ApCoCoA:SB.IsInSubalgebra

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<command>

 <title>SB.IsInSubalgebra</title>
 <short_description>Subalgebra membership test of a polynomial.</short_description>
 

<syntax> SB.IsInSubalgebra(Gens:LIST of POLY, F:POLY):LIST </syntax>

 <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 <ref>SB.NFS</ref> and <ref>SB.SubalgebraPoly</ref>). <par/> Important: This function works only, if a finite SAGBI-basis of S is existing!

<itemize>

 <item>@param Gens A list of polynomials, which are the generators of the current subalgebra.</item>
 <item>@param F A polynomial.</item>
 <item>@return A list consisting of an boolean value and if it exists the subalgebra representation.</item>

</itemize>

<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> <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> <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.


</example>

 </description>

<see>SB.NFS</see> <see>SB.SubalgebraPoly</see>

 <types>
   <type>sagbi</type>
   <type>poly</type>
 </types>
 <key>isinsubalgebra</key>
 <key>sb.isinsubalgebra</key>
 <key>sagbi.isinsubalgebra</key>
 <wiki-category>Package_sagbi</wiki-category>

</command>