ApCoCoA-1:SB.NFS

SB.NFS

Computes the subalgebra normal form of a polynomial with respect to subalgebra generators.

Syntax

SB.NFS(Polys:LIST of POLY, F:POLY):POLY
SB.NFS(Polys:LIST of POLY, F:POLY, ReprType:INT):POLY
SB.NFS(Polys:LIST of POLY, F:POLY, SubAlgRepr:BOOL):POLY or LIST
SB.NFS(Polys:LIST of POLY, F:POLY, ReprType:INT, SubAlgRepr:BOOL):POLY or LIST

Description

This function computes the subalgebra normal form of the polynomial F with respect to the polynomials in the list Polys which generate a subalgebra S of the current algebra. The optional parameter ReprType gives the possibility to choose between different ways of getting a term representation (see SB.TermRepr). With the optional parameter SubAlgRepr it is possible to control the form of the output. If SubAlgRepr=FALSE (which is also the default value) only the normal form will be returned, otherwise a list with the normal form NFS(F) of F and a subalgebra representation of F - NFS(F) will be returned. Example: Let G = [g_1, g_2, g_3] and let [[1,0,3,-1],[4,2,1,2]] be the returned representation of F - NFS(F). Then it follows

F-NFS(F) = -1*(g_1)^1(g_2)^0(g_3)^3 + 2*(g_1)^4(g_2)^2(g_3)^1

That means that the last entry of every list in the representation gives the coefficients and the other entries the exponents.

• @param Polys A list of polynomials, which are the generators of the current subalgebra.

• @param F A polynomial.

• @return Depending on the optional parameter SubAlgRepr either a polynomial or a list including a polynomial and a list of integers.

The following parameters are optional:

• @param ReprType Either 0,1 or 2. With this parameter it is possible to choose between different ways of getting the representation: By ReprType=0 a toric ideal is used to compute the representation. This is also the default value. By ReprType=1 algebra homomorphisms are used, by ReprType=2 a system of diophantine equations is used to compute the representation.

• @param SubAlgRepr A boolean value. The default value is FALSE. If SubAlgRepr=FALSE only the normal form will be returned, otherwise a list with two entries will be returned: The first one is the normal form NFS(F) of F, the second one is the subalgebra representation of the polynomial F - NFS(F) which always lies in the given subalgebra.

Example

Use R::=QQ[x,y];

G:=[x-y,x+y];

SB.NFS(G,x^2-y^2);
SB.NFS(G,x^2-y^2,TRUE);

-------------------------------------------------------
-- output:

-- Interpretation:
-- x^2-y^2 - (-2xy-2y^2) = x^2+2xy+y^2 = 1*(x+y)^2 is in K[G] 

-2xy - 2y^2
-------------------------------
[-2xy - 2y^2, [[0, 2, 1]]]
-------------------------------
-- Done.
-------------------------------

Example

Use R::=QQ[x,y], DegLex;

F:=x^4y^2+x^2y^4;
G:=[x^2-1,y^2-1];
SB.NFS(G,F);
SB.NFS(G,F,TRUE);

-------------------------------------------------------
-- output:

-- Interpretation:
-- F is in K[G] with 
-- F = 1*G[1]^2G[2] + 1*G[1]G[2]^2 + 1*G[1]^2 + 4*G[1]G[2] + 1*G[2]^2
--      + 3*G[1] + 3*G[2] + 2.

0
-------------------------------
[0, [[2, 1, 1], [1, 2, 1], [2, 0, 1], [1, 1, 4], [0, 2, 1], [1, 0, 3], [0, 1, 3], [0, 0, 2]]]
-------------------------------
-- Done.
-------------------------------

SB.TermRepr

SB.Sagbi