New Bounds on Quotient Polynomials with Applications to Exact Divisibility and Divisibility Testing of Sparse Polynomials

Abstract

A sparse polynomial (also called a lacunary polynomial) is a polynomial that has relatively few terms compared to its degree. The sparse-representation of a polynomial represents the polynomial as a list of its non-zero terms (coefficient-degree pairs). In particular, the degree of a sparse polynomial can be exponential in the sparse-representation size. We prove that for monic polynomials $f, g \in \mathbb{C}[x]$ such that $g$ divides $f$, the $\ell_2$-norm of the quotient polynomial $f/g$ is bounded by $\lVert f \rVert_1 \cdot \tilde{O}(\lVert{g}\rVert_0^3\text{deg}^2{ f})^{\lVert{g}\rVert_0 - 1}$. This improves upon the exponential (in $\text{deg}{ f}$) bounds for general polynomials and implies that the trivial long division algorithm runs in time quasi-linear in the input size and number of terms of the quotient polynomial $f/g$, thus solving a long-standing problem on exact divisibility of sparse polynomials. We also study the problem of bounding the number of terms of $f/g$ in some special cases. When $f, g \in \mathbb{Z}[x]$ and $g$ is a cyclotomic-free (i.e., it has no cyclotomic factors) trinomial, we prove that $\lVert{f/g}\rVert_0 \leq O(\lVert{f}\rVert_0 \text{size}({f})^2 \cdot \log^6{\text{deg}{ g}})$. When $g$ is a binomial with $g(\pm 1) \neq 0$, we prove that the sparsity is at most $O(\lVert{f}\rVert_0 ( \log{\lVert{f}\rVert_0} + \log{\lVert{f}\rVert_{\infty}}))$. Both upper bounds are polynomial in the input-size. We leverage these results and give a polynomial time algorithm for deciding whether a cyclotomic-free trinomial divides a sparse polynomial over the integers. As our last result, we present a polynomial time algorithm for testing divisibility by pentanomials over small finite fields when $\text{deg}{ f} = \tilde{O}(\text{deg}{ g})$