A New Bound on Cofactors of Sparse Polynomials
Ido Nahshon, Amir Shpilka
TL;DR
The paper addresses bounding the cofactor $h$ in a sparse polynomial factorization $f=gh$ with $f(0)\neq 0$, showing that classical height bounds ( Gel'fond–Mahler–Mignotte ) which are exponential in $\deg f$ can be substantially improved when $g$ is sparse. It employs a Fourier-analytic framework, evaluating at primitive $p$-th roots of unity and leveraging Parseval’s identity to relate $\|h\|_2$ to $|f(\theta)|/|g(\theta)|$, while proving robust lower bounds for $|g(e^{i\alpha})|$ outside a small bad set via an inductive argument on the sparsity $\|g\|_0$ and prime-number theory. The main contributions include an explicit bound on $\|h\|_2$ that depends primarily on the sparsity of $g$ (with corollaries for monic or integral coefficients), the implication that exact division can be performed in quasi-linear time in the input size and quotient sparsity, and a quadratic separation between exact and non-exact divisibility. These results resolve a long-standing open problem on exact divisibility of sparse polynomials and have potential to substantially speed up exact-division algorithms in computer algebra.
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 $\text{deg }{(f)}$.