We establish upper bounds on the size of the largest subset of \(\{1,2,\dots ,N\}\) lacking nonzero differences of the form \(h(p_1,\dots ,p_{\ell })\) , where \(h\in {\mathbb {Z}}[x_1,\dots ,x_{\ell }]\) is a fixed polynomial satisfying appropriate conditions and \(p_1,\dots ,p_{\ell }\) are prime. The bounds are of the type established in the unrestricted integer input setting by Bloom-Maynard and Arala for \(\ell =1\) , and by the authors for \(\ell \ge 2\) .
