Low-degree minimal generating sets of polynomial ideals

Abstract

We obtain tight bounds for the minimal number of generators of an ideal with bounded-degree generators in a polynomial ring $K[X_1,\dots,X_n],$ as well as a sharp quantification of the maximum possible size of a minimal generating set of bounded degree. Our bounds are sharp for all fields of size greater than the degree. Moreover, we provide explicit constructions reaching the tightness constraints for all fields of characteristic 0, and for all sufficiently large fields in the one- and two-variable case. Additionally, we fully solve the one-variable case, and conjecture the asymptotics in the multivariate case, for all finite fields.