Intrinsic properties determining extendability of approximating polynomials

Identify intrinsic structural properties of a real polynomial p that approximates a partial Boolean function f on its domain which determine whether there exists a total completion F extending f whose approximate degree is polynomially related to the degree of p (i.e., whether p admits an extendable completion without superpolynomial blowup).

Background

The paper shows that not all partial functions admit completions that preserve approximate degree up to polynomial factors, implying that not all approximating polynomials are "extendable."

Clarifying which intrinsic features of p (e.g., smoothness, influence, sparsity, margin) guarantee extendability would sharpen the completion framework and delineate when superpolynomial quantum speedups are impossible.

References

We list some open problems that would further the usefulness of our techniques. Since the completion of approximate degree captures the gap between deterministic and approximate degree -- and consequently, between deterministic and quantum complexity -- but not all functions exhibit quantum speedups, it follows that not all polynomials are extendable. What intrinsic properties of a polynomial determine whether it admits such a completion?

— From Promises to Totality: A Framework for Ruling Out Quantum Speedups  (2603.29256 - Huffstutler et al., 31 Mar 2026) in Discussion and further work (Introduction)