Inclusion-exclusion polynomials with large coefficients

TL;DR

This paper demonstrates that inclusion-exclusion polynomials can have arbitrarily large coefficients, with explicit lower bounds depending on the parameters, for any fixed number of terms.

Contribution

It establishes the existence of inclusion-exclusion polynomials with exponentially large coefficients for any number of terms, providing explicit lower bounds.

Findings

Existence of inclusion-exclusion polynomials with large coefficients

Explicit lower bounds on polynomial heights

Coefficients grow exponentially with parameters

Abstract

We prove that for every positive integer $k$ there exist an inclusion-exclusion polynomial $Q_{\{q_1,q_2,...,q_k\}}$ with the height at least $c^{2^k}\prod_{j=1}^{k-2}q_j^{2^{k-j-1}-1}$, where $c$ is a positive constant and $q_1<q_2<...<q_k$ are pairwise coprime and arbitrary large.