Superpolynomially Large Support in Every Irreducible Factor of Lacunary Polynomials

Abstract

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.