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 $\text{deg}{ f} = \tilde{O}(\text{deg}{ g})$.

翻译：稀疏多项式（也称为缺项多项式）是指项数远少于其次数的多项式。稀疏表示将多项式表示为其非零项（系数-次数对）的列表。特别地，稀疏多项式的次数可能关于稀疏表示规模呈指数增长。我们证明，对于首一多项式 $f, g \in \mathbb{C}[x]$ 且 $g$ 整除 $f$，商多项式 $f/g$ 的 $\ell_2$ 范数受限于 $\lVert f \rVert_1 \cdot \tilde{O}(\lVert{g}\rVert_0^3\text{deg}^2{ f})^{\lVert{g}\rVert_0 - 1}$。这一结果改进了关于一般多项式的指数级（关于 $\text{deg}{ f}$）上界，并表明朴素长除法算法在输入规模和商多项式 $f/g$ 的项数上运行时间为拟线性时间，从而解决了关于稀疏多项式精确整除性的一个长期未决问题。我们还研究了在某些特殊情况下 $f/g$ 项数的上界问题。当 $f, g \in \mathbb{Z}[x]$ 且 $g$ 为无分圆因子的三项式（即不含分圆多项式因子）时，我们证明 $\lVert{f/g}\rVert_0 \leq O(\lVert{f}\rVert_0 \text{size}({f})^2 \cdot \log^6{\text{deg}{ g}})$。当 $g$ 为满足 $g(\pm 1) \neq 0$ 的二项式时，我们证明稀疏度至多为 $O(\lVert{f}\rVert_0 ( \log{\lVert{f}\rVert_0} + \log{\lVert{f}\rVert_{\infty}}))$。这两个上界均为输入规模的多项式量级。我们利用这些结果，给出了一个多项式时间算法，用于判定整数上无分圆因子的三项式是否整除一个稀疏多项式。作为最后一项结果，我们提出了一个多项式时间算法，用于在小型有限域上，当 $\text{deg}{ f} = \tilde{O}(\text{deg}{ g})$ 时，测试五项式的整除性。