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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
Abstract

Let a polynomial f [X1,,Xn] be given. The square sieve can provide an upper bound for the number of integral x [B,B]n such that f(x) is a perfect square. Recently this has been generalized substantially: first to a power sieve, counting x [B,B]n for which f(x) = yr is solvable for y ; then to a polynomial sieve, counting x [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 [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 [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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