Improve or match the exponent in the degree–size relation

Improve the exponent in the degree–size relation for resolution over bounded-degree polynomials, or construct formula families whose proof-size lower bounds demonstrate that the existing exponent is tight.

Background

The main theorem gives a quantitative lower bound of the form ln⁡S≥D(D/(16n))(p−1)r/(32p2r)\ln S\ge D( D/(16n))^{(p-1)r}/(32p^2r) for resolution over degree-rr polynomials over Fp\mathbb F_p. The conclusion leaves open whether this exponent reflects the true strength of the degree–size tradeoff.

A resolution of this problem could proceed either by strengthening the lower-bound analysis or by identifying explicit families with matching upper bounds, thereby establishing tightness.

References

It would also be interesting to either improve the exponent in the relation~eq:main or find families witnessing its tightness, in light of the results for resolution and PC/PCR.

eq:main:

ln⁡S≥D32p2r(D16n)(p−1)r.\ln S\ge\frac{D}{32p^2r}\left(\frac{D}{16n}\right)^{(p-1)r}.

— A Degree--Size Relation for Resolution over Polynomials  (2610.00837 - Pang, 30 Sep 2026) in Section 7, paragraph titled “Open problems”