getLocalAlgebraBasis -- produces a basis for a local finitely generated algebra over a field or finite étale algebra

Usage:

getLocalAlgebraBasis(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 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 algebra $Q_p(f):=k[x_{1},\dots,x_{n}]_{\mathfrak{m}_{p}}/(f_{1},\dots,f_{n})$ where $\mathfrak{m}_{p}$ is the maximal ideal corresponding to the closed point $p$ by computing a normal basis for $(I:(I:p^{\infty}))$ (see [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 : getLocalAlgebraBasis(f, p)

o4 = {1, x}

o4 : List

References

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

Ways to use getLocalAlgebraBasis:

getLocalAlgebraBasis(List,Ideal)

For the programmer

The object getLocalAlgebraBasis is a method function.

The source of this document is in A1BrouwerDegrees/Documentation/ArithmeticMethodsDoc.m2:57:0.

getLocalAlgebraBasis -- produces a basis for a local finitely generated algebra over a field or finite étale algebra

Description

References

See also

Ways to use getLocalAlgebraBasis:

For the programmer