Application of a polynomial sieve: beyond separation of variables

Bonolis, Dante; Pierce, Lillian B.

doi:10.2140/ant.2024.18.1515

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Application of a polynomial sieve: beyond separation of variables

Dante Bonolis and Lillian B. Pierce

Vol. 18 (2024), No. 8, 1515–1556

DOI: 10.2140/ant.2024.18.1515

Abstract

Let a polynomial $f \in ℤ [X_{1}, \dots, X_{n}]$ be given. The square sieve can provide an upper bound for the number of integral $x \in {[- B, B]}^{n}$ such that $f (x)$ is a perfect square. Recently this has been generalized substantially: first to a power sieve, counting $x \in {[- B, B]}^{n}$ for which $f (x) = y^{r}$ is solvable for $y \in ℤ$ ; then to a polynomial sieve, counting $x \in {[- B, B]}^{n}$ for which $f (x) = g (y)$ is solvable, for a given polynomial $g$ . Formally, a polynomial sieve lemma can encompass the more general problem of counting $x \in {[- B, B]}^{n}$ for which $F (y, x) = 0$ is solvable, for a given polynomial $F$ . Previous applications, however, have only succeeded in the case that $F (y, x)$ exhibits separation of variables, that is, $F (y, x)$ takes the form $f (x) - g (y)$ . In the present work, we present the first application of a polynomial sieve to count $x \in {[- B, B]}^{n}$ such that $F (y, x) = 0$ is solvable, in a case for which $F$ does not exhibit separation of variables. Consequently, we obtain a new result toward a question of Serre, pertaining to counting points in thin sets.

Keywords

thin sets, polynomial sieve

Mathematical Subject Classification

Primary: 11D45, 11D85, 11N36

Milestones

Received: 6 September 2022

Revised: 19 June 2023

Accepted: 31 October 2023

Published: 18 September 2024

Authors

Dante Bonolis
Mathematics Department Duke University Durham, NC 27708 United States

Lillian B. Pierce
Mathematics Department Duke University Durham, NC 27708 United States

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Vol. 18, No. 8, 2024