Changeset 28734e in git

Timestamp:

Feb 21, 2009, 4:26:42 PM (14 years ago)

Author:

Viktor Levandovskyy <levandov@…>

Branches:

(u'spielwiese', '828514cf6e480e4bafc26df99217bf2a1ed1ef45')

Children:

019a78d9b792cb277c1ba4090e9edfa00c9d19fe

Parents:

40b31de81774f2cb2360815a1a5e3b8c608dd76f

Message:

*levandov: stuff for rational Groebner bases updated


git-svn-id: file:///usr/local/Singular/svn/trunk@11426 2c84dea3-7e68-4137-9b89-c4e89433aadc

Location:

Singular/LIB

Files:

• dmodapp.lib (modified) (7 diffs)
• ratgb.lib (modified) (8 diffs)

Singular/LIB/dmodapp.lib

@@ -1 +1 @@
 //////////////////////////////////////////////////////////////////////////////
-version="$Id: dmodapp.lib,v 1.18 2009-02-12 20:25:23 levandov Exp $";
+version="$Id: dmodapp.lib,v 1.19 2009-02-21 15:26:41 levandov Exp $";
 category="Noncommutative";
 info="
@@ -595 +595 @@
 {
   // Appel F1, d = b', SST p.48
-  ring @r = (0,a,b,c,d),(x,y,Dx,Dy),(a(0,0,1,1),dp);
+  ring @r = (0,a,b,c,d),(x,y,Dx,Dy),(a(0,0,1,1),a(0,0,1,0),dp);
   matrix @D[4][4];
   @D[1,3]=1; @D[2,4]=1;
@@ -611 +611 @@
 {
   "EXAMPLE:"; echo = 2;
-  ring r = 0,x,dp;
   def A = appelF1();
   setring A;
@@ -626 +625 @@
 {
   // Appel F2, c = b', SST p.85
-  ring @r = (0,a,b,c),(x,y,Dx,Dy),(a(0,0,1,1),dp);
+  ring @r = (0,a,b,c),(x,y,Dx,Dy),(a(0,0,1,1),a(0,0,1,0),dp);
   matrix @D[4][4];
   @D[1,3]=1; @D[2,4]=1;
@@ -641 +640 @@
 {
   "EXAMPLE:"; echo = 2;
-  ring r = 0,x,dp;
   def A = appelF2();
   setring A;
@@ -656 +654 @@
 {
   // Appel F4, d = c', SST, p. 39
-  ring @r = (0,a,b,c,d),(x,y,Dx,Dy),(a(0,0,1,1),dp);
+  ring @r = (0,a,b,c,d),(x,y,Dx,Dy),(a(0,0,1,1),a(0,0,1,0),dp);
   matrix @D[4][4];
   @D[1,3]=1; @D[2,4]=1;
@@ -671 +669 @@
 {
   "EXAMPLE:"; echo = 2;
-  ring r = 0,x,dp;
   def A = appelF4();
   setring A;

Singular/LIB/ratgb.lib

@@ -1 +1 @@
 //////////////////////////////////////////////////////////////////////////////
-version="$Id: ratgb.lib,v 1.13 2009-01-14 16:07:05 Singular Exp $";
+version="$Id: ratgb.lib,v 1.14 2009-02-21 15:26:42 levandov Exp $";
 category="Noncommutative";
 info="
-LIBRARY: ratgb.lib  Groebner bases in Ore localizations
+LIBRARY: ratgb.lib  Groebner bases in Ore localizations of noncommutative G-algebras
 AUTHOR: Viktor Levandovskyy,     levandov@risc.uni-linz.ac.at
+
+THEORY: Let A be an operator algebra with R = K[x1,.,xN] as subring. The operators
+are usually denoted by {d1,..,dM}. Assume, that A is a G-algebra, then the set S=R-{0}
+is multiplicatively closed and is an Ore set in A. That is, for any s in S and a in A,
+there exist t in S and b in A, such that sa=bt. In other words one can transform
+any left fraction into the right fraction. The algebra A_S is entermed an Ore
+localization of A with respect to S. This library provides Groebner basis
+procedure for A_S, performing polynomial computations only.
 
 PROCEDURES:
@@ -13 +21 @@
 
 LIB "poly.lib";
+LIB "dmodapp.lib"; // for Appel1, Appel2, Appel4
 
 //static
@@ -53 +62 @@
 @* the first block of length n contains the elements
 @*     with respect to which one localizes,
-@* the basering is equipped with the elimination ordering
+@* the basering is equipped with the elimination block ordering
 @*     for the variables in the second block
 NOTE: the output ring O is commutative. The ideal rGBid in O
@@ -60 +69 @@
 vector space D-dimension of I are computed and printed out.
 @* Setting optional integer eng to 1, slimgb is taken as Groebner engine
-TRACING: In order to see the steps of the computation, set printlevel to >=2
+DISPLAY: In order to see the steps of the computation, set printlevel to >=2
 EXAMPLE: example ratstd; shows examples
 "
@@ -150 +159 @@
   RL[2] = tmp1;
 
-  // assume the ordering is the block with (a(0:is),ORD)
-  // set up ORD as the ordering
+  // old: assume the ordering is the block with (a(0:is),ORD)
+  // old :set up ORD as the ordering
   //  L; "RL:"; RL;
   if (size(L[3]) != 3)
   {
-    "note: strange ordering";
-  }
-  kill tmp2; list tmp2;
-  tmp2[1] = L[3][2];
-  tmp2[2] = L[3][3];
-  RL[3]   = tmp2;
-
+    //"note: strange ordering";
+    // NEW assume: ordering is the antiblock with (a(0:is),a(*1),a(*), ORD)
+    // get the a() parts after is => they should form a complete D-ordering
+    list L3 = L[3]; list NL3; kill tmp3; list tmp3;
+    int @sl = size(L3);
+    int w=1; int z; intvec va,vb;
+    while(L3[w][1] == "a")
+    {
+      va = L3[w][2];
+      for(z=1;z<=nvars(save)-is;z++)
+      { 
+        vb[z] = va[is+z]; 
+      }
+      tmp3[1] = "a";
+      tmp3[2] = vb;
+      NL3[w] = tmp3; tmp3=0;
+      w++;
+    }
+    // check for completeness: must be >= nvars(save)-is rows
+    if (w < nvars(save)-is)
+    {
+      "note: ordering is incomplete on D. Adding lower Dp block";
+      // adding: positive things like Dp
+      tmp3[1]= "Dp";
+      for (z=1; z<=nvars(save)-is; z++)
+      {
+        va[is+z] = 1;
+      }
+      tmp3[2] = va;
+      NL3[w] = tmp3; tmp3 = 0;
+      w++;
+    }
+    NL3[w] = L3[@sl]; // module ord?
+    RL[3] = NL3;
+  }
+  else
+  {
+    kill tmp2; list tmp2;
+    tmp2[1] = L[3][2];
+    tmp2[2] = L[3][3];
+    RL[3]   = tmp2;
+  }
   // factor ideal is ignored
   RL[4] = ideal(0);
+
+  //  "ringlist:"; RL;
 
   def @RAT = ring(RL);
@@ -284 +330 @@
 {
   "EXAMPLE:"; echo = 2;
-  ring r = 0,(k,n,K,N),(a(0,1),dp);
-  matrix D[4][4];
-  D[1,3] = K;
-  D[2,4] = N;
+  ring r = 0,(k,n,K,N),(a(0,0,1,1),a(0,0,1,0),dp);
+  matrix D[4][4]; D[1,3] = K; D[2,4] = N;
   def S = nc_algebra(1,D);
-  setring S;
+  setring S; // S is the 2nd shift algebra
   ideal I = (k+1)*K - (n-k), (n-k+1)*N - (n+1);
-  int is = 1;
+  int is = 2; // hence 1st and 2nd variables treated as units
   def A  = ratstd(I,is);
   pGBid; // polynomial form
@@ -298 +342 @@
 }
 
-static proc exParam()
+/*
+proc exParamAppelF4()
 {
   // Appel F4
   LIB "ratgb.lib";
-  ring r = (0,a,b,c,d),(x,y,Dx,Dy),(a(0,0,1,1),dp);
+  ring r = (0,a,b,c,d),(x,y,Dx,Dy),(a(0,0,1,1),a(0,0,1,0),dp);
   matrix @D[4][4];
   @D[1,3]=1; @D[2,4]=1;
@@ -314 +359 @@
   setring A;
   rGBid; // rational form
-}
+  // hence, the K(x,y) basis is {1,Dx,Dy,Dy^2}
+}
+
+// more examples:
+
+// F1 is hard
+appel F1
+{
+  LIB "dmodapp.lib";
+  LIB "ratgb.lib";
+  def A = appelF1();
+  setring A;
+  IAppel1;
+  def F1 = ratstd(IAppel1,2); 
+  lead(pGBid); 
+  setring F1; rGBid;
+}
+
+// F2 is much easier
+appel F2
+{
+  LIB "dmodapp.lib";
+  LIB "ratgb.lib";
+  def A = appelF2();
+  setring A;
+  IAppel2;
+  def F1 = ratstd(IAppel2,2); 
+  lead(pGBid); 
+  setring F1; rGBid;
+}
+
+// F4 is feasible
+appel F4
+{
+  LIB "dmodapp.lib";
+  LIB "ratgb.lib";
+  def A = appelF4();
+  setring A;
+  IAppel4;
+  def F1 = ratstd(IAppel4,2); 
+  lead(pGBid); 
+  setring F1; rGBid;
+}
+
+
+*/