Changeset 590998 in git

Timestamp:

Dec 12, 2000, 2:50:08 PM (23 years ago)

Author:

Anne Frühbis-Krüger <anne@…>

Branches:

(u'fieker-DuVal', '117eb8c30fc9e991c4decca4832b1d19036c4c65')(u'spielwiese', '38dfc5131670d387a89455159ed1e071997eec94')

Children:

8893c4b18dc23caf911d7ec697a8a89d60b36248

Parents:

ebbe4a2df3136602d7e171ea5a1f83cca4c3e013

Message:

*anne: added mod2id id2mod and subrInterred


git-svn-id: file:///usr/local/Singular/svn/trunk@4875 2c84dea3-7e68-4137-9b89-c4e89433aadc

File:

• Singular/LIB/poly.lib (modified) (3 diffs)

Singular/LIB/poly.lib

@@ -1 @@
-// $Id: poly.lib,v 1.26 2000-05-12 12:17:16 krueger Exp $
+// $Id: poly.lib,v 1.27 2000-12-12 13:50:08 anne Exp $
 //(GMG, last modified 22.06.96)
 //(obachman: 17.12.97 -- added katsura)
-///////////////////////////////////////////////////////////////////////////////
-
-version="$Id: poly.lib,v 1.26 2000-05-12 12:17:16 krueger Exp $";
+//(anne: 11.12.2000 -- added mod2id, id2mod, subrInterred)
+///////////////////////////////////////////////////////////////////////////////
+
+version="$Id: poly.lib,v 1.27 2000-12-12 13:50:08 anne Exp $";
 info="
 LIBRARY:  poly.lib      PROCEDURES FOR MANIPULATING POLYS, IDEALS, MODULES
@@ -25 +26 @@
  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
+ subrInterred(i1,i2,iv);interred w.r.t. a subset of variables
           (parameters in square brackets [] are optional)
 ";
@@ -718 +722 @@
   denominator(n);
 }
+////////////////////////////////////////////////////////////////////////
+
+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
+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
+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
+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
+NOTE:      * use this procedure only makes sense if the ideal contains
+             all var(vpos[i])*var(vpos[j]) as monomial generators and
+             all other generators 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)
+RETURN:    module corresponding to the ideal by replacing var(vpos[i]) by
+           gen(i) and omitting all generators var(vpos[i])*var(vpos[j])
+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
+                one 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]
+               (interreduced system of generators of sm seen as a submodule
+                of 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);
+  motemp=interred(motemp);
+  setring r;
+  kill motemp;
+  def motemp=imap(rtemp,motemp);
+  list ret=monmat,motemp,monmat*motemp;
+  for(i=1;i<=ncols(ret[2]);i++)
+  {
+    ret[2][i]=cleardenom(ret[2][i]);
+  }
+  for(i=1;i<=ncols(ret[3]);i++)
+  {
+    ret[3][i]=cleardenom(ret[3][i]);
+  }
+  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);
+}
+////////////////////////////////////////////////////////////////////////