localAlgebraBasis -- produces a basis for a local finitely generated algebra over a field k

Synopsis

• Usage:

  localAlgebraBasis(L,p)

• Inputs:
  • L, a list, of polynomials $f=(f_1, \dots ,f_n)$ over the same ring
  • p, an ideal, a prime ideal of an isolated zero
• Outputs:
  • a list, of basis elements of the local $k$-algebra $Q_p(f)$

Description

Given an endomorphism of affine space, $f=(f_1,\dots ,f_n)$, given as a list of polynomials called L and the prime ideal of an isolated zero, this command returns a list of basis elements of the local k-algebra $Q_p(f)$ by computing a normal basis for $(I:(I:p^{\infty}))$ (vis. [S02, Proposition 2.5]).

i1 : QQ[x,y];

i2 : f = {x^2+1-y,y};

i3 : p = ideal(x^2+1,y);

o3 : Ideal of QQ[x..y]

i4 : localAlgebraBasis(f,p) 

o4 = {1, x}

o4 : List

Citations:

• [S02] B. Sturmfels, Solving Systems of Polynomial Equations, American Mathematical Society, 2002.

See also

• localA1Degree -- computes a local A1-Brouwer degree of a list of n polynomials in n variables over a field k at a prime ideal in the zero locus