A classical proof that the algebraic homotopy class of a rational   function is the residue pairing
  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. 
Jesse Leo Kass and Kirsten Wickelgren
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