ApCoCoA-1:SB.SubalgebraPoly

This article is about a function from ApCoCoA-1.

SB.SubalgebraPoly

Computes a subalgebra polynomial from a subalgebra representation.

Syntax

SB.SubalgebraPoly(Gens:LIST of POLY, SARepr:LIST of LIST of INT):POLY

Description

This function computes from a given representation of a polynomial as a list of logarithms (see also SB.NFS) the polynomial in the current subalgebra, which is generated by the polynomials of the list Gens. Example: Let Gens=[g_1,g_2,g_3] be the list of subalgebra generators, let S=K[y_1,y_2,y_3] be the current subalgebra and SARepr=[[0,2,3,-1],[2,3,1,4]] the given representation. Then the polynomial

-1*(y_2)^2(y_3)^3 + 4*(y_1)^2(y_2)^3(y_3)

in the ring S will be returned.

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

• @param SARepr A list of lists with integers as entries.

• @return A polynomial in the current subalgebra.

Example

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

F:=x^4+x^3y+x^2y^2+y^4;
G:=[x^2-y^2,x^2y,x^2y^2-y^4,x^2y^4,y^6x^2y^6-y^8];
L:=SB.NFS(G,F,TRUE);
L;

SB.SubalgebraPoly(G,L[2]);

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

[x^3y + 3x^2y^2, [[2, 0, 0, 0, 0, 1]]]
-------------------------------
SARing :: y[1]^2
-------------------------------
-- Done.
-------------------------------

Example

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

F:=x^3+x^2y;
G:=[x+y,xy];
L:=SB.NFS(G,F,TRUE);
L;

SB.SubalgebraPoly(G,L[2]);

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

[-xy^2 - y^3, [[3, 0, 1], [1, 1, -2]]]
-------------------------------
SARing :: y[1]^3 - 2y[1]y[2]
-------------------------------
-- Done.
-------------------------------

Example

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

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

SB.SubalgebraPoly(G,L[2]);

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

[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]]]
-------------------------------
SARing :: y[1]^2y[2] + y[1]y[2]^2 + y[1]^2 + 4y[1]y[2] + y[2]^2 + 3y[1] + 3y[2] + 2
-------------------------------
-- Done.
-------------------------------

SB.NFS