Exact values of low-degree Hilbert numbers

Determine the exact value of the Hilbert number H(d) for each polynomial degree d, including the presently unresolved case H(2).

Background

The paper surveys known lower bounds for H(d), including explicit constructions for degrees 2 through 8. These bounds establish only that at least a specified number of limit cycles can occur; they do not establish matching upper bounds or exact values.

The authors explicitly note that even the quadratic Hilbert number remains unresolved. Thus, determining H(2) is a concrete low-degree instance of the broader problem of computing the exact Hilbert numbers.