The Hilbert polynomial of a homogeneous complete intersection is determined by the degrees of the generators of the defining ideal. The degrees of the generators are not, in general, determined by the Hilbert polynomial — but sometimes they are. When? We give some general criteria and completely answer the question up to codimension 6.
"Complete intersections with given Hilbert polynomials." J. Commut. Algebra 13 (1) 29 - 39, Spring 2021. https://doi.org/10.1216/jca.2021.13.29