On exact division and divisibility testing for sparse polynomials

TL;DR

This paper introduces a randomized quasi-linear algorithm for computing quotients of sparse polynomials and proposes a polynomial-time divisibility test for certain structured divisors, advancing the understanding of sparse polynomial divisibility.

Contribution

It presents a new randomized algorithm for quotient computation and a polynomial-time divisibility test for structured divisors, addressing key challenges in sparse polynomial division.

Findings

Quasi-linear complexity algorithm for quotient computation.

Polynomial-time divisibility test for structured divisors.

Identification of divisor patterns enabling efficient testing.

Abstract

No polynomial-time algorithm is known to test whether a sparse polynomial G divides another sparse polynomial $F$. While computing the quotient Q=F quo G can be done in polynomial time with respect to the sparsities of F, G and Q, this is not yet sufficient to get a polynomial-time divisibility test in general. Indeed, the sparsity of the quotient Q can be exponentially larger than the ones of F and G. In the favorable case where the sparsity #Q of the quotient is polynomial, the best known algorithm to compute Q has a non-linear factor #G#Q in the complexity, which is not optimal. In this work, we are interested in the two aspects of this problem. First, we propose a new randomized algorithm that computes the quotient of two sparse polynomials when the division is exact. Its complexity is quasi-linear in the sparsities of F, G and Q. Our approach relies on sparse interpolation and it works over any finite field or the ring of integers. Then, as a step toward faster divisibility testing, we provide a new polynomial-time algorithm when the divisor has a specific shape. More precisely, we reduce the problem to finding a polynomial S such that QS is sparse and testing divisibility by S can be done in polynomial time. We identify some structure patterns in the divisor G for which we can efficiently compute such a polynomial~S.