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version="version sagbi.lib 4.0.0.0 Jun_2013 "; // $Id: 1a57ac39190e7a55f6e7ea39972d0eb26c02b191 $
category="Commutative Algebra";
info="
LIBRARY: sagbi.lib Compute SAGBI basis (subalgebra bases analogous to Groebner bases for ideals) of a subalgebra
AUTHORS: Jan Hackfeld, Jan.Hackfeld@rwth-aachen.de
Gerhard Pfister, pfister@mathematik.uni-kl.de
Viktor Levandovskyy, levandov@math.rwth-aachen.de
OVERVIEW:
SAGBI stands for 'subalgebra bases analogous to Groebner bases for ideals'.
SAGBI bases provide important tools for working with finitely presented
subalgebras of a polynomial ring. Note, that in contrast to Groebner
bases, SAGBI bases may be infinite.
REFERENCES:
Ana Bravo: Some Facts About Canonical Subalgebra Bases,
MSRI Publications 51, p. 247-254
PROCEDURES:
sagbiSPoly(A [,r,m]); computes SAGBI S-polynomials of A
sagbiReduce(I,A [,t,mt]); performs subalgebra reduction of I by A
sagbi(A [,m,t]); computes SAGBI basis for A
sagbiPart(A,k[,m]); computes partial SAGBI basis for A
algebraicDependence(I,it); performs iterations of SAGBI for algebraic dependencies of I
SEE ALSO: algebra_lib
";
LIB "elim.lib";
LIB "toric.lib";
LIB "algebra.lib";
LIB "ring.lib";
//////////////////////////////////////////////////////////////////////////////
static proc assumeQring()
{
if (ideal(basering) != 0)
{
ERROR("This function has not yet been implemented over qrings.");
}
}
static proc uniqueVariableName (string variableName)
{
//Adds character "@" at the beginning of variableName until this name ist unique
//(not contained in the names of the ring variables or description of the coefficient field)
string ringVars = charstr(basering) + "," + varstr(basering);
while (find(ringVars,variableName) <> 0)
{
variableName="@"+variableName;
}
return(variableName);
}
static proc extendRing(def r, ideal leadTermsAlgebra, int method) {
/* Extends ring r with additional variables. If k=ncols(leadTermsAlgebra) and
* r contains already m additional variables @y, the procedure adds k-m variables
* @y(m+1)...@y(k) to the ring.
* The monomial ordering of the extended ring depends on method.
* Important: When calling this function, the basering (where algebra is defined) has to be active
*/
def br=basering;
int i;
ideal varsBasering=maxideal(1);
int numTotalAdditionalVars=ncols(leadTermsAlgebra);
string variableName=uniqueVariableName("@y");
//get a variable name different from existing variables
//-------- extend current baserring r with new variables @y,
// one for each new element in ideal algebra -------------
setring r;
list l = ringlist(r);
for (i=nvars(r)-nvars(br)+1; i<=numTotalAdditionalVars;i++)
{
l[2][i+nvars(br)]=string(variableName,"(",i,")");
}
if (method>=0 && method<=1)
{
if (nvars(r)==nvars(br))
{ //first run of spolynomialGB in sagbi construction algorithms
l[3][size(l[3])+1]=l[3][size(l[3])]; //save module ordering
l[3][size(l[3])-1]=list("dp",intvec(1:numTotalAdditionalVars));
}
else
{ //overwrite existing order for @y(i) to only get one block for the @y
l[3][size(l[3])-1]=list("dp",intvec(1:numTotalAdditionalVars));
}
}
// VL : todo noncomm case: correctly use l[5] and l[6]
// that is update matrices
// at the moment this is troublesome, so use nc_algebra call
// see how it done in algebraicDependence proc // VL
def rNew=ring(l);
setring br;
return(rNew);
}
static proc stdKernPhi(ideal kernNew, ideal kernOld, ideal leadTermsAlgebra,int method)
{
/* Computes Groebner basis of kernNew+kernOld, where kernOld already is a GB
* and kernNew contains elements of the form @y(i)-leadTermsAlgebra[i] added to it.
* The techniques chosen is specified by the integer method
*/
ideal kern;
attrib(kernOld,"isSB",1);
if (method==0)
{
kernNew=reduce(kernNew,kernOld);
kern=kernOld+kernNew;
kern=std(kern);
//kern=std(kernOld,kernNew); //Found bug using this method.
// TODO Change if bug is removed
//this call of std return Groebner Basis of ideal kernNew+kernOld
// given that kernOld is a Groebner basis
}
if (method==1)
{
kernNew=reduce(kernNew,kernOld);
kern=slimgb(kernNew+kernOld);
}
return(kern);
}
static proc spolynomialsGB(ideal algebra,def r,int method)
{
/* This procedure does the actual S-polynomial calculation using Groebner basis methods and is
* called by the procedures sagbiSPoly,sagbi and sagbiPart. As this procedure is called
* at each step of the SAGBI construction algorithm, we can reuse the information already calculated
* which is contained in the ring r. This is done in the following order
* 1. If r already contain m additional variables and m'=number of elements in algebra, extend r with variables @y(m+1),...,@y(m')
* 2. Transfer all objects to this ring, kernOld=kern is the Groebnerbasis already computed
* 3. Define ideal kernNew containing elements of the form leadTermsAlgebra(m+1)-@y(m+1),...,leadTermsAlgebra(m')-@y(m')
* 4. Compute Groebnerbasis of kernOld+kernNew
* 5. Compute the new algebraic relations
*/
int ppl = printlevel-voice+3; //variable for additional printlevel-dependend information
dbprint(ppl,"//Spoly-1- initialisation and precomputation");
def br=basering;
ideal varsBasering=maxideal(1);
ideal leadTermsAlgebra=lead(algebra);
//save leading terms as ordering in ring extension
//may not be compatible with ordering in basering
int numGenerators=ncols(algebra);
def rNew=extendRing(r,leadTermsAlgebra,method);
// important: br has to be active here
setring r;
if (!defined(kern))
//only true for first run of spolynomialGB in sagbi construction algorithms
{
ideal kern=0;
ideal algebraicRelations=0;
}
setring rNew;
//-------------------------- transfer object to new ring rNew ----------------------
ideal varsBasering=fetch(br,varsBasering);
ideal kernOld,algebraicRelationsOld;
kernOld=fetch(r,kern); //kern is Groebner basis of the kernel of the map Phi:r->K[x_1,...,x_n], x(i)->x(i), @y(i)->leadTermsAlgebra(i)
algebraicRelationsOld=fetch(r,algebraicRelations);
ideal leadTermsAlgebra=fetch(br,leadTermsAlgebra);
ideal listOfVariables=maxideal(1);
//---------define kernNew containing elements to be added to the ideal kern --------
ideal kernNew;
for (int i=nvars(r)-nvars(br)+1; i<=numGenerators; i++)
{
kernNew[i-nvars(r)+nvars(br)]=leadTermsAlgebra[i]-listOfVariables[i+nvars(br)];
}
//--------------- calculate kernel of Phi depending on method chosen ---------------
dbprint(ppl,"//Spoly-2- Groebner basis computation");
attrib(kernOld,"isSB",1);
ideal kern=stdKernPhi(kernNew,kernOld,leadTermsAlgebra,method);
dbprint(ppl-2,"//Spoly-2-1- ideal kern",kern);
//-------------------------- calulate algebraic relations -----------------------
dbprint(ppl,"//Spoly-3- computing new algebraic relations");
ideal algebraicRelations=nselect(kern,1..nvars(br));
attrib(algebraicRelationsOld,"isSB",1);
ideal algebraicRelationsNew=reduce(algebraicRelations,algebraicRelationsOld);
/* canonicalizing: */
algebraicRelationsNew=canonicalform(algebraicRelationsNew);
dbprint(ppl-2,"//Spoly-3-1- ideal of new algebraic relations",algebraicRelationsNew);
/* algebraicRelationsOld is a groebner basis by construction (as variable
* ordering is
* block ordering we have an elemination ordering for the varsBasering)
* Therefore, to only get the new algebraic relations, calculate
* <algebraicRelations>\<algebraicRelationsOld> using groebner reduction
*/
kill kernOld,kernNew,algebraicRelationsOld,listOfVariables;
export algebraicRelationsNew,algebraicRelations,kern;
setring br;
return(rNew);
}
static proc spolynomialsToric(ideal algebra) {
/* This procedure does the actual S-polynomial calculation using toric.lib for
* computation of a Groebner basis for the toric ideal kern(phi), where
* phi:K[y_1,...,y_m]->K[x_1,...,x_n], y_i->leadmonom(algebra[i])
* By suitable substitutions we obtain the kernel of the map
* K[y_1,...,y_m]->K[x_1,...,x_n], x(i)->x(i), @y(i)->leadterm(algebra[i])
*/
int ppl = printlevel-voice+3; //variable for additional printlevel-dependend information
dbprint(ppl,"//Spoly-1- initialisation and precomputation");
def br=basering;
int m=ncols(algebra);
int n=nvars(basering);
intvec tempVec;
int i,j;
ideal leadCoefficients;
for (i=1;i<=m; i++)
{
leadCoefficients[i]=leadcoef(algebra[i]);
}
dbprint(ppl-2,"//Spoly-1-1- Vector of leading coefficients",leadCoefficients);
int k=1;
for (i=1;i<=n;i++)
{
for (j=1; j<=m; j++)
{
tempVec[k]=leadexp(algebra[j])[i];
k++;
}
}
//The columns of the matrix A are now the exponent vectors
//of the leadings monomials in algebra.
intmat A[n][m]=intmat(tempVec,n,m);
dbprint(ppl-2,"//Spoly-1-2- Matrix A",A);
//Create the preimage ring K[@y(1),...,@y(m)], where m=ncols(algebra).
string variableName=uniqueVariableName("@y");
list l = ringlist(basering);
for (i=1; i<=m;i++)
{
l[2][i]=string(variableName,"(",i,")");
}
l[3][2]=l[3][size(l[3])];
l[3][1]=list("dp",intvec(1:m));
def rNew=ring(l);
setring rNew;
//Use toric_ideal to compute the kernel
dbprint(ppl,"//Spoly-2- call of toric_ideal");
ideal algebraicRelations=toric_ideal(A,"ect");
//Suitable substitution
dbprint(ppl,"//Spoly-3- substitutions");
ideal leadCoefficients=fetch(br,leadCoefficients);
for (i=1; i<=m; i++)
{
if (leadCoefficients[i]!=0)
{
algebraicRelations=subst(algebraicRelations,var(i),1/leadCoefficients[i]*var(i));
}
}
dbprint(ppl-2,"//Spoly-3-1- algebraic relations",algebraicRelations);
export algebraicRelations;
return(rNew);
}
static proc reductionGB(ideal F, ideal algebra,def r, int tailreduction,int method,int parRed)
{
/* This procedure does the actual SAGBI/subalgebra reduction using GB methods and is
* called by the procedures sagbiReduce,sagbi and sagbiPart
* If r already is an extension of the basering
* and contains the ideal kern needed for the subalgebra reduction,
* the reduction can be started directly, at each reduction step using the fact that
* p=reduce(leadF,kern) in K[@y(1),...,@y(m)] <=> leadF in K[lead(algebra)]
* Otherwise some precomputation has to be done, outlined below.
* When using sagbiReduce,sagbi and sagbiPart the integer parRed will always be zero. Only the procedure
* algebraicDependence causes this procedure to be called with parRed<>0. The only difference when parRed<>0
* is that the reduction algorithms returns the non-zero constants it attains (instead of just returning zero as the
* correct remainder), as they will be expressions in parameters for an algebraic dependence.
*/
int ppl = printlevel-voice+3; //variable for additional printlevel-dependend information
dbprint(ppl,"//Red-1- initialisation and precomputation");
def br=basering;
int numVarsBasering=nvars(br);
ideal varsBasering=maxideal(1);
int i;
if (numVarsBasering==nvars(r))
{
dbprint(ppl-1,"//Red-1-1- Groebner basis computation");
/* Case that ring r is the same ring as the basering. Using proc extendRing,
* stdKernPhi
* one construct the extension of the current baserring with new variables @y, one for each element
* in ideal algebra and calculates the kernel of Phi, where
* Phi: r---->br, x_i-->x_i, y_i-->f_i,
* algebra={f_1,...f_m}, br=K[x1,...,x_n] und r=K[x1,...x_n,@y1,...@y_m]
* This is similarly dones
* (however step by step for each run of the SAGBI construction algorithm)
* in the procedure spolynomialsGB
*/
ideal leadTermsAlgebra=lead(algebra);
kill r;
def r=extendRing(br,leadTermsAlgebra,method);
setring r;
ideal listOfVariables=maxideal(1);
ideal leadTermsAlgebra=fetch(br,leadTermsAlgebra);
ideal kern;
for (i=1; i<=ncols(leadTermsAlgebra); i++)
{
kern[i]=leadTermsAlgebra[i]-listOfVariables[numVarsBasering+i];
}
kern=stdKernPhi(kern,0,leadTermsAlgebra,method);
dbprint(ppl-2,"//Red-1-1-1- Ideal kern",kern);
}
setring r;
poly p,leadF;
ideal varsBasering=fetch(br,varsBasering);
setring br;
map phi=r,varsBasering,algebra;
poly p,normalform,leadF;
intvec tempExp;
//-------------algebraic reduction for each polynomial F[i] ------------------------
dbprint(ppl,"//Red-2- reduction, polynomial by polynomial");
for (i=1; i<=ncols(F);i++)
{
dbprint(ppl-1,"//Red-2-"+string(i)+"- starting with new polynomial");
dbprint(ppl-2,"//Red-2-"+string(i)+"-1- Polynomial before reduction",F[i]);
normalform=0;
while (F[i]!=0)
{
leadF=lead(F[i]);
if(leadmonom(leadF)==1)
{
//K is always contained in the subalgebra,
//thus the remainder is zero in this case
if (parRed)
{
//If parRed<>0 save non-zero constants the reduction algorithms attains.
break;
}
else
{
F[i]=0;
break;
}
}
//note: as the ordering in br and r might not be compatible
//it can be that lead(F[i]) in r is
//different from lead(F[i]) in br.
//To take the "correct" leading term therefore take lead(F[i])
//in br and transfer it to the extension r
setring r;
leadF=fetch(br,leadF);
p=reduce(leadF,kern);
if (leadmonom(p)<varsBasering[numVarsBasering])
{
//as chosen ordering is a block ordering,
//lm(p) in K[y_1...y_m] is equivalent to lm(p)<x_n
//Needs to be changed, if no block ordering is used!
setring br;
F[i]=F[i]-phi(p);
}
else
{
if (tailreduction)
{
setring br;
normalform=normalform+lead(F[i]);
F[i]=F[i]-lead(F[i]);
}
else
{
setring br;
break;
}
}
}
if (tailreduction)
{
F[i] = normalform;
}
dbprint(ppl-2,"//Red-2-"+string(i)+"-2- Polynomial after reduction",F[i]);
}
return(F);
}
static proc reduceByMonomials(ideal algebra)
/*This procedures uses the sagbiReduce procedure to reduce all polynomials in algebra,
* which are not monomials, by the subset of all monomials.
*/
{
ideal monomials;
int i;
for (i=1; i<=ncols(algebra);i++)
{
if(size(algebra[i])==1)
{
monomials[i]=algebra[i];
algebra[i]=0;
}
else
{
monomials[i]=0;
}
}
//Monomials now contains the subset of all monomials in algebra,
//algebra contains the non-monomials.
if(size(monomials)>0)
{
algebra=sagbiReduce(algebra,monomials,1);
for (i=1; i<=ncols(algebra);i++)
{
if(size(monomials[i])==1)
{
//Put back monomials into algebra.
algebra[i]=monomials[i];
}
}
}
return(algebra);
}
static proc sagbiConstruction(ideal algebra, int iterations, int tailreduction, int method,int parRed)
/* This procedure is the SAGBI construction algorithm and does the actual computation
* both for the procedure sagbi and sagbiPart.
* - If the sagbi procedure calls this procedure, iterations==-1
* and this procedure only stops
* if all S-Polynomials reduce to zero
* (criterion for termination of SAGBI construction algorithm).
* - If the sagbiPart procedure calls this procedure, iterations>=0
* and iterations specifies the
* number of iterations. A degree boundary is not used here.
* When this method is called via the procedures sagbi and sagbiPart the integer parRed
* will always be zero. Only the procedure algebraicDependence calls this procedure with
* parRed<>0. The only difference when parRed<>0 is that the reduction algorithms returns
* the non-zero constants it attains (instead of just returning zero as the correct
* remainder), as they will be expressions in parameters for an algebraic dependence.
* These constants are saved in the ideal reducedParameters.
*/
{
int ppl = printlevel-voice+3; //variable for additional printlevel-dependend information
dbprint(ppl,"// -0- initialisation and precomputation");
def br=basering;
int i=1;
ideal reducedParameters;
int numReducedParameters=1; //number of elements plus one in reducedParameters
int j;
if (parRed==0) //if parRed<>0 the algebra does not contain monomials and normalisation should be avoided
{
algebra=reduceByMonomials(algebra);
algebra=simplify(simplify(algebra,3),4);
}
// canonicalizing the gen's:
algebra = canonicalform(algebra);
ideal P=1;
//note: P is initialized this way, so that the while loop is entered.
//P gets overriden there, anyhow.
ideal varsBasering=maxideal(1);
map phi;
ideal spolynomialsNew;
def r=br;
while (size(P)>0)
{
dbprint(ppl,"// -"+string(i)+"- interation of SAGBI construction algorithm");
dbprint(ppl-1,"// -"+string(i)+"-1- Computing algebraic relations");
def rNew=spolynomialsGB(algebra,r,method); /* canonicalizing inside! */
kill r;
def r=rNew;
kill rNew;
phi=r,varsBasering,algebra;
dbprint(ppl-1,"// -"+string(i)+"-2- Substituting into algebraic relations");
spolynomialsNew=simplify(phi(algebraicRelationsNew),6);
//By construction spolynomialsNew only contains the spolynomials,
//that have not already
//been calculated in the steps before.
dbprint(ppl-1,"// -"+string(i)+"-3- SAGBI reduction");
dbprint(ppl-2,"// -"+string(i)+"-3-1- new S-polynomials before reduction",spolynomialsNew);
P=reductionGB(spolynomialsNew,algebra,r,tailreduction,method,parRed);
if (parRed)
{
for(j=1; j<=ncols(P); j++)
{
if (leadmonom(P[j])==1)
{
reducedParameters[numReducedParameters]=P[j];
P[j]=0;
numReducedParameters++;
}
}
}
if (parRed==0)
{
P=reduceByMonomials(P);
//Reducing with monomials is cheap and can only result in less terms
P=simplify(simplify(P,3),4);
//Avoid that zeros are added to the bases or one element in P more than once
}
else
{
P=simplify(P,6);
}
/* canonicalize ! */
P = canonicalform(P);
dbprint(ppl-2,"// -"+string(i)+"-3-1- new S-polynomials after reduction",P);
algebra=algebra,P;
//Note that elements and order of elements must in algebra must not be changed,
//otherwise the already calculated
//ideal in r will give wrong results. Thus it is important to use a komma here.
i=i+1;
if (iterations!=-1 && i>iterations) //When iterations==-1 the number of iterations is unlimited
{
break;
}
}
if (iterations!=-1)
{ //case that sagbiPart called this procedure
if (size(P)==0)
{
dbprint(4-voice,
"//SAGBI construction algorithm terminated after "+string(i-1)
+" iterations, as all SAGBI S-polynomials reduced to 0.
//Returned generators therefore are a SAGBI basis.");
}
else
{
dbprint(4-voice,
"//SAGBI construction algorithm stopped as it reached the limit of "
+string(iterations)+" iterations.
//In general the returned generators are no SAGBI basis for the given algebra.");
}
}
kill r;
if (parRed)
{
algebra=algebra,reducedParameters;
}
algebra = simplify(algebra,6);
algebra = canonicalform(algebra);
return(algebra);
}
proc sagbiSPoly(ideal algebra,list #)
"USAGE: sagbiSPoly(A[, returnRing, meth]); A is an ideal, returnRing and meth are integers.
RETURN: ideal or ring
ASSUME: basering is not a qring
PURPOSE: Returns SAGBI S-polynomials of the leading terms of a given ideal A if returnRing=0.
@* Otherwise returns a new ring containing the ideals algebraicRelations
@* and spolynomials, where these objects are explained by their name.
@* See the example on how to access these objects.
@format The other optional argument meth determines which method is
used for computing the algebraic relations.
- If meth=0 (default), the procedure std is used.
- If meth=1, the procedure slimgb is used.
- If meth=2, the prodecure uses toric_ideal.
@end format
EXAMPLE: example sagbiSPoly; shows an example"
{
assumeQring();
int returnRing;
int method=0;
def br=basering;
ideal spolynomials;
if (size(#)>=1)
{
if (typeof(#[1])=="int")
{
returnRing=#[1];
}
else
{
ERROR("Type of first optional argument needs to be int.");
}
}
if (size(#)==2)
{
if (typeof(#[2])=="int")
{
if (#[2]<0 || #[2]>2)
{
ERROR("Type of second optional argument needs to be 0,1 or 2.");
}
else
{
method=#[2];
}
}
else
{
ERROR("Type of second optional argument needs to be int.");
}
}
if (method>=0 and method<=1)
{
ideal varsBasering=maxideal(1);
def rNew=spolynomialsGB(algebra,br,method);
map phi=rNew,varsBasering,algebra;
spolynomials=simplify(phi(algebraicRelationsNew),7);
}
if(method==2)
{
def r2=spolynomialsToric(algebra);
map phi=r2,algebra;
spolynomials=simplify(phi(algebraicRelations),7);
def rNew=extendRing(br,lead(algebra),0);
setring rNew;
ideal algebraicRelations=imap(r2,algebraicRelations);
export algebraicRelations;
setring br;
}
if (returnRing==0)
{
return(spolynomials);
}
else
{
setring rNew;
ideal spolynomials=fetch(br,spolynomials);
export spolynomials;
setring br;
return(rNew);
}
}
example
{ "EXAMPLE:"; echo = 2;
ring r= 0,(x,y),dp;
ideal A=x*y+x,x*y^2,y^2+y,x^2+x;
//------------------ Compute the SAGBI S-polynomials only
sagbiSPoly(A);
//------------------ Extended ring is to be returned, which contains
// the ideal of algebraic relations and the ideal of the S-polynomials
def rNew=sagbiSPoly(A,1); setring rNew;
spolynomials;
algebraicRelations;
//----------------- Now we verify that the substitution of A[i] into @y(i)
// results in the spolynomials listed above
ideal A=fetch(r,A);
map phi=rNew,x,y,A;
ideal spolynomials2=simplify(phi(algebraicRelations),1);
spolynomials2;
}
proc sagbiReduce(def idealORpoly, ideal algebra, list #)
"USAGE: sagbiReduce(I, A[, tr, mt]); I, A ideals, tr, mt optional integers
RETURN: ideal of remainders of I after SAGBI reduction by A
ASSUME: basering is not a qring
PURPOSE:
@format
The optional argument tr=tailred determines whether tail reduction will be performed.
- If (tailred=0), no tail reduction is done.
- If (tailred<>0), tail reduction is done.
The other optional argument meth determines which method is
used for Groebner basis computations.
- If mt=0 (default), the procedure std is used.
- If mt=1, the procedure slimgb is used.
@end format
EXAMPLE: example sagbiReduce; shows an example"
{
assumeQring();
int tailreduction=0; //Default
int method=0; //Default
ideal I;
if(typeof(idealORpoly)=="ideal")
{
I=idealORpoly;
}
else
{
if(typeof(idealORpoly)=="poly")
{
I[1]=idealORpoly;
}
else
{
ERROR("Type of first argument needs to be an ideal or polynomial.");
}
}
if (size(#)>=1)
{
if (typeof(#[1])=="int")
{
tailreduction=#[1];
}
else
{
ERROR("Type of optional argument needs to be int.");
}
}
if (size(#)>=2 )
{
if (typeof(#[2])=="int")
{
if (#[2]<0 || #[2]>1)
{
ERROR("Type of second optional argument needs to be 0 or 1.");
}
else
{
method=#[2];
}
}
else
{
ERROR("Type of optional arguments needs to be int.");
}
}
def r=basering;
I=simplify(reductionGB(I,algebra,r,tailreduction,method,0),1);
if(typeof(idealORpoly)=="ideal")
{
return(I);
}
else
{
if(typeof(idealORpoly)=="poly")
{
return(I[1]);
}
}
}
example
{ "EXAMPLE:"; echo = 2;
ring r=0,(x,y,z),dp;
ideal A=x2,2*x2y+y,x3y2;
poly p1=x^5+x2y+y;
poly p2=x^16+x^12*y^5+6*x^8*y^4+x^6+y^4+3;
ideal P=p1,p2;
//---------------------------------------------
//SAGBI reduction of polynomial p1 by algebra A.
//Default call, that is, no tail-reduction is done.
sagbiReduce(p1,A);
//---------------------------------------------
//SAGBI reduction of set of polynomials P by algebra A,
//now tail-reduction is done.
sagbiReduce(P,A,1);
}
proc sagbi(ideal algebra, list #)
"USAGE: sagbi(A[, tr, mt]); A ideal, tr, mt optional integers
RETURN: ideal, a SAGBI basis for A
ASSUME: basering is not a qring
PURPOSE: Computes a SAGBI basis for the subalgebra given by the generators in A.
@format
The optional argument tr=tailred determines whether tail reduction will be performed.
- If (tailred=0), no tail reduction is performed,
- If (tailred<>0), tail reduction is performed.
The other optional argument meth determines which method is
used for Groebner basis computations.
- If mt=0 (default), the procedure std is used.
- If mt=1, the procedure slimgb is used.
@end format
EXAMPLE: example sagbi; shows an example"
{
assumeQring();
int tailreduction=0; //default value
int method=0; //default value
if (size(#)>=1)
{
if (typeof(#[1])=="int")
{
tailreduction=#[1];
}
else
{
ERROR("Type of optional argument needs to be int.");
}
}
if (size(#)>=2 )
{
if (typeof(#[2])=="int")
{
if (#[2]<0 || #[2]>1)
{
ERROR("Type of second optional argument needs to be 0 or 1.");
}
else
{
method=#[2];
}
}
else
{
ERROR("Type of optional arguments needs to be int.");
}
}
ideal a;
a=sagbiConstruction(algebra,-1,tailreduction,method,0);
a=simplify(a,7);
// a=interreduced(a);
return(a);
}
example
{ "EXAMPLE:"; echo = 2;
ring r= 0,(x,y,z),dp;
ideal A=x2,y2,xy+y;
//Default call, no tail-reduction is done.
sagbi(A);
//---------------------------------------------
//Call with tail-reduction and method specified.
sagbi(A,1,0);
}
proc sagbiPart(ideal algebra, int iterations, list #)
"USAGE: sagbiPart(A, k,[tr, mt]); A is an ideal, k, tr and mt are integers
RETURN: ideal
ASSUME: basering is not a qring
PURPOSE: Performs k iterations of the SAGBI construction algorithm for the subalgebra given by the generators given by A.
@format
The optional argument tr=tailred determines if tail reduction will be performed.
- If (tailred=0), no tail reduction is performed,
- If (tailred<>0), tail reduction is performed.
The other optional argument meth determines which method is
used for Groebner basis computations.
- If mt=0 (default), the procedure std is used.
- If mt=1, the procedure slimgb is used.
@end format
EXAMPLE: example sagbiPart; shows an example"
{
assumeQring();
int tailreduction=0; //default value
int method=0; //default value
if (size(#)>=1)
{
if (typeof(#[1])=="int")
{
tailreduction=#[1];
}
else
{
ERROR("Type of optional argument needs to be int.");
}
}
if (size(#)>=2 )
{
if (typeof(#[2])=="int")
{
if (#[2]<0 || #[2]>3)
{
ERROR("Type of second optional argument needs to be 0 or 1.");
}
else
{
method=#[2];
}
}
else
{
ERROR("Type of optional arguments needs to be int.");
}
}
if (iterations<0)
{
ERROR("Number of iterations needs to be non-negative.");
}
ideal a;
a=sagbiConstruction(algebra,iterations,tailreduction,method,0);
a=simplify(a,6);
// a=interreduced(a);
return(a);
}
example
{ "EXAMPLE:"; echo = 2;
ring r= 0,(x,y,z),dp;
//The following algebra does not have a finite SAGBI basis.
ideal A=x,xy-y2,xy2;
//---------------------------------------------------
//Call with two iterations, no tail-reduction is done.
sagbiPart(A,2);
//---------------------------------------------------
//Call with three iterations, tail-reduction and method 0.
sagbiPart(A,3,1,0);
}
proc algebraicDependence(ideal I,int iterations)
"USAGE: algebraicDependence(I,it); I an an ideal, it is an integer
RETURN: ring
ASSUME: basering is not a qring
PURPOSE: Returns a ring containing the ideal @code{algDep}, which contains possibly
@* some algebraic dependencies of the elements of I obtained through @code{it}
@* iterations of the SAGBI construction algorithms. See the example on how
@* to access these objects.
EXAMPLE: example algebraicDependence; shows an example"
{
assumeQring();
int ppl = printlevel-voice+3; //variable for additional printlevel-dependend information
dbprint(ppl,"//AlgDep-1- initialisation and precomputation");
def br=basering;
int i;
I=simplify(I,2); //avoid that I contains zeros
//Create two polynomial rings, which both are extensions of the current basering.
//The first ring will contain the additional paramteres @c(1),...,@c(m), the second one
//will contain the additional variables @c(1),...,@c(m), where m=ncols(I).
string parameterName=uniqueVariableName("@c");
list l = ringlist(basering);
list parList;
for (i=1; i<=ncols(I);i++)
{
parList[i]=string(parameterName,"(",i,")");
}
l[1]=list(l[1],parList,list(list("dp",1:ncols(I)))); //add @c(i) to the ring as paramteres
ideal temp=0;
l[1][4]=temp;
// addition VL: noncomm case
int isNCcase = 0; // default for comm
// if (size(l)>4)
// {
// // that is we're in the noncomm algebra
// isNCcase = 1; // noncomm
// matrix @C@ = l[5];
// matrix @D@ = l[6];
// l = l[1],l[2],l[3],l[4];
// }
def parameterRing=ring(l);
string extendVarName=uniqueVariableName("@c");
list l2 = ringlist(basering);
for (i=1; i<=ncols(I);i++)
{
l2[2][i+nvars(br)]=string(extendVarName,"(",i,")"); //add @c(i) to the rings as variables
}
l2[3][size(l2[3])+1]=l2[3][size(l2[3])];
l2[3][size(l2[3])-1]=list("dp",intvec(1:ncols(I)));
// if (isNCcase)
// {
// // that is we're in the noncomm algebra
// matrix @C@2 = l2[5];
// matrix @D@2 = l2[6];
// l2 = l2[1],l2[2],l2[3],l2[4];
// }
def extendVarRing=ring(l2);
setring extendVarRing;
// VL : this requires extended matrices
// let's forget it for the moment
// since this holds only for showing the answer
// if (isNCcase)
// {
// matrix C2=imap(br,@C@2);
// matrix D2=imap(br,@D@2);
// def er2 = nc_algebra(C2,D2);
// setring er2;
// def extendVarRing=er2;
// }
setring parameterRing;
// if (isNCcase)
// {
// matrix C=imap(br,@C@);
// matrix D=imap(br,@D@);
// def pr = nc_algebra(C,D);
// setring pr;
// def parameterRing=pr;
// }
//Compute a partial SAGBI basis of the algebra generated by I[1]-@c(1),...,I[m]-@c(m),
//where the @c(n) are parameters
ideal I=fetch(br,I);
ideal algebra;
for (i=1; i<=ncols(I);i++)
{
algebra[i]=I[i]-par(i);
}
dbprint(ppl,"//AlgDep-2- call of SAGBI construction algorithm");
algebra=sagbiConstruction(algebra, iterations,0,0,1);
dbprint(ppl,"//AlgDep-3- postprocessing of results");
int j=1;
//If K[x_1,...,x_n] was the basering, then algebra is in K(@c(1),...,@c(m))[x_1,...x_n]. We intersect
//elements in algebra with K(@c(1),..,@c(n)) to get algDep. Note that @c(i) can only appear in the numerator,
//as the SAGBI construction algorithms just multiplies and substracts polynomials. So actually we have
//algDep=algebra intersect K[@c(1),...,@c(m)]
ideal algDep;
for (i=1; i<= ncols(algebra); i++)
{
if (leadmonom(algebra[i])==1) //leadmonom(algebra[i])==1 iff algebra[i] in K[@c(1),...,@c(m)]
{
algDep[j]=algebra[i];
j++;
}
}
//Transfer algebraic dependencies to ring where @c(i) are not parameters, but now variables.
setring extendVarRing;
ideal algDep=imap(parameterRing,algDep);
ideal algebra=imap(parameterRing,algebra);
//Now get rid of constants in K that may have been added to algDep.
for (i=1; i<=ncols(algDep); i++)
{
if(leadmonom(algDep[i])==1)
{
algDep[i]=0;
}
}
algDep=simplify(algDep,2);
export algDep,algebra;
setring br;
return(extendVarRing);
}
example
{ "EXAMPLE:"; echo = 2;
ring r= 0,(x,y),dp;
//The following algebra does not have a finite SAGBI basis.
ideal I=x^2, xy-y2, xy2;
//---------------------------------------------------
//Call with two iterations
def DI = algebraicDependence(I,2);
setring DI; algDep;
// we see that no dependency has been seen so far
//---------------------------------------------------
//Call with two iterations
setring r; kill DI;
def DI = algebraicDependence(I,3);
setring DI; algDep;
map F = DI,x,y,x^2, xy-y2, xy2;
F(algDep); // we see that it is a dependence indeed
}
static proc interreduced(ideal I)
{
/* performs subalgebra interreduction of a set of subalgebra generators */
int ppl = printlevel-voice+3; //variable for additional printlevel-dependend information
dbprint(ppl,"//Interred-1- starting interreduction");
ideal J,B;
int i,j,k;
poly f;
for(k=1;k<=ncols(I);k++)
{
dbprint(ppl-1,"//Interred-1-"+string(k)+"- reducing next poly");
f=I[k];
I[k]=0;
f=sagbiReduce(f,I,1);
I[k]=f;
}
I=simplify(I,2);
dbprint(ppl,"//Interred-2- interreduction completed");
return(I);
}
///////////////////////////////////////////////////////////////////////////////
proc sagbiReduction(poly p,ideal dom,list #)
"USAGE: sagbiReduction(p,dom[,n]); p poly , dom ideal
RETURN: polynomial, after one step of subalgebra reduction
PURPOSE:
@format
Three algorithm variants are used to perform subalgebra reduction.
The positive interger n determines which variant should be used.
n may take the values 0 (default), 1 or 2.
@end format
NOTE: works over both polynomial rings and their quotients
EXAMPLE: example sagbiReduction; shows an example"
{
def bsr=basering;
ideal B=ideal(bsr);//When the basering is quotient ring this type casting
// gives the quotient ideal.
int b=size(B);
int n=nvars(bsr);
//In quotient rings, SINGULAR, usually does not reduce polynomials w.r.t the
//quotient ideal,therefore we should first reduce,
//when it is necessary for computations,
// to have a uniquely determined representant for each equivalent
//class,which is the case of this algorithm.
if(b !=0) //means that the basering is a quotient ring
{
p=reduce(p,std(0));
dom=reduce(dom,std(0));
}
int i,choose;
int z=ncols(dom);
if((size(#)>0) && (typeof(#[1])=="int"))
{
choose = #[1];
}
if (size(#)>1)
{
choose =#[2];
}
//=======================first algorithm(default)=========================
if ( choose == 0 )
{
list L = algebra_containment(lead(p),lead(dom),1);
if( L[1]==1 )
{
// the ring L[2] = char(bsr),(x(1..nvars(bsr)),y(1..z)),(dp(n),dp(m)),
// contains poly check s.t. LT(p) is of the form check(LT(f1),...,LT(fr))
def s1 = L[2];
map psi = s1,maxideal(1),dom;
poly re = p - psi(check);
// divide by the maximal power of #[1]
if ( (size(#)>0) && (typeof(#[1])=="poly") )
{
while ((re!=0) && (re!=#[1]) &&(subst(re,#[1],0)==0))
{
re=re/#[1];
}
}
return(re);
}
return(p);
}
//======================2end variant of algorithm=========================
//It uses two different commands for elimaination.
//if(choose==1):"elimainate"command.
//if (choose==2):"nselect" command.
else
{
poly v=product(maxideal(1));
//------------- change the basering bsr to bsr[@(0),...,@(z)] ----------
def s=addNvarsTo(basering,z+1,,"@",0); setring s;
// Ev hier die Reihenfolge der Vars aendern. Dazu muss unten aber entsprechend
// geaendert werden:
// execute("ring s="+charstr(basering)+",(@(0..z),"+varstr(basering)+"),dp;");
//constructs the leading ideal of dom=(p-@(0),dom[1]-@(1),...,dom[z]-@(z))
ideal dom=imap(bsr,dom);
for (i=1;i<=z;i++)
{
dom[i]=lead(dom[i])-var(nvars(bsr)+i+1);
}
dom=lead(imap(bsr,p))-@(0),dom;
//---------- eliminate the variables of the basering bsr --------------
//i.e. computes dom intersected with K[@(0),...,@(z)].
if(choose==1)
{
ideal kern=eliminate(dom,imap(bsr,v));//eliminate does not need a
//standard basis as input.
}
if(choose==2)
{
ideal kern= nselect(groebner(dom),1..n);//"nselect" is combinatorial command
//which uses the internal command
// "simplify"
}
//--------- test wether @(0)-h(@(1),...,@(z)) is in ker ---------------
// for some poly h and divide by maximal power of q=#[1]
poly h;
z=size(kern);
for (i=1;i<=z;i++)
{
h=kern[i]/@(0);
if (deg(h)==0)
{
h=(1/h)*kern[i];
// define the map psi : s ---> bsr defined by @(i) ---> p,dom[i]
setring bsr;
map psi=s,maxideal(1),p,dom;
poly re=psi(h);
// divide by the maximal power of #[1]
if ((size(#)>0) && (typeof(#[1])== "poly") )
{
while ((re!=0) && (re!=#[1]) &&(subst(re,#[1],0)==0))
{
re=re/#[1];
}
}
return(re);
}
}
setring bsr;
return(p);
}
}
example
{"EXAMPLE:"; echo = 2;
ring r= 0,(x,y),dp;
ideal dom =x2,y2,xy-y;
poly p=x4+x3y+xy2-y2;
sagbiReduction(p,dom);
sagbiReduction(p,dom,2);
// now let us see the action over quotient ring
ideal I = xy;
qring Q = std(I);
ideal dom = imap(r,dom); poly p = imap(r,p);
sagbiReduction(p,dom,1);
}
proc sagbiNF(id,ideal dom,int k,list#)
"USAGE: sagbiNF(id,dom,k[,n]); id either poly or ideal,dom ideal, k and n positive intergers.
RETURN: same as type of id; ideal or polynomial.
PURPOSE:
@format
The integer k determines what kind of s-reduction is performed:
- if (k=0) no tail s-reduction is performed.
- if (k=1) tail s-reduction is performed.
Three Algorithm variants are used to perform subalgebra reduction.
The positive integer n determines which variant should be used.
n may take the values (0 or default),1 or 2.
@end format
NOTE: sagbiNF works over both rings and quotient rings
EXAMPLE: example sagbiNF; show example "
{
ideal rs;
if (ideal(basering) == 0)
{
rs = sagbiReduce(id,dom,k) ;
}
else
{
rs = sagbiReduction(id,dom,k) ;
}
if (typeof(id)=="poly") { return (rs[1]); }
return(rs);
}
example
{"EXAMPLE:"; echo = 2;
ring r=0,(x,y),dp;
poly p=x4+x2y+y;
ideal dom =x2,x2y+y,x3y2;
sagbiNF(p,dom,1);
ideal I= x2-xy;
qring Q=std(I); // we go to the quotient ring
poly p=imap(r,p);
NF(p,std(0)); // the representative of p has changed
ideal dom = imap(r,dom);
print(matrix(NF(dom,std(0)))); // dom has changed as well
sagbiNF(p,dom,0); // no tail reduction
sagbiNF(p,dom,1);// tail subalgebra reduction is performed
}
static proc canonicalform(ideal I)
{
/* placeholder for the canonical form of a set of gen's */
/* for the time being we agree on content(p)=1; that is coeffs with no fractions */
int i; ideal J=I;
for(i=ncols(I); i>=1; i--)
{
J[i] = canonicalform_poly(I[i]);
}
return(J);
}
static proc canonicalform_poly(poly p)
{
/* placeholder for the canonical form of a poly */
/* for the time being we agree on content(p)=1; that is coeffs with no fractions */
number n = content(p);
return( p/content(p) );
}
/*
ring r= 0,(x,y),dp;
//The following algebra does not have a finite SAGBI basis.
ideal J=x^2, xy-y2, xy2, x^2*(x*y-y^2)^2 - (x*y^2)^2*x^4 + 11;
//---------------------------------------------------
//Call with two iterations
def DI = algebraicDependence(J,2);
setring DI; algDep;
*/
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