ApCoCoA-1:SB.IsInSubalgebra
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This article is about a function from ApCoCoA-1. If you are looking for the ApCoCoA-2 version of it, see Package sagbi/SB.IsInSubalgebra. |
SB.IsInSubalgebra
Subalgebra membership test of a polynomial.
Syntax
SB.IsInSubalgebra(Gens:LIST of POLY, F:POLY):LIST
Description
This functions checks the membership of the polynomial F in the subalgebra S generated by the polynomials of the list Gens. If F is not a member of S the function will return a list with the corresponding boolean value and NULL, otherwise TRUE and the subalgebra representation of F will be returned (see also SB.NFS and SB.SubalgebraPoly).
Important: This function works only, if a finite SAGBI-basis of S is existing!
@param Gens A list of polynomials, which are the generators of the current subalgebra.
@param F A polynomial.
@return A list consisting of an boolean value and if it exists the subalgebra representation.
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]; SB.IsInSubalgebra(G,F); ------------------------------------------------------- -- output: [FALSE, NULL] ------------------------------- -- Done. -------------------------------
Example
Use R::=QQ[x,y], DegLex; F:=x^3+x^2y; G:=[x+y,xy]; SB.IsInSubalgebra(G,F); ------------------------------------------------------- -- output: [FALSE, NULL] ------------------------------- -- Done. -------------------------------
Example
Use R::=QQ[x,y], DegLex; F:=x^4y^2+x^2y^4; G:=[x^2-1,y^2-1]; L:=SB.IsInSubalgebra(G,F); L; SB.SubalgebraPoly(G,L[2]); ------------------------------------------------------- -- output: [TRUE, [[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. -------------------------------