Changeset 1e33d5 in git for Singular/LIB/sagbi.lib

Timestamp:

Oct 8, 2010, 11:09:04 PM (14 years ago)

Author:

Viktor Levandovskyy <levandov@…>

Branches:

(u'spielwiese', '2a584933abf2a2d3082034c7586d38bb6de1a30a')

Children:

943eb600edc10daf68e4769172bb646154ff07d8

Parents:

fa85ef15a2e50da0844c23c3d3de984b6865f70c

Message:

*levandov: many updates of doc, examples, additions

git-svn-id: file:///usr/local/Singular/svn/trunk@13440 2c84dea3-7e68-4137-9b89-c4e89433aadc

File:

• Singular/LIB/sagbi.lib (modified) (14 diffs)

Singular/LIB/sagbi.lib

@@ -6 +6 @@
 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'.
+It is an important tools in working with finitely presented subalgebras,
+many properties of SAGBI bases are analogously important to Groebner bases
+of ideals. Note, that due to absence of Noetherian property, SAGBI basis
+of a finite number of generators of a subalgebra may be infinite.
+Hence we provide procedures, which perform a given number of steps in
+computations.
 
 PROCEDURES:
- sagbiSPoly(A [,r,m]); SAGBI S-polynomials of A
- sagbiReduce(I,A);     subalgebra reduction of I by A
- sagbi(A [,m,t]);      SAGBI basis for A
- sagbiPart(A,k[,m]);   partial SAGBI basis for A
- algebraicDependence(I,it); iteration of SAGBI for algebraic dependencies of I
+ sagbiSPoly(A [,r,m]); computes SAGBI S-polynomials of A
+ sagbiNF(id,dom,k[,n]); computes SAGBI normal form of id wrt dom
+ sagbiReduce(I,A [,t,mt]);  performs subalgebra reduction of I by A
+ sagbiReduction(I,A [,n]);  performs subalgebra reduction of I by A in a quotient ring
+ 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
@@ -32 +44 @@
 //////////////////////////////////////////////////////////////////////////////
 
+
+//static 
+proc tst_sagbi()
+{
+  example sagbiSPoly;
+  example sagbiNF;
+  example sagbiReduce;
+  example sagbiReduction;
+  example sagbi;
+  example sagbiPart;
+  example algebraicDependence;
+}
 
 
@@ -444 +468 @@
 proc sagbiSPoly(ideal algebra,list #)
 "USAGE:   sagbiSPoly(A[, returnRing, meth]);  A is an ideal, returnRing and meth are integers.
-RETURN:   Returns SAGBI S-polynomials of the leading terms of 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.
+RETURN:   ideal or ring 
+ASSUME: basering is not a qring
+PURPOSE: Returns SAGBI S-polynomials of the leading terms of 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"
@@ -543 +569 @@
 proc sagbiReduce(idealORpoly, ideal algebra, list #)
 "USAGE:   sagbiReduce(I, A[, tr, mt]); I, A ideals, tr, mt optional integers
-RETURN:   Returns the remainder(s) of I after SAGBI reduction by A
+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.
@@ -635 +663 @@
 proc sagbi(ideal algebra, list #)
 "USAGE:   sagbi(A[, tr, mt]); A ideal, tr, mt optional integers
-RETURN: A SAGBI basis for the subalgebra given by the generators in A.
+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.
@@ -678 +708 @@
     }
   }
-  algebra=sagbiConstruction(algebra,-1,tailreduction,method,0);
-  algebra=simplify(algebra,7);
-  algebra=interreduced(algebra);
-  return(algebra);
+  ideal a;
+  a=sagbiConstruction(algebra,-1,tailreduction,method,0);
+  a=simplify(a,7);
+  a=interreduced(a);
+  return(a);
 }
 example
@@ -695 +726 @@
 
 proc sagbiPart(ideal algebra, int iterations, list #)
-"USAGE:   sagbiPart(A, k,[tr, mt]); A is an ideal, k, tr and mt are integers
-RETURN: Performs k iterations of the SAGBI construction algorithm for the subalgebra given by the generators in A.
+"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.
@@ -743 +776 @@
     ERROR("Number of iterations needs to be non-negative.");
   }
-  algebra=sagbiConstruction(algebra,iterations,tailreduction,method,0);
-  algebra=simplify(algebra,7);
-  algebra=interreduced(algebra);
-  return(algebra);
+  ideal a;
+  a=sagbiConstruction(algebra,iterations,tailreduction,method,0);
+  a=simplify(a,7);
+  a=interreduced(a);
+  return(a);
 }
 example
@@ -762 +796 @@
 
 
-//DONE? 1nsen als werden noch mit ausgegeben, aus algebraischen AbhÀngigkeiten entfernen
+
 // VL: finished the documentation
-// TO comapre with algDependent from algebra_lib
+// TO compare with algDependent from algebra_lib
+// TODO : remove constants from algebraic dependencies
 proc algebraicDependence(ideal I,int iterations)
 "USAGE: algebraicDependence(I,it); I an an ideal, it is an integer
-RETURN: In 'it' iterations, compute algebraic dependencies between elements of I
+RETURN: ring
+ASSUME: basering is not a qring
+PURPOSE: In @code{it} iterations, compute algebraic dependencies between elements of I
 EXAMPLE: example algebraicDependence; shows an example"
 {
@@ -900 +937 @@
 proc sagbiReduction(poly p,ideal dom,list #)
 "USAGE: sagbiReduction(p,dom[,n]); p poly , dom  ideal
-RETURN: a polynomial, after one step subalgebra reduction
+RETURN: polynomial, after one step of subalgebra reduction
+PURPOSE:
 @format
     Three algorithm variants are used to perform subalgebra reduction.
@@ -906 +944 @@
     n may take the values 0 (default), 1 or 2.
 @end format
-EXAMPLE: sagbiReduction; shows an example"
+NOTE: works over both polynomial rings and their quotients
+EXAMPLE: example sagbiReduction; shows an example"
 {
   def bsr=basering;
@@ -1032 +1071 @@
   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);
-  sagbiReduction(p,dom,2);
 }
 
@@ -1039 +1082 @@
 "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.
+ASSUME: basering is not a qring
+PURPOSE:
 @format
     The integer k determines what kind of s-reduction is performed:
@@ -1047 +1092 @@
     n may take the values (0 or default),1 or 2.
 @end format
-NOTE: computation of subalgebra normal forms may be performed in polynomial rings or quotients
-      thereof
 EXAMPLE: example sagbiNF; show example "
 {