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

Journal Article (Academic article)

© 2020 Elsevier Inc. 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.

Full Text

Duke Authors

Cited Authors

  • Kass, JL; Wickelgren, K

Published Date

  • June 15, 2020

Published In

Volume / Issue

  • 595 /

Start / End Page

  • 157 - 181

International Standard Serial Number (ISSN)

  • 0024-3795

Digital Object Identifier (DOI)

  • 10.1016/j.laa.2019.12.041