稀疏多项式每个不可约因子的超多项式大支撑
Superpolynomially Large Support in Every Irreducible Factor of Lacunary Polynomials
- The University of Memphis（孟菲斯大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明存在无穷多个仅含m个非零系数的有理系数多项式，其每个不可约因子都含指数级（exp(Ω(√(m/log m)))）非零系数，表明乘法与幂、复合等运算的稀疏性原理截然不同，并给出无条件、与次数无关的稀疏分解障碍。
AI中文摘要：
我们证明存在无穷多个多项式 $F\in\mathbb{Q}[x]$，其恰好有 $m$ 个非零系数，使得 $F$ 在 $\mathbb{Q}$ 上的每个不可约因子都具有 $\exp\\!\bigl(\Omega(\sqrt{m/\log m})\bigr)$ 个非零系数。这使得乘法与幂、复合及其他已知满足反向稀疏性原理的代数运算截然不同。它还给出了一个无条件的、与次数无关的稀疏分解障碍，补充了那些对系数高度、指数位置、倒数结构或次数界限施加假设的正向因子稀疏性结果。
英文摘要：
We show that there exist infinitely many polynomials $F\in\mathbb{Q}[x]$ with exactly $m$ nonzero coefficients such that every irreducible factor of \(F\) over \(\mathbb{Q}\) has \(\exp\) \(\!\bigl(Ω(\sqrt{m/\log m})\bigr)\) nonzero coefficients. This places multiplication in a sharply different category from powers, composition, and other algebraic operations for which reverse-sparsity principles are known. It also gives an unconditional, degree-free obstruction to sparse factorization, complementing positive factor-sparsity results that impose hypotheses on coefficient height, exponent positions, reciprocal structure, or degree bounds.
