A classical proof that the algebraic homotopy class of a rational function is the residue pairing

TL;DR

This paper provides a new proof linking the algebraic homotopy class of rational functions to the residue pairing, connecting classical results with modern algebraic topology and extending to multivariable cases.

Contribution

It offers an alternative proof of Cazanave's result using Hurwitz's classical approach and interprets it through the residue pairing, extending the understanding to multiple variables.

Findings

Alternative proof of Cazanave's algebraic homotopy class result

Connection between homotopy class and residue pairing

Extension of results to functions in multiple variables

Abstract

Cazanave has identified the algebraic homotopy class of a rational function of $1$ variable with an explicit nondegenerate symmetric bilinear form. Here we show that Hurwitz's proof of a classical result about real rational functions essentially gives an alternative proof of the stable part of Cazanave's result. We also explain how this result can be interpreted in terms of the residue pairing and that this interpretation relates the result to the signature theorem of Eisenbud, Khimshiashvili, and Levine, showing that Cazanave's result answers a question posed by Eisenbud for polynomial functions in $1$ variable. Finally, we announce results answering this question for functions in an arbitrary number of variables.