Improve approximation error-elimination rate or reduce approximation degree

Determine whether errors in the Razborov–Smolensky approximations used for resolution over bounded-degree polynomials can be eliminated with a better rate than the parameter \(\eta_K(\delta)\), or construct approximations whose degree grows more slowly.

Background

The degree–size relation is obtained by constructing Razborov–Smolensky approximators and a common multiplier that annihilates approximation errors modulo bounded-degree polynomial-calculus consequences. The quantitative strength of the argument depends on the rank-shrinkage parameter $\eta_K(\delta)=(1-1/p)N_n(K-\delta)/N_n(K)$ and on the degree required by the approximators.

The conclusion identifies the tradeoff between approximation degree and rank shrinkage as a bottleneck, especially when the input polynomial degree at one layer causes an exponential loss in the shrinkage rate at the next layer. Improving either component could extend the method to stronger lower bounds and deeper proof systems.