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

Published online by Cambridge University Press:  07 February 2019

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

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.

Type
Article
Copyright
© Canadian Mathematical Society 2019

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References

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