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# 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.

## See also

- getLocalA1Degree -- computes a local $\mathbb{A}^{1}$-Brouwer degree of a list of $n$ polynomials in $n$ variables over a field $k$ at a prime ideal in the zero locus

## Ways to use getLocalAlgebraBasis:

- `getLocalAlgebraBasis(List,Ideal)`

## For the programmer

The object getLocalAlgebraBasis is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/A1BrouwerDegrees/Documentation/ArithmeticMethodsDoc.m2:57:0.