Application of a polynomial sieve: beyond separation of variables
Dante Bonolis, Lillian B. Pierce · Algebra & Number Theory · 2024
Let a polynomial f 2 ޚŒX 1 ; : : : ; X n be given.The square sieve can provide an upper bound for the number of integral x 2 Œ B; B n such that f .x/ is a perfect square.Recently this has been generalized substantially: first to a power sieve, counting x 2 Œ B; B n for which f .x/D y r is solvable for y 2 ;ޚ then to a polynomial sieve, counting x 2 Œ B; B n for which f .x/D g.y/ is solvable, for a given polynomial g.Formally, a polynomial sieve lemma can encompass the more general problem of counting x 2 Œ B; B n for which F.y; x/ D 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 2 Œ B; B n such that F.y; x/ D 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.