly(1);
jmp p2;
p2.h=x^2;
p2.t=poly(0);
jmp p3;
p3.h=x;
p3.t=poly(0);
 GJmpMins(p1, p2);
 GJmpMins(p2, p3);
GJmpMins(p1,p1);
}
////////////////////////////////////////////////////////////////////
proc TernCompare(list A, list B, list G)
"USAGE:    TernCompare(A,B,C); A list, B list, G list
RETURN:   int: d
NOTE:     A and B are terns, while G is the given list of
J-marked polynomials.
EXAMPLE:  example TernCompare; shows an example"
{
int d=-1;
//Start: A<B
if(A[1]==B[1])
  {
   if(A[2]==B[2]&& A[3]==B[3])
     {
      //print("Uguali");
      d=0;
     }
   else
     {
jmp g1=G[A[2]][A[3]];
jmp g2=G[B[2]][B[3]];
       if(GJmpMins(g1, g2)==1)
          {
            //print("Maggiore per il G");
            d=1;
          }
     }
  }
else
  {
    if(A[1]>B[1])
      {
//the ordering MUST be ip
        //print("Maggiore per Lex");
        d=1;
      }
  }
return(d);
}
example
{ "EXAMPLE:"; echo = 2;
 ring r=0, (x,y,z), ip;
jmp r1;
r1.h=z^3;
r1.t=poly(0);
jmp r2;
r2.h=z^2*y;
r2.t=poly(0);
jmp r3;
r3.h=z*y^2 ;
r3.t=-x^2*y;
jmp r4;
r4.h=y^5;
r4.t=poly(0);
list G2F=list(list(r1,r2,r3),list(r4));
TernCompare([1,1,1],[x,1,1],G2F);
}
////////////////////////////////////////////////////////////////////
proc MinOfV(list V, list G)
"USAGE:    Minimal(V,G); V list, G list
RETURN:   int: R
NOTE:     Input=lista(terne), G.
EXAMPLE:  example Minimal; shows an example"
{
//Minimal element for a given degree
list R=list();
list MIN=V[1];
int h=1;
int i;
for(i=2; i<=size(V); i++)
    {
//I consider the first as minimum
//If I find something smaller I change minimum
     if(TernCompare(V[i],MIN,G)<=0)
       {
        MIN=V[i];
        h=i;
       }
    }
//Return: [minimum,position of the minimum]
R=MIN,h;
return(R);
}
example
{ "EXAMPLE:"; echo = 2;
 ring r=0, (x,y,z), ip;
jmp r1;
r1.h=z^3;
r1.t=poly(0);
jmp r2;
r2.h=z^2*y;
r2.t=poly(0);
jmp r3;
r3.h=z*y^2 ;
r3.t=-x^2*y;
jmp r4;
r4.h=y^5;
r4.t=poly(0);
list G2F=list(list(r1,r2,r3),list(r4));
MinOfV(VConst(G2F,4,basering)[1],G2F);
}
////////////////////////////////////////////////////////////////////
proc OrderingV(list V,list G,list R)
"USAGE:   OrderingV(V,G,R); V list, G list, R list
RETURN:   list: R
NOTE:     Input: Vm,G,emptylist
EXAMPLE:  example OrderingV; shows an example"
{
//Order V[m]
//R  will contain results but at the beginning it is empty
list M=list();
if(size(V)==1)
  {
    R=insert(R,V[1],size(R));
  }
else
  {
   M=MinOfV(V,G);
   R=insert(R,M[1],size(R));
   V=delete(V,M[2]);
//recursive call
   R=OrderingV(V,G,R);
  }
return(R);
}
example
{ "EXAMPLE:"; echo = 2;
 ring r=0, (x,y,z), ip;
jmp r1;
r1.h=z^3;
r1.t=poly(0);
jmp r2;
r2.h=z^2*y;
r2.t=poly(0);
jmp r3;
r3.h=z*y^2;
r3.t=-x^2*y;
jmp r4;
r4.h=y^5;
r4.t=poly(0);
list G2F=list(list(r1,r2,r3),list(r4));
OrderingV(VConst(G2F,4,basering)[1],G2F,list());
}
////////////////////////////////////////////////////////////////////
proc StartOrderingV(list V,list G)
"USAGE:   StartOrdina(V,G); V list, G list
RETURN:   list: R
NOTE:     Input Vm,G. This procedure uses OrderingV to get
the ordered polynomials as in [BCLR].
EXAMPLE:  example StartOrderingV; shows an example"
{
 return(OrderingV(V,G, list()));
}
example
{ "EXAMPLE:"; echo = 2;
 ring r=0, (x,y,z), ip;
jmp r1;
r1.h=z^3;
r1.t=poly(0);
jmp r2;
r2.h=z^2*y;
r2.t=poly(0);
jmp r3;
r3.h=z*y^2;
r3.t=-x^2*y;
jmp r4;
r4.h=y^5;
r4.t=poly(0);
list G2F=list(list(r1,r2,r3),list(r4));
StartOrderingV(VConst(G2F,4,basering)[1],G2F);
}
////////////////////////////////////////////////////////////////////
proc Multiply(list L, list G)
"USAGE:    moltiplica(L,G); L list, G list
RETURN:   jmp: K
NOTE:     Input: a 3-ple,G. It performs the product associated
to the 3-uple.
EXAMPLE:  example Multiply; shows an example"
{
jmp g=G[L[2]][L[3]];
jmp K;
K.h=L[1]*g.h;
K.t=L[1]*g.t;
 return(K);
}
example
{ "EXAMPLE:"; echo = 2;
 ring r=0, (x,y,z), ip;
 list P=x^2,1,1;
jmp r1;
r1.h=z^3;
r1.t=poly(0);
jmp r2;
r2.h=z^2*y;
r2.t=poly(0);
jmp r3;
r3.h=z*y^2 ;
r3.t=-x^2*y;
jmp r4;
r4.h=y^5;
r4.t=poly(0);
list G2F=list(list(r1,r2,r3),list(r4));
Multiply(P,G2F);
}
////////////////////////////////////////////////////////////////////
proc IdealOfV(list V)
"USAGE:    IdealOfV(V); V list
RETURN:   ideal: I
NOTES: this procedure takes a list of Vm's of a certain degree
and construct their ideal, multiplying the head by the weighted
variable t.
EXAMPLE:  example IdealOfV; shows an example"
{
ideal I=0;
int i;
if (size(V)!=0)
  {
   list M=list();
jmp g;
   for(i=1; i<= size(V); i++)
       {
         g=V[i];
         g.h=t*g.h;
         M[i]=g.h+g.t;
       }
   I=M[1..size(M)];
  //I=std(I);
  }
return(I);
}
example
{ "EXAMPLE:"; echo = 2;
 ring r=0, (x,y,z,t), ip;
jmp r1;
r1.h=z^3;
r1.t=poly(0);
jmp r2;
r2.h=z^2*y;
r2.t=poly(0);
jmp r3;
r3.h=z*y^2 ;
r3.t=-x^2*y;
jmp r4;
r4.h=y^5;
r4.t=poly(0);
list G2F=list(list(r1,r2,r3),list(r4));
IdealOfV(G2F[1]);
}
////////////////////////////////////////////////////////////////////
proc NewWeight(int n)
"USAGE:    NewWeight(n); n int
RETURN:   intvec: u
EXAMPLE:  example NewWeight; shows an example"
{
intvec u=0;
u[n]=1;
return(u);
}
example
{ "EXAMPLE:"; echo = 2;
 NewWeight(3);
}
////////////////////////////////////////////////////////////////////
proc FinalVm(list V1 , list G1 ,def r)
"USAGE:     FinalVm(V1, G1, r);  V1 list,  G1 list , r
RETURN:   intvec: u
EXAMPLE:  example NewWeight; shows an example"
{
  //multiply and reduce, degree by degree
  intvec u=NewWeight(nvars(r)+1);
  list L=ring_list(r);
  L[2]=insert(L[2],"t",size(L[2]));
  list ordlist="a",u;
  L[3]=insert(L[3],ordlist,0);
  def H=ring(L);
  list M=list();
  jmp p;
  list N;
  poly q;
  poly s;
  int i;
  int j;
  for(i=1; i<=size(G1); i++)
  {
    N=list();
    for(j=1; j<=size(G1[i]); j++)
    {
      p=G1[i][j];
      q=p.h;
      s=p.t;
      N[j]=list(q,s);
    }
    M[i]=N;
  }
  p.h=poly(0);
  p.t=poly(0);
  setring H;
  list R=list();
  list S=list();
  //print("anello definito");
  def V=imap(r,V1);
  //def G=imap(r,G1);
  //print(V);
  def MM=imap(r,M);
  list G=list();
  list N=list();
  for(i=1; i<=size(MM); i++)
  {
    for(j=1; j<=size(MM[i]); j++)
    {
      p.h=MM[i][j][1];
      p.t=MM[i][j][2];
      N[j]=p;
    }
    G[i]=N;
  }
  ideal I=0;
  jmp LL;
  jmp UU;
  for(i=1; i<=size(V);i++)
  {
    R[i]=list();
    S[i]=list();
    I=0;
    for(j=1;j<=size(V[i]); j++)
    {
      LL=Multiply(V[i][j],G);
      LL.t=reduce(t*LL.t,I);
      //I only reduce the tail
      LL.t=subst(LL.t,t,1);
      S[i]=insert(S[i],LL,size(S[i]));
      LL.h=t*LL.h;
      R[i]=insert(R[i],LL,size(R[i]));
      UU=R[i][j];
      I=I+ideal(UU.h+UU.t);
      attrib(I,"isSB",1);
    }
  }
  list M=list();
  poly q;
  poly s;
  for(i=1; i<=size(S); i++)
  {
    N=list();
    for(j=1; j<=size(S[i]); j++)
    {
      p=S[i][j];
      q=p.h;
      s=p.t;
      N[j]=list(q,s);
    }
    M[i]=N;
  }
  p.h=poly(0);
  p.t=poly(0);
  setring r;
  def MM=imap(H,M);
  list MMM=list();
  for(i=1; i<=size(MM); i++)
  {
    N=list();
    for(j=1; j<=size(MM[i]); j++)
    {
      p.h=MM[i][j][1];
      p.t=MM[i][j][2];
      N[j]=p;
    }
    MMM[i]=N;
  }
  return(MMM);
}
example
{ "EXAMPLE:"; echo = 2;
 ring r=0, (x,y,z), ip;
 jmp r1;
 r1.h=z^3;
 r1.t=poly(0);
 jmp r2;
 r2.h=z^2*y;
 r2.t=poly(0);
 jmp r3;
 r3.h=z*y^2 ;
 r3.t=-x^2*y;
 jmp r4;
 r4.h=y^5;
 r4.t=poly(0);
 list G2F=list(list(r1,r2,r3),list(r4));
 FinalVm(VConst(G2F,6,r) , G2F, r);
}
////////////////////////////////////////////////////////////////////
proc ConstructorMain(list G, int c,def r)
"USAGE:    Costruttore(G,c); G list, c int
RETURN:   list: R
NOTE:     At the end separated by degree.
EXAMPLE:  example Costruttore; shows an example"
{
list V=list();
V= VConst(G,c);
//V non ordered
list L=list();
list R=list();
int i;
// head, position
//order the different degrees
for(i=1; i<=size(V); i++)
   {
    L[i]=StartOrderingV(V[i], G);
   }
//multiply and reduce
//print("Ordinare");
R=FinalVm(L, G, r);
//print("FinalVm");
return(R);
}
example
{ "EXAMPLE:"; echo = 2;
 ring r=0, (x,y,z), ip;
jmp r1;
r1.h=z^3;
r1.t=poly(0);
jmp r2;
r2.h=z^2*y;
r2.t=poly(0);
jmp r3;
r3.h=z*y^2 ;
r3.t=-x^2*y;
jmp r4;
r4.h=y^5;
r4.t=poly(0);
list G2F=list(list(r1,r2,r3),list(r4));
ConstructorMain(G2F,6,r);
}
////////////////////////////////////////////////////////////////////
proc EKCouples(jmp A, jmp B)
"USAGE:    CoppiaEK(A,B); A list, B list
RETURN:   list: L
NOTE:     At the end the monomials involved by EK.
EXAMPLE:  example EKCouples; shows an example"
{
poly E;
list L=0,0;
string s=varstr(basering);
list VVV=varstr(basering);
//L will contain results
poly h=Minimus(variables(A.h));
int l=findvars(h,1)[2][1];
if(l!=nvars(basering))
 {
  for(int j=l+1;j<=nvars(basering); j++)
   {
    E=var(j)*A.h/B.h;
//Candidate for * product
    if(E!=0)
      {
        //print("primo if passato");
        if(Minimus(variables(B.h))>=Maximus(variables(E)))
           {
//Does it work with * ?
            //print("secondo if passato");
            L[1]=j;
            L[2]=E;
            break;
           }
      }
   }
  }
return (L);
}
example
{ "EXAMPLE:"; echo = 2;
 ring r=0, (x,y,z), ip;
jmp A;
A.h=y*z^2;
A.t=poly(0);
jmp B;
B.h=y^2*z;
B.t=poly(0);
EKCouples(A,B);
EKCouples(B,A);
}
////////////////////////////////////////////////////////////////////
proc EKPolys(list G)
"USAGE:    PolysEK(G); G list
RETURN:   list: EK, list: D
NOTE:     At the end EK polynomials and their degrees

EXAMPLE:  example PolysEK; shows an example"
{
list D=list();
list C=list();
list N=0,0;
list EK=list();
int i;
int j;
int k;
int l;
jmp p;
for(i=1; i<=size(G); i++)
   {
     for(j=1; j<=size(G[i]); j++)
        {
         for(k=1; k<=size(G); k++)
             {
               for(l=1; l<=size(G[k]); l++)
                  {
                     if(i!=k||j!=l)
                       {
//Loop on polynomials
                        C=EKCouples(G[i][j], G[k][l]);
                        if(C[2]!=0)
                          {
                           C=insert(C,list(i,j,k,l),size(C));
                           EK=insert(EK,C,size(EK));
                           p=G[k][l];
                           D=insert(D,deg(C[2]*p.h),size(D));
                          }
                       }
                  }
             }
        }
   }
//Double Return
return(EK, D);
}
example
{ "EXAMPLE:"; echo = 2;
  ring r=0, (x,y,z), ip;
jmp r1;
r1.h=z^3;
r1.t=poly(0);
jmp r2;
r2.h=z^2*y;
r2.t=poly(0);
jmp r3;
r3.h=z*y^2;
r3.t=-x^2*y;
jmp r4;
r4.h=y^5;
r4.t=poly(0);
list G2F=list(list(r1,r2,r3),list(r4));
EKPolys(G2F);
}
////////////////////////////////////////////////////////////////////
proc EKPolynomials(list EK, list G)
"USAGE:     EKPolynomials(EK,G); EK list, G list
RETURN:   list: p
NOTE:     At the end I obtain the EK polynomials and
their degrees.
EXAMPLE:  example SpolyEK; shows an example"
{
jmp u=G[EK[3][1]][EK[3][2]];
jmp q=G[EK[3][3]][EK[3][4]];
return(var(EK[1])*(u.h+u.t)-EK[2]*(q.h+q.t));
}
example
{ "EXAMPLE:"; echo = 2;
 ring r=0, (x,y,z), ip;
jmp r1;
r1.h=z^3;
r1.t=poly(0);
jmp r2;
r2.h=z^2*y;
r2.t=poly(0);
jmp r3;
r3.h=z*y^2;
r3.t=-x^2*y;
jmp r4;
r4.h=y^5;
r4.t=poly(0);
list G2F=list(list(r1,r2,r3),list(r4));
list EK,D=EKPolys(G2F);
EKPolynomials(EK[1],G2F);
}
////////////////////////////////////////////////////////////////////
proc TestJMark(list G1,def r)
"USAGE:    TestJMark(G);  G list
RETURN:    int: i
NOTE:
This procedure performs J-marked basis test.@*
The input is a list of J-marked polynomials (jmp) arranged
by degree, so G1 is a list of list.@*
The output is a boolean evaluation:
True=1/False=0
EXAMPLE:  example TestJMark; shows an example"
{
  int flag=1;
  if(size(G1)==1 && size(G1[1])==1)
  {
    //Hypersurface
    print("Only One Polynomial");
    flag=1;
  }
  else
  {
    int d=0;
    list EK,D=EKPolys(G1);
    //I found EK couples
    int massimo=Max(D);
    list V1=ConstructorMain(G1,massimo,r);
    jmp mi=V1[1][1];
    int minimo=Min(deg(mi.h));
    intvec u=NewWeight(nvars(r)+1);
    list L=ring_list(r);
    L[2]=insert(L[2],"t",size(L[2]));
    list ordlist="a",u;
    L[3]=insert(L[3],ordlist,0);
    def H=ring(L);
    list JJ=list();
    jmp pp;
    jmp qq;
    int i;
    int j;
    list NN;
    for(i=size(V1);i>0;i--)
    {
      NN=list();
      for(j=size(V1[i]);j>0;j--)
      {
        pp=V1[i][j];
        NN[j]=list(pp.h,pp.t);
      }
      JJ[i]=NN;
    }
    list KK=list();
    list UU=list();
    //jmp qq;
    for(i=size(G1);i>0;i--)
    {
      for(j=size(G1[i]);j>0;j--)
      {
        qq=G1[i][j];
        UU[j]=list(qq.h,qq.t);
      }
      KK[i]=UU;
    }
    setring H;
    //I defined the new ring with the weighted
    //variable t
    poly p;
    //print("anello definito");
    def JJJ=imap(r,JJ);
    def EK=imap(r,EK);
    list V=list();
    jmp fp;
    //int i;
    //int j;
    list N;
    for(i=size(JJJ); i>0; i--)
    {
      N=list();
      for(j=size(JJJ[i]); j>0; j--)
      {
        fp.h=JJJ[i][j][1];
        fp.t=JJJ[i][j][2];
        N[j]=fp;
      }
      V[i]=N;
    }
    def KKJ=imap(r,KK);
    list G=list();
    list U=list();
    for(i=1; i<=size(KKJ); i++)
    {
      for(j=1; j<=size(KKJ[i]); j++)
      {
        fp.h=KKJ[i][j][1];
        fp.t=KKJ[i][j][2];
        U[j]=fp;
      }
      G[i]=U;
    }
    //I imported in H everything I need
    poly q;
    ideal I;
    for(j=1; j<=size(EK);j++)
    {
      d=D[j];
      p=EKPolynomials(EK[j],G);
      I=IdealOfV(V[d-minimo+1]);
      attrib(I,"isSB",1);
      q=reduce(t*p,I);
      q=subst(q,t,1);
      //I reduce all the EK polynomials
      //       q=RiduzPoly(V[d-minimo+1], p);
      if(q!=0)
      {
        //check whether reduction is 0
        print("NOT A BASIS");
        flag=0;
        break;
      }
    }
  }
  setring r;
  //typeof(flag);
  return(flag);
}
example
{ "EXAMPLE:"; echo = 2;
 ring r=0, (x,y,z), ip;
 jmp r1;
 r1.h=z^3;
 r1.t=poly(0);
 jmp r2;
 r2.h=z^2*y;
 r2.t=poly(0);
 jmp r3;
 r3.h=z*y^2 ;
 r3.t=-x^2*y;
 jmp r4;
 r4.h=y^5;
 r4.t=poly(0);
 list G2F=list(list(r1,r2,r3),list(r4));
 TestJMark(G2F,r);
}