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∈ Z[x₁,,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.