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Global index of real polynomials

Part of: Fiber spaces and bundles Curves Theory of singularities and catastrophe theory Differential topology

Published online by Cambridge University Press:  18 March 2024

Gabriel E. Monsalve  and
Mihai Tibăr
Gabriel E. Monsalve
Affiliation:
Universidade de São Paulo-ICMC, São Carlos, Brasil ([email protected]) Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France
Mihai Tibăr
Affiliation:
Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France ([email protected])
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Abstract

We develop two methods for expressing the global index of the gradient of a 2 variable polynomial function $f$: in terms of the atypical fibres of $f$, and in terms of the clusters of Milnor arcs at infinity. These allow us to derive upper bounds for the global index, in particular refining the one that was found by Durfee in terms of the degree of $f$.

Keywords

polynomial functionsatypical fibresindex of the gradient

MSC classification

Primary: 57R70: Critical points and critical submanifolds
Secondary: 58K60: Deformation of singularities 55R55: Fiberings with singularities 14H20: Singularities, local rings
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 29
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Copyright
Copyright © The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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