CoCoA:GBasis5, and more

GBasis5, and more

calculations using the CoCoAServer

Description

These functions are implemented in CoCoALib (requires an active CoCoAServer).

Mostly the same as <ttref>GBasis</ttref>, <ttref>Intersection</ttref>,

<ttref>Elim</ttref>, but computed using the CoCoAServer.

All the 5 functions behave like the corresponding

function in CoCoA-4, i.e. should give the same output!

All the 5x functions allow extra options to be set, such

as (twin)floating point computations, parameters, shifts, different

orderings, gradings...

NB if you set incompatible options with the CurrentRing

(e.g. a different grading or ordering) the resulting GBasis will be

incompatible with CurrentRing... of course!

Example

  ----[  parameters ]-----------------------------------------------
  M := $cocoa5.AddParamOrdMat(DegRevLexMat(3), 1); // compatible Term Ordering
  Use Z/(32003)[a, x,y,z], Ord(M);
  
  I := Ideal((a-1)x+(a^2+a)y,(a+1)x + y);
  GBasis5(I);
  GBasis5x(I, Record(NumParams=1));
  
  ----[  Twin floats  ]--------------------------------------------------
  Use Q[x[0..5]];
  L := [Randomized(DensePoly(4)) | I In 1..2];
  Time LT(Ideal(L));
  LT5x(Ideal(L), Record(FloatPrecision=128));
  
  ----[  Elimination  ]----------------
  Use Q[x,y,z,w[3..5]], Weights([7, 4, 3, 1, 1, 1]);
  
  I := Ideal(
  	   x - 7413431 w[4]^7 - 9162341 w[3]*w[4]*w[5]^5,
  	   y - 6521443 w[4]^4 - 2312257 w[3]^2*w[4]*w[5],
  	   z - 5329421 w[4]^3 - 2122414 w[3]*w[5]^2
  	   );
  
  Time E := Elim([w[3],w[4]], I);
  E5 := Elim5([w[3],w[4]], I);  // usually much faster than CoCoA4
  E = E5;
  
  ----[  Intersection  ]-------------------------------
  Use Q[x,y,z], Weights(1,2,1);
  I := Ideal(xy, z^2);
  J := Ideal(yz, x-z);
  Intersection5(I, J);
  // with parameters
  M := CoCoA5.AddParamOrdMat(DegRevLexMat(3), 2); -- 2 parameters
  Use Z/(32003)[a,b, x,y,z], Ord(M);
  II := Ideal(x-y);
  I := (a-b+1) * x * II;
  J := (a+1) * y * II;
  Intersection5x(I,J, Record(NumParams=2)); -- 2 parameters
  
  ----[  Syzygies  ]--------------------------------------
  Use Z/(101)[x,y,z], Weights(1,2,4);
  L := [Vector(x^2,x^2),Vector(xy,xy)];
  S5 := SyzOfGens5(L);
  
  ----[  Modules  ]---------------------------------------------------------
  Use Q[x,y,z], PosTo;
  M := Module([x,z], [z,y]);
  ReducedGBasis(M);
  ReducedGBasis5(M);
  ReducedGBasis5x(M, Record(IsPosTo=False));
  
  Use Q[x,y,z], Weights([1,2,4]);
  M := Module([x^2-y,1],[x^4-z,y^2]);
  Info5 := Record();
  Info5.ModuleShifts := Mat([[0,2]]);
  GBasis5x(M, Info5);
  
  // Grading given via OrdMat and Grading Dim
  OrdMat := Mat([[1,1,1],[2,1,1],[1,1,0]]); // Ring Grading (first 2 Rows)
                                            // Plus order (last row)
  Use Z/(101)[x,y,z], Ord(OrdMat), ToPos;
  M := Module([y-x,0,0], [x,0,z], [0,y^2-z^2,0]);
  X := ReducedGBasis5(Module(Gens(M)));
  // GBasis on a module with shifts
  Info5 := Record();
  Info5.OrdMat := OrdMat;
  Info5.GradingDim := 2;
  Info5.ModuleShifts := Mat([[3,1,2],[2,2,5]]); // GrDim rows!!
  ReducedGBasis5x(M, Info5);

Syntax

-operation-5(same as -operation-)
-operation-5x(same as -operation-, Info5: RECORD)
GBasis5(M:IDEAL, MODULE):LIST
GBasis5x(M:IDEAL, MODULE,  Info5: RECORD):LIST

Introduction to Groebner Bases in CoCoA

OpenSocket

   <type>groebner</type>
   <type>groebner-basic</type>
   <type>ideal</type>
   <type>module</type>
   <type>quotient</type>
   <type>cocoaserver</type>