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:
- a Grothendieck Witt Class, 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

Ways to use localA1Degree :

localA1Degree(List,Ideal)

For the programmer

The object localA1Degree is a method function.

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

Description

See also

Ways to use localA1Degree :

For the programmer