Interpolation of Shifted-Lacunary Polynomials

TL;DR

This paper presents a polynomial-time algorithm for interpolating rational polynomials in a shifted sparse power basis from black box evaluations, efficiently handling high-degree sparse polynomials.

Contribution

It introduces a novel algorithm that computes the sparsest shifted power basis representation with minimal sparsity, combining existing techniques for sparse interpolation and shift computation.

Findings

Algorithm runs in polynomial time relative to the sparse representation size.

Handles polynomials with extremely high degrees efficiently.

Achieves minimal sparsity in the shifted power basis representation.

Abstract

Given a "black box" function to evaluate an unknown rational polynomial f in Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t, the shift s (a rational), the exponents 0 <= e1 < e2 < ... < et, and the coefficients c1,...,ct in Q\{0} such that f(x) = c1(x-s)^e1+c2(x-s)^e2+...+ct(x-s)^et. The computed sparsity t is absolutely minimal over any shifted power basis. The novelty of our algorithm is that the complexity is polynomial in the (sparse) representation size, and in particular is logarithmic in deg(f). Our method combines previous celebrated results on sparse interpolation and computing sparsest shifts, and provides a way to handle polynomials with extremely high degree which are, in some sense, sparse in information.