Hilbert polynomials for arbitrary graphs

Determine the Hilbert polynomial of \(H_i(B\Gamma)\) for an arbitrary graph \(\Gamma\), beyond the cases for which explicit formulas are currently known.

Background

The paper defines the Hilbert polynomial PΓi(k)P^i_\Gamma(k) by eventual agreement with the Betti numbers dim⁡FHi(BkΓ)\dim_F H_i(B_k\Gamma). It derives complete formulas for bunches of grapes using one-bridge decompositions and local graphs, but notes that only computational results are available for some small graphs. The general determination of the Hilbert polynomial remains unresolved.

References

However, the Hilbert polynomial is unknown in general.

— Hilbert polynomials of configuration spaces over graphs of circumference at most 1  (2505.24416 - An et al., 30 May 2025) in Section 2, subsection “Edge stabilization and the Hilbert polynomial,” Remark following Definition “Hilbert polynomials”