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 previously known exponential (in $\text{deg}{ f}$) bounds for general polynomials. Our results implies that the trivial long division algorithm runs in quasi-linear time relative to 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：我们证明，对于满足 $g$ 整除 $f$ 的首一多项式 $f, g \in \mathbb{C}[x]$，商多项式 $f/g$ 的 $\ell_2$-范数以 $\lVert f \rVert_1 \cdot \tilde{O}(\lVert{g}\rVert_0^3\text{deg}^2{ f})^{\lVert{g}\rVert_0 - 1}$ 为上界。这改进了先前对一般多项式已知的关于 $\text{deg}{ f}$ 的指数型上界。我们的结果表明，相对于输入规模与商多项式 $f/g$ 的项数，朴素的长除法算法可在拟线性时间内运行，从而解决了稀疏多项式精确整除性这一长期存在的问题。我们还研究了在某些特殊情形下 $f/g$ 的项数上界问题。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，我们证明其稀疏性至多为 $O(\lVert{f}\rVert_0 ( \log{\lVert{f}\rVert_0} + \log{\lVert{f}\rVert_{\infty}}))$。这两个上界均关于输入规模为多项式量级。我们利用这些结果，给出了一个多项式时间算法，用于判定一个无分圆因子的三项式是否整除整数环上的稀疏多项式。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 $\text{deg}{ f} = \tilde{O}(\text{deg}{ g})$ 时测试小有限域上五元多项式的整除性。