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D.4.21.5 algebraicDependence

Procedure from library sagbi.lib (see sagbi_lib).

Usage:
algebraicDependence(I,it); I an an ideal, it is an integer

Return:
ring

Assume:
basering is not a qring

Purpose:
In it iterations, compute algebraic dependencies between elements of I

Example:
 
LIB "sagbi.lib";
ring r= 0,(x,y),dp;
//The following algebra does not have a finite SAGBI basis.
ideal I=x^2, xy-y2, xy2;
//---------------------------------------------------
//Call with two iterations
def DI = algebraicDependence(I,2);
==> //SAGBI construction algorithm stopped as it reached the limit of 2 itera\
   tions.
==> //In general the returned generators are no SAGBI basis for the given alg\
   ebra.
setring DI; algDep;
==> algDep[1]=0
// we see that no dependency has been seen so far
//---------------------------------------------------
//Call with two iterations
setring r; kill DI;
def DI = algebraicDependence(I,3);
==> //SAGBI construction algorithm stopped as it reached the limit of 3 itera\
   tions.
==> //In general the returned generators are no SAGBI basis for the given alg\
   ebra.
setring DI; algDep;
==> algDep[1]=-2*@c(1)^2*@c(2)^4+2*@c(1)^3*@c(3)^2-8*@c(1)^2*@c(2)*@c(3)^2+4*\
   @c(1)*@c(2)^2*@c(3)^2-2*@c(3)^4
map F = DI,x,y,x^2, xy-y2, xy2;
F(algDep); // we see that it is a dependence indeed
==> _[1]=0

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