Macaulay2 » Documentation
Packages » A1BrouwerDegrees :: getLocalA1Degree
next | previous | forward | backward | up | index | toc

getLocalA1Degree -- 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

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 = getGlobalA1Degree 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 = getLocalA1Degree(f1, q)

o6 = | 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 = getLocalA1Degree(f1, r)

o7 = | -3/5 0     |
     | 0    -17/3 |

o7 : GrothendieckWittClass
i8 : f1LDsum = addGW(f1LDq,f1LDr)

o8 = | 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 : isIsomorphicForm(f1GD,f1LDsum)

o9 = true

See also

Ways to use getLocalA1Degree:

  • getLocalA1Degree(List,Ideal)

For the programmer

The object getLocalA1Degree is a method function.


The source of this document is in A1BrouwerDegrees/Documentation/LocalGlobalDegreesDoc.m2:95:0.