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Zeroes of Polynomials With Prime Inputs and Schmidt’s $h$-invariant

Part of: Diophantine equations Additive number theory; partitions

Published online by Cambridge University Press:  07 February 2019

Stanley Yao Xiao  and
Shuntaro Yamagishi
Stanley Yao Xiao
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON, Canada Email: syxiao@math.toronto.edu
Shuntaro Yamagishi
Affiliation:
Department of Mathematics & Statistics, Queen’s University, Kingston, ON K7L 3N6, Canada Email: sy46@queensu.ca
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Abstract

In this paper we show that a polynomial equation admits infinitely many prime-tuple solutions, assuming only that the equation satisfies suitable local conditions and the polynomial is sufficiently non-degenerate algebraically. Our notion of algebraic non-degeneracy is related to the $h$-invariant introduced by W. M. Schmidt. Our results prove a conjecture by B. Cook and Á. Magyar for hypersurfaces of degree 3.

Keywords

Circle methodh-invariantHardy–Littlewoodprime numbers

MSC classification

Primary: 11D72: Equations in many variables
Secondary: 11P32: Goldbach-type theorems; other additive questions involving primes
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 3 , June 2020 , pp. 805 - 833
Copyright
© Canadian Mathematical Society 2019

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References

Birch, B. J., Forms in many variables. Proc. Roy. Soc. Ser. A 265(1961/1962), 245–263. https://doi.org/10.1098/rspa.1962.0007Google Scholar
Cook, B. and Magyar, Á., Diophantine equations in the primes. Invent. Math. 198(2014), 701–737. https://doi.org/10.1007/s00222-014-0508-1Google Scholar
Hua, L. K., Additive theory of prime numbers. (Translations of Mathematical Monographs, 13), American Mathematical Society, Providence, RI, 1965.Google Scholar
Schmidt, W. M., The density of integer points on homogeneous varieties. Acta Math. 154(1985), 3–4, 243–296. https://doi.org/10.1007/978-1-4939-3201-6_9Google Scholar