A New Bound on Cofactors of Sparse Polynomials

TL;DR

This paper establishes a new bound on the cofactor of sparse polynomials, leading to improved algorithms for polynomial division and resolving a key open problem in the field.

Contribution

It provides the first sub-exponential bound on cofactors of sparse polynomials and demonstrates quasi-linear time algorithms for exact division.

Findings

Bound on the $ ext{l}_2$-norm of cofactors of sparse polynomials.

Polynomial division runs in quasi-linear time under exact divisibility.

Shows a quadratic separation between exact and non-exact divisibility complexities.

Abstract

We prove that for polynomials $f, g, h \in \mathbb{Z}[x]$ satisfying $f = gh$ and $f(0) \neq 0$, the $\ell_2$-norm of the cofactor $h$ is bounded by $\|h\|_2 \leq \|f\|_1 \cdot\left( \widetilde{O}\left(\|g\|_0^3 \frac{\text{deg }{(f)}^2}{\sqrt{\text{deg }{(g)}}}\right)\right)^{\|g\|_0 - 1}$, where $\|g\|_0$ is the number of nonzero coefficients of $g$ (its sparsity). We also obtain similar results for polynomials over $\mathbb{C}$. This result significantly improves upon previously known exponential bounds (in $\text{deg }{(f)}$) for general polynomials. It further implies that, under exact division, the polynomial division algorithm runs in quasi-linear time with respect to the input size and the number of terms in the quotient $h$. This resolves a long-standing open problem concerning the exact divisibility of sparse polynomials. In particular, our result demonstrates a quadratic separation between the runtime (and representation size) of exact and non-exact divisibility by sparse polynomials. Notably, prior to our work, it was not even known whether the representation size of the quotient polynomial could be bounded by a sub-quadratic function of its number of terms, specifically of $\text{deg }{(f)}$.