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A classical proof that the algebraic homotopy class of a rational function is the residue pairing. (English) Zbl 1437.14030
The authors of this paper have a substantial research project based around the concept of “degree” in \(\mathbb{A}^1\)-homotopy theory. This paper is a smaller companion of a larger paper of theirs [Duke Math. J. 168, No. 3, 429–469 (2019; Zbl 1412.14014)]. Fix a field \(k\). The main theorem of the major paper identifies the local \(\mathbb{A}^1\)-degree at \(0\) of a map \(f: \mathbb{A}_k^n \to \mathbb{A}_k^n\) having an isolated zero at the origin with the class in the Grothendieck-Witt group \(GW(k)\) of a specific bilinear form depending on \(f\).
This paper concerns maps \(f/g: \mathbb{P}_k^1 \to \mathbb{P}_k^1\). A construction by Hurwitz from 1895 calculates the topological degree of a rational map \(f/g: S^1=\mathbb{R}P^1 \to \mathbb{R}P^1=S^1\) as the signature of an explicitly-associated bilinear form. For general \(k\), the variety \(\mathbb{P}_k^1\) is an \(\mathbb{A}^1\)-homotopy sphere, and the set of stable \(\mathbb{A}^1\)-homotopy self-maps of \(\mathbb{P}_k^1\) is \(GW(k)\), by virtue of work of F. Morel [in: Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures. Zürich: European Mathematical Society (EMS). 1035–1059 (2006; Zbl 1097.14014)].
A paper of C. Cazanave [C. R., Math., Acad. Sci. Paris 346, No. 3–4, 129–133 (2008; Zbl 1151.14016)] gives an explicit description of the unstable \(\mathbb{A}^1\)-homotopy classes of self-maps of \(\mathbb{P}^1_k\), and in particular, given any rational map \(f/g\), one can produce an associated stable-degree class in \(GW(k)\), the class of a completely explicit “Bézout matrix”.
Fix a rational map \(f/g : \mathbb{P}^1_k \to \mathbb{P}^1_k\). This paper uses the authors’ work [Duke Math. J. 168, No. 3, 429–469 (2019; Zbl 1412.14014)] to produce Cazanave’s class for \(f/g\) by the classical topological method of summation over local degrees. The result is somewhat weaker than can be obtained by Cazanave’s methods, because the invariants considered here are only in the stable \(\mathbb{A}^1\)-homotopy groups, not the unstable. Nonetheless, it is extremely pleasant to see that the machinery of the \(\mathbb{A}^1\)-degree producing a completely explicit, calculable invariant of self-maps of \(\mathbb{P}^1_k\) by classical homotopy-theoretic methods.
The paper is short, and much of it is exposition. This is sure to be of great help to anyone who wishes to learn about the \(\mathbb{A}^1\)-degree, which might be daunting to the newcomer. Most of what is not exposition is concerned with the details required to make the \(\mathbb{A}^1\)-degree machinery work in this context. It is highly recommended to anyone wanting to see how this theory actually works in practice. What remains is some brief results required to show certain matrices are congruent to one another, and a final Section 3 that discusses the connection between the work at hand and the work of D. Eisenbud and H. I. Levine [Ann. Math. (2) 106, 19–44 (1977; Zbl 0398.57020)] and G. N. Khimshiashvili [Soobshch. Akad. Nauk Gruz. SSR 85, 309–312 (1977; Zbl 0346.55008); Proc. A. Razmadze Math. Inst. 125, 1–121 (2001; Zbl 1059.58027)]. The final section may function as motivation and background for the longer paper: [loc. cit.].

MSC:
14F42 Motivic cohomology; motivic homotopy theory
14B05 Singularities in algebraic geometry
55M25 Degree, winding number
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[1] Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N., Classification of critical points, caustics and wave fronts, (Singularities of Differentiable Maps, vol. 1. Singularities of Differentiable Maps, vol. 1, Modern Birkhäuser Classics (2012), Birkhäuser/Springer: Birkhäuser/Springer New York), translated from the Russian by Ian Porteous based on a previous translation by Mark Reynolds, reprint of the 1985 edition · Zbl 1290.58001
[2] Basu, S.; Pollack, R.; Roy, M.-F., Algorithms in Real Algebraic Geometry, Algorithms and Computation in Mathematics, vol. 10 (2006), Springer-Verlag: Springer-Verlag Berlin
[3] Cazanave, C., Classes d’homotopie de fractions rationnelles, C. R. Math. Acad. Sci. Paris, 346, 3-4, 129-133 (2008) · Zbl 1151.14016
[4] Cazanave, C., Algebraic homotopy classes of rational functions, Ann. Sci. Éc. Norm. Supér. (4), 45, 4, 511-534 (2013), 2012 · Zbl 1419.14025
[5] Conrad, K., Galois descent (2019), online
[6] Eisenbud, D., An algebraic approach to the topological degree of a smooth map, Bull. Am. Math. Soc., 84, 5, 751-764 (1978) · Zbl 0425.55003
[7] Elman, R.; Karpenko, N.; Merkurjev, A., The Algebraic and Geometric Theory of Quadratic Forms, American Mathematical Society Colloquium Publications, vol. 56 (2008), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1165.11042
[8] Eisenbud, D.; Levine, H. I., An algebraic formula for the degree of a \(C^\infty\) map germ, Ann. Math. (2), 106, 1, 19-44 (1977), with an appendix by Bernard Teissier, “Sur une inégalité à la Minkowski pour les multiplicités” · Zbl 0398.57020
[9] Fuhrmann, P. A., A Polynomial Approach to Linear Algebra, Universitext (2012), Springer: Springer New York · Zbl 1239.15001
[10] Gantmacher, F. R., The Theory of Matrices, vol. 2 (1964), Chelsea Publishing Co.: Chelsea Publishing Co. New York, translated by K.A. Hirsch
[11] Hurwitz, A., Ueber die Bedingungen, unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt, Math. Ann., 46, 2, 273-284 (1895) · JFM 26.0119.03
[12] Kass, J. L.; Wickelgren, J., The class of Eisenbud-Khimshiashvili-Levine is the local \(\mathbf{A}^1\)-Brouwer degree, Duke Math. J., 168, 3, 429-469 (2019) · Zbl 1412.14014
[13] Khimshiashvili, G., The local degree of a smooth mapping, Sakharth. SSR Mecn. Akad. Moambe, 85, 2, 309-312 (1977) · Zbl 0346.55008
[14] Khimshiashvili, G., Signature formulae for topological invariants, Proc. A. Razmadze Math. Inst., 125, 1-121 (2001) · Zbl 1059.58027
[15] Knus, M.-A., Quadratic and Hermitian Forms over Rings, Grundlehren der Mathematischen Wissenschaften, vol. 294 (1991), Springer-Verlag: Springer-Verlag Berlin, with a foreword by I. Bertuccioni · Zbl 0756.11008
[16] Kreĭn, M. G.; Naĭmark, M. A., The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations, Linear Multilinear Algebra, 10, 4, 265-308 (1981), translated from the Russian by O. Boshko and J.L. Howland · Zbl 0584.12018
[17] Lam, T. Y., Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics, vol. 67 (2005), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 1068.11023
[18] Lam, T. Y., Serre’s Problem on Projective Modules, Springer Monographs in Mathematics (2006), Springer-Verlag: Springer-Verlag Berlin · Zbl 1101.13001
[19] Levine, M., Motivic homotopy theory, Milan J. Math., 76, 165-199 (2008) · Zbl 1191.14024
[20] Milnor, J.; Husemoller, D., Symmetric Bilinear Forms, Ergebnisse der Mathematik und ihrer Grenzgebiete (1973), Springer-Verlag: Springer-Verlag New York-Heidelberg · Zbl 0292.10016
[21] Morel, F., On the motivic \(\pi_0\) of the sphere spectrum, (Axiomatic, Enriched and Motivic Homotopy Theory. Axiomatic, Enriched and Motivic Homotopy Theory, NATO Sci. Ser. II Math. Phys. Chem., vol. 131 (2004), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht), 219-260 · Zbl 1130.14019
[22] Morel, F., \( \mathbb{A}^1\)-algebraic topology, (International Congress of Mathematicians, vol. II (2006), Eur. Math. Soc.: Eur. Math. Soc. Zürich), 1035-1059 · Zbl 1097.14014
[23] Morel, F.; Voevodsky, V., \( \mathbb{A}^1\)-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math., 90, 45-143 (2001), 1999 · Zbl 0983.14007
[24] Palamodov, V. P., The multiplicity of a holomorphic transformation, Funkc. Anal. Priložen., 1, 3, 54-65 (1967)
[25] Rahman, Q. I.; Schmeisser, G., Analytic Theory of Polynomials, London Mathematical Society Monographs. New Series, vol. 26 (2002), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press Oxford · Zbl 1072.30006
[26] Scheja, G.; Storch, U., Über Spurfunktionen bei vollständigen Durchschnitten, J. Reine Angew. Math., 278, 279, 174-190 (1975) · Zbl 0316.13003
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