In this article, we study the problem of deterministic factorization of sparse polynomials. We show that if f ∈ F[ x 1 , x 2 ,… , x n ] is a polynomial with s monomials, with individual degrees of its variables bounded by d , then f can be deterministically factored in time s poly( d )log n . Prior to our work, the only efficient factoring algorithms known for this class of polynomials were randomized, and other than for the cases of d =1 and d =2, only exponential time-deterministic factoring algorithms were known. A crucial ingredient in our proof is a quasi-polynomial sparsity bound for factors of sparse polynomials of bounded individual degree. In particular, we show that if f is an s -sparse polynomial in n variables, with individual degrees of its variables bounded by d , then the sparsity of each factor of f is bounded by s (9 d 2 log n ) . This is the first non-trivial bound on factor sparsity for d > 2. Our sparsity bound uses techniques from convex geometry, such as the theory of Newton polytopes and an approximate version of the classical Carathéodory’s Theorem. Our work addresses and partially answers a question of von zur Gathen and Kaltofen [1985] who asked whether a quasi-polynomial bound holds for the sparsity of factors of sparse polynomials.

中文翻译：

---

有界个体度的稀疏多项式的确定性分解

在本文中，我们研究了稀疏多项式的确定性分解问题。我们证明如果F∈ F[X 1,X 2,…,X n ] 是具有 s 个单项式的多项式，其变量的各个度数由d， 然后F可以在时间上确定性地分解s 聚(d）日志n . 在我们的工作之前，此类多项式已知的唯一有效因式分解算法是随机的，除了d=1 和d= 2，仅已知指数时间确定性分解算法。我们证明中的一个关键成分是拟多项式稀疏性，它适用于有界个体度数的稀疏多项式的因子。特别是，我们证明如果F是一个s-稀疏多项式n变量，其变量的各个程度由d，那么每个因子的稀疏度F由s （9d 2日志n). 这是因子稀疏性的第一个非平凡界限d> 2. 我们的稀疏界使用凸几何技术，例如牛顿多面体理论和经典卡拉西奥多里定理的近似版本。我们的工作解决并部分回答了 von zur Gathen 和 Kaltofen [1985] 提出的问题，他们询问准多项式边界是否适用于稀疏多项式因子的稀疏性。