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The polynomial method over varieties

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Abstract

We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic varieties. As a first application, we provide a general incidence estimate that is tight in its dependence on the size, degree and dimension of the varieties involved.

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Acknowledgements

Part of this work was carried while the author was a Clay Research Fellow and a Fellow of Merton College at the University of Oxford. The author would like to thank Cosmin Pohoata and Martin Sombra for pointing out some typos in an earlier version of this manuscript, as well as an anonymous referee for some helpful suggestions.

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  1. Departamento de Matemática e IMAS-CONICET, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428, Buenos Aires, Argentina

    Miguel N. Walsh

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  1. Miguel N. Walsh
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Correspondence to Miguel N. Walsh.

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Walsh, M.N. The polynomial method over varieties. Invent. math. 222, 469–512 (2020). https://doi.org/10.1007/s00222-020-00975-6

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  • DOI: https://doi.org/10.1007/s00222-020-00975-6

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