The polynomial method over varieties
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The polynomial method over varieties Miguel N. Walsh1
Received: 6 April 2019 / Accepted: 30 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . 2.1 Notation . . . . . . . . . . . . . . . . . . . 2.2 Algebraic preliminaries . . . . . . . . . . . 3 Polynomial partitioning for varieties . . . . . . . 4 Siegel’s Lemma for varieties . . . . . . . . . . . 5 Estimating the partial degrees . . . . . . . . . . 6 Envelopes and full covers . . . . . . . . . . . . 6.1 Envelopes . . . . . . . . . . . . . . . . . . 6.2 Full covers . . . . . . . . . . . . . . . . . . 7 Bounding the number of connected components 7.1 A result of Barone and Basu . . . . . . . . 7.2 Proof of Theorem 1.4 . . . . . . . . . . . . 7.3 Proof of Theorem 1.5 . . . . . . . . . . . .
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B Miguel N. Walsh
[email protected]
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Departamento de Matemática e IMAS-CONICET, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
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M. N. Walsh 8 The incidence geometry of hypersurfaces 8.1 Free configurations . . . . . . . . . 8.2 Proof of Theorem 1.6 . . . . . . . . 8.3 Sharp constructions . . . . . . . . . References . . . . . . . . . . . . . . . . . .
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1 Introduction The polynomial method is a powerful tool for establishing results in a wide variety of areas by means of the construction of an adequate polynomial [18,37,39]. By its very nature, many of the arguments involving this method require us to inductively study what happens inside the varieties produced by this polynomial. Because of this, one may be lead to the study of how the polynomial method adapts to ge
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