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# 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

## Synopsis

• Usage:
locallA1Degree(L,p)
• Inputs:
• L, a list, of polynomials $f = (f_1, \ldots, f_n)$ in the polynomial ring $k[x_1,\ldots,x_n]$ over a field $k$
• p, an ideal, a prime ideal $p \trianglelefteq k[x_1,\ldots,x_n]$ in the zero locus $V(f)$
• Outputs:
• , the class $\text{deg}_p^{\mathbb{A}^1}(f)$ in the Grothendieck-Witt ring $\text{GW}(k)$

## Description

Given an endomorphism of affine space $f=(f_1,\dots ,f_n) \colon \mathbb{A}^n_k \to \mathbb{A}^n_k$ and an isolated zero $p\in V(f)$, we may compute its local $\mathbb{A}^1$-Brouwer degree valued in the Grothendieck-Witt ring $\text{GW}(k)$.

For historical and mathematical background, see global A1-degrees.

 i1 : T1 = QQ[z_1..z_2]; i2 : f1 = {(z_1-1)*z_1*z_2, (3/5)*z_1^2 - (17/3)*z_2^2}; i3 : f1GD = globalA1Degree(f1); i4 : q=ideal {z_1,z_2}; o4 : Ideal of T1 i5 : r=ideal {z_1-1,z_2^2-(9/85)}; o5 : Ideal of T1 i6 : f1LDq= localA1Degree(f1,q) o6 = GrothendieckWittClass{cache => CacheTable{} } matrix => | 0 0 0 17/3 | | 0 3/5 0 -17/3 | | 0 0 17/3 0 | | 17/3 -17/3 0 0 | o6 : GrothendieckWittClass i7 : f1LDr= localA1Degree(f1,r) o7 = GrothendieckWittClass{cache => CacheTable{} } matrix => | -3/5 0 | | 0 -17/3 | o7 : GrothendieckWittClass i8 : f1LDsum = gwAdd(f1LDq, f1LDr) o8 = GrothendieckWittClass{cache => CacheTable{} } matrix => | 0 0 0 17/3 0 0 | | 0 3/5 0 -17/3 0 0 | | 0 0 17/3 0 0 0 | | 17/3 -17/3 0 0 0 0 | | 0 0 0 0 -3/5 0 | | 0 0 0 0 0 -17/3 | o8 : GrothendieckWittClass

The sum of the local A1-degrees is equal to the global A1-degree:

 i9 : gwIsomorphic(f1GD,f1LDsum) o9 = true