Distribution of Hilbert polynomials and average-case complexity

Characterize the distribution of Hilbert polynomials within the polynomial ring with rational coefficients Q[x] in order to enable derivation of the average-case runtime complexity of the Discrete Derivative Recovery Algorithm presented in Section 4.

Background

The paper introduces an efficient algorithm based on discrete derivatives to recover the integer partition λ from a given Hilbert polynomial and to decide whether a polynomial is a Hilbert polynomial. The authors derive the worst-case complexity of this algorithm.

However, the authors explicitly note that the average-case complexity cannot be derived because the distribution of Hilbert polynomials among polynomials in the ambient polynomial ring is not precisely known. Establishing this distribution would allow analysis of expected performance and average-case bounds for the recovery algorithm.