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

• Usage:

  getLocallA1Degree(L, p)

• Inputs:
  • L, a list, of polynomials $f=(f_{1}, \ldots, f_{n})$ in the polynomial ring $k[x_{1},\dots,x_{n}]$ where $k$ is $\mathbb{C}$, $\mathbb{Q}$, or a finite field of characteristic not 2. Over $\mathbb{R}$, the user is prompted to instead do the computation over $\mathbb{Q}$ and then base change to $\mathbb{R}$.
  • p, an ideal, a prime ideal $p\subset k[x_{1},\dots,x_{n}]$ corresponding to a point 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 $\text{GW}(k)$

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 : q = ideal(z_1, z_2);

o3 : Ideal of T1

i4 : r = ideal(z_1-1, z_2^2 - 9/85);

o4 : Ideal of T1

i5 : f1LDq = getLocalA1Degree(f1, q)

o5 = | 0    0     0    17/3  |
     | 0    3/5   0    -17/3 |
     | 0    0     17/3 0     |
     | 17/3 -17/3 0    0     |

o5 : GrothendieckWittClass

i6 : f1LDr = getLocalA1Degree(f1, r)

o6 = | -3/5 0     |
     | 0    -17/3 |

o6 : GrothendieckWittClass

i7 : f1LDsum = addGW(f1LDq,f1LDr);

i8 : f1GD = getGlobalA1Degree f1;

i9 : isIsomorphicForm(f1GD,f1LDsum)

o9 = true