getGlobalUnstableA1Degree -- computes the global unstable $\mathbb{A}^{1}$-Brouwer degree of a pointed rational function $f/g:\mathbb{P}^{1}_{k}\to\mathbb{P}^{1}_{k}$

Given a pointed rational function $f/g:\mathbb{P}^{1}_{k}\to\mathbb{P}^{1}_{k}$ (where $(f/g)(\infty)=\infty$), we may compute its global unstable $\mathbb{A}^{1}$-Brouwer degree valued in the unstable Grothendieck-Witt group $\text{GW}^{u}(k):=\text{GW}(k)\times_{k^{\times}/(k^{\times})^{2}}k^{\times}$.

Morel's $\mathbb{A}^{1}$-Brouwer degree generalizes the classical Brouwer degree by associating to an endomorphism of the sphere a class in the Grothendieck-Witt ring of non-degenerate symmetric bilinear forms. While this morphism is an isomorphism in dimensions two and above, it is only surjective in dimension one [M12]. In this case, a computation of Morel [M12] and Cazanave [C12] furnish an isomorphism $[\mathbb{P}^{1}_{k},\mathbb{P}^{1}_{k}]\cong\text{GW}^{u}(k)$.

Building on Cazanave's work, Kass and Wickelgren [KW20] and later Igieobo and coauthors [I+24] prove that there is an explicit bilinear form associated to the rational function $f/g$ in both the local and global cases, and provide a local-to-global formula for the degree dependent on the configuration of the zeroes of the rational function.

For additional historical and mathematical background about $\mathbb{A}^{1}$-degrees and its relationship to $\mathbb{A}^{1}$-algebraic topology more generally, see global A1-degrees.

In the case of the global unstable $\mathbb{A}^{1}$-Brouwer degree, the class is represented by a variant of the Bézoutian bilinear form [C12, Theorem 3.6].

The rank of this form is of rank five, which agrees with the number of zeroes of the rational function counted with multiplicity over the complex numbers.

In the unstable setting, however, the global unstable $\mathbb{A}^{1}$-Brouwer degree is not computed as the sum of local $\mathbb{A}^{1}$-Brouwer degrees at the zeroes of the rational function in the unstable Grothendieck-Witt ring $\text{GW}^{u}(k)$. Instead, it is computed as the divisorial sum which depends on the divisor of points given by the zeroes of the rational function [I+24].

Since $f/g$ is assumed to be reduced, $f$ and $g$ do not share a common factor. If the user chooses functions $f$ and $g$ with a common factor, the reduction is computed, checked for pointedness, and the degree computation run on the reduction. Over $\mathbb{Q}$ and $\mathbb{F}_{q}$, rational functions are automatically reduced, and computing getGlobalUnstableA1Degree(f/g) agrees with computing getGlobalUnstableA1Degree(f,g) as the computation is done on the underlying rational function. Over $\mathbb{C}$, we consider roots of $f$ and $g$ to be equal if the absolute value of their difference is less than that of the linearTolerance. The user may specify a tolerance for this cancellation via the option linearTolerance, with the default being 1e-6.

[C12] C. Cazanave, "Algebraic homotopy classes of rational functions," Ann. Scient. Ec. Norm. Sup., 2012.

[I+24] J. Igieobo, et. al., "Motivic configurations on the line," arXiv: 2411.15347, 2024.

[KW20] J. Kass, K. Wickelgren, "A Classical Proof that the Algebraic Homotopy Class of a Rational Function is the Residue Pairing," Linear Algebra Appl., 2020.

[M12] F. Morel, "$\mathbb{A}^{1}$-Algebraic topology over a field," Springer Lecture Notes in Mathematics, 2012.