Quantitative concatenation for polynomial box norms

Using PET and quantitative concatenation techniques, we es- tablish box-norm control with the “expected” directions for counting operators for general multidimensional polynomial progressions, with at most polynomial losses in the parame- ters. Such results are often useful first steps towards obtaining explicit upper bounds on sets lacking instances of given such progressions. In the companion paper [20], we complete this program for sets in [N ]2 lacking nondegenerate progressions of the form (x, y), (x + P (z), y), (x, y + P (z)), where P ∈ Z[z] is any fixed polynomial with an integer root of multiplicity 1.