Soft-optimal interpolation with unbalanced exponents

Determine whether an O(s) (soft-optimal) time algorithm exists for interpolating a univariate integer polynomial with unbalanced coefficients and large, unbalanced exponents, improving upon the current O(s log D) expected runtime where D bounds the degree and s is the total bit-length of coefficients and exponents.

Background

The paper introduces algorithms for interpolation and multiplication of unbalanced polynomials, achieving expected O(s log D) time where s is the total bit-length and D the degree bound. While these results are quasi-linear in s, the authors highlight a remaining barrier: handling large, unbalanced exponents without incurring the extra log D factor.

Section 7 presents a targeted open question on whether the runtime can be reduced to O(s) for interpolation in the presence of unbalanced exponents, outlining why both top-down and bottom-up strategies currently fail to eliminate the log D term under their MDBB-based framework.