关于具有有界个体次数的稀疏多项式因式分解
On Factorization of Sparse Polynomials of Bounded Individual Degree
AI总结：
本文提出了一种多项式时间算法，用于稀疏多项式因式分解，并改进了现有算法的时间复杂度。
摘要：
我们研究具有有界个体次数的稀疏多项式及其因式，获得以下结构和算法结果。\n1. 一个确定性多项式时间算法，用于找到具有有界个体次数的稀疏多项式的全部稀疏因式，并给出此类多项式非单项式不可约因式的上界。\n2. 一个运行时间为 poly(n,s^{d log ℓ}) 的算法，可以从黑盒访问其（不一定稀疏）乘积中恢复 ℓ 个不可约的 s-稀疏多项式，其个体次数至多为 d。这在一定程度上回答了 Dutta-Sinhababu-Thierauf（RANDOM 2024）提出的问题。特别是，当 ℓ=O(1) 时，该算法运行时间为多项式时间。\n3. 从黑盒访问中对 s-稀疏多项式的产品进行因式分解的确定性算法。在特征为零或足够大的特征的域上，运行时间为 poly(n,s^{d^3 log n})；在任意域上，运行时间为 poly(n,(d^2)! ,s^{d^5 log n})。这改进了 Bhargava-Saraf-Volkovich（JACM 2020）的结果，后者为单个稀疏多项式提供了 poly(n,s^{d^7 log n}) 的时间复杂度。对于单个稀疏输入，我们获得了 poly(n,s^{d^2 log n}) 的时间复杂度。\n4. 给定对稀疏多项式因式乘积的黑盒访问权限，我们给出一个确定性多项式时间算法，用于找到所有不可约的稀疏多二次因式及其重数。这扩展了 Volkovich（RANDOM 2015, 2017）的算法，并延伸了 Bisht-Volkovich（CC 2025）的完全幂测试。\n为了处理任意域，我们引入了原始因数的概念，从而去除了大多数算法中的特征假设。
英文摘要：
We study sparse polynomials with bounded individual degree and their factors, obtaining the following structural and algorithmic results. 1. A deterministic polynomial-time algorithm to find all sparse divisors of a sparse polynomial of bounded individual degree, together with the first upper bound on the number of non-monomial irreducible factors of such polynomials. 2. A $\mathrm{poly}(n,s^{d\log \ell})$-time algorithm that recovers $\ell$ irreducible $s$-sparse polynomials of individual degree at most $d$ from blackbox access to their (not necessarily sparse) product. This partially resolves a question of Dutta-Sinhababu-Thierauf (RANDOM 2024). In particular, if $\ell=O(1)$ the algorithm runs in polynomial time. 3. Deterministic algorithms for factoring a product of $s$-sparse polynomials of individual degree $d$ from blackbox access. Over fields of characteristic zero or sufficiently large characteristic the runtime is $\mathrm{poly}(n,s^{d^3\log n})$; over arbitrary fields it is $\mathrm{poly}(n,(d^2)!,s^{d^5\log n})$. This improves Bhargava-Saraf-Volkovich (JACM 2020), which gives $\mathrm{poly}(n,s^{d^7\log n})$ time for a single sparse polynomial. For a single sparse input we obtain $\mathrm{poly}(n,s^{d^2\log n})$ time. 4. Given blackbox access to a product of factors of sparse polynomials of bounded individual degree, we give a deterministic polynomial-time algorithm to find all irreducible sparse multiquadratic factors with multiplicities. This generalizes the algorithms of Volkovich (RANDOM 2015, 2017) and extends the complete-power test of Bisht-Volkovich (CC 2025). To handle arbitrary fields we introduce a notion of primitive divisors that removes characteristic assumptions from most of our algorithms.
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