Application of a polynomial sieve: Beyond separation of variables

Let a polynomial 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) = y^r is solvable for; 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.

Duke Scholars

Author Lillian Beatrix Pierce Mathematics