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

Publication
, Journal Article

Kass, JL; Wickelgren, K

Published in: Linear Algebra and Its Applications

June 15, 2020

© 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.

### Duke Scholars

## Published In

Linear Algebra and Its Applications

## DOI

## ISSN

0024-3795

## Publication Date

June 15, 2020

## Volume

595

## Start / End Page

157 / 181

## Related Subject Headings

- Numerical & Computational Mathematics
- 49 Mathematical sciences
- 40 Engineering
- 09 Engineering
- 08 Information and Computing Sciences
- 01 Mathematical Sciences

### Citation

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Chicago

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MLA

NLM

Kass, J. L., & Wickelgren, K. (2020). A classical proof that the algebraic homotopy class of a rational function is the residue pairing.

*Linear Algebra and Its Applications*,*595*, 157–181. https://doi.org/10.1016/j.laa.2019.12.041Kass, J. L., and K. Wickelgren. “A classical proof that the algebraic homotopy class of a rational function is the residue pairing.”

*Linear Algebra and Its Applications*595 (June 15, 2020): 157–81. https://doi.org/10.1016/j.laa.2019.12.041.Kass JL, Wickelgren K. A classical proof that the algebraic homotopy class of a rational function is the residue pairing. Linear Algebra and Its Applications. 2020 Jun 15;595:157–81.

Kass, J. L., and K. Wickelgren. “A classical proof that the algebraic homotopy class of a rational function is the residue pairing.”

*Linear Algebra and Its Applications*, vol. 595, June 2020, pp. 157–81.*Manual*, doi:10.1016/j.laa.2019.12.041.Kass JL, Wickelgren K. A classical proof that the algebraic homotopy class of a rational function is the residue pairing. Linear Algebra and Its Applications. 2020 Jun 15;595:157–181.

## Published In

Linear Algebra and Its Applications

## DOI

## ISSN

0024-3795

## Publication Date

June 15, 2020

## Volume

595

## Start / End Page

157 / 181

## Related Subject Headings

- Numerical & Computational Mathematics
- 49 Mathematical sciences
- 40 Engineering
- 09 Engineering
- 08 Information and Computing Sciences
- 01 Mathematical Sciences