getLocalUnstableA1Degree -- computes a local unstable $\mathbb{A}^{1}$-Brouwer degree of a pointed rational function $f/g:\mathbb{P}^{1}_{k}\to\mathbb{P}^{1}_{k}$ at a root $p\in\mathbb{P}^{1}_{k}$

• Usage:

  getLocalUnstableA1Degree(q, p)

  getLocalUnstableA1Degree(f, g, p)

• Inputs:
  • q, a ring element, a pointed rational function $f/g$ where $f,g\in k[x]$ are coprime polynomials over a field $k$ of characteristic not 2 and $g$ not identically zero. Over $\mathbb{C}$ the two-input variant is to be used as fraction fields are not supported over $\mathbb{C}$. Over $\mathbb{R}$, the user is prompted to instead do the computation over $\mathbb{Q}$ and then base change to $\mathbb{R}$.
  • f, a ring element, a polynomial $f\in k[x]$ where $k$ is a field of characteristic not 2
  • g, a ring element, a polynomial $g\in k[x]$ where $k$ is a field of characteristic not 2 and $g$ is not identically zero, such that $f/g$ is a pointed rational function
  • p, a ring element, a point $p\in\mathbb{P}^{1}_{k}$ corresponding to a root of the rational function $f/g$ in the field $k$
  • p, a number, a point $p\in\mathbb{P}^{1}_{k}$ corresponding to a root of the rational function $f/g$ in the field $k$
• Optional inputs:
  • linearTolerance => a real number, default value .000001, a positive real number used to determine whether roots of f and g are considered equal when working over the inexact field $\mathbb{C}$, used to put rational functions over $\mathbb{C}$ in reduced form by cancelling roots that agree up to the tolerance
• Outputs:
  • an Unstable Grothendieck-Witt Class, the class $\text{deg}_{p}^{\mathbb{A}^{1}}(f/g)$ in the unstable Grothendieck-Witt group $\text{GW}^{u}(k)$

i1 : frac QQ[x];

i2 : q = (x^2 + x - 2)/(3*x + 5);

i3 : getLocalUnstableA1Degree(q, -2)

               1
o3 = (| 1/3 |, -)
               3

o3 : UnstableGrothendieckWittClass