The Furstenberg-S\'ark\"ozy theorem for polynomials in one or more prime variables

We establish upper bounds on the size of the largest subset of \1, 2, , N\ lacking nonzero differences of the form h (p₁, , p_), where h Zx₁, , x_ is a fixed polynomial satisfying appropriate conditions and p₁, , p_ are prime. The bounds are of the same type as the best-known analogs for unrestricted integer inputs, due to Bloom-Maynard and Arala for =1, and to the authors for 2.