Faster Interpolation Algorithms for Sparse Multivariate Polynomials Given by Straight-Line Programs\

TL;DR

This paper introduces new deterministic and Monte Carlo algorithms for interpolating sparse multivariate polynomials represented by straight-line programs, improving efficiency and complexity bounds over previous methods.

Contribution

It presents the first Monte Carlo algorithms with linear complexity in the product of variables and terms, and a deterministic algorithm with better complexity than existing methods.

Findings

Deterministic algorithm is quadratic in n,T and cubic in log D.

Monte Carlo algorithms have linear complexity in nT.

Algorithms are optimal in n and T in the Soft-Oh sense.

Abstract

In this paper, we propose new deterministic and Monte Carlo interpolation algorithms for sparse multivariate polynomials represented by straight-line programs. Let $f$ be an $n$-variate polynomial given by a straight-line program, which has a degree bound $D$ and a term bound $T$. Our deterministic algorithm is quadratic in $n,T$ and cubic in $\log D$ in the Soft-Oh sense, which has better complexities than existing deterministic interpolation algorithms in most cases. Our Monte Carlo interpolation algorithms have better complexities than existing Monte Carlo interpolation algorithms and are the first algorithms whose complexities are linear in $nT$ in the Soft-Oh sense. Since $nT$ is a factor of the size of $f$, our Monte Carlo algorithms are optimal in $n$ and $T$ in the Soft-Oh sense.