Product representations of polynomials over finite fields
Hyunwoo Lee, Chi Hoi Yip, Semin Yoo
TL;DR
The paper studies a finite-field analogue of Verstraëte's polynomial product problem by defining $F_k(q;h)$ as the maximal size of a subset of ${\mathbb F}_q^*$ avoiding $k$ distinct elements whose product lies in the value set of a non-constant polynomial $h\in {\mathbb F}_q[x]$. It establishes a main asymptotic formula $F_k(q;h)=\frac{m(k,n;s)}{n}\,q+O(\sqrt{q})$, where $h=Cf^{\ell}$ with $n=\gcd(\ell,q-1)$ and $s$ determined by $C\in g^sH$, and $H$ is the index-$n$ subgroup of ${\mathbb F}_q^*$. The proof combines a Weil-bound/character-sum framework (via a Gyarmati-type lemma) to obtain an upper bound, with a constructive lower bound showing that extremal sets are essentially unions of $m(k,n;s)$ cosets of $H$ when $m(k,n;s)>0$. The paper then provides explicit estimates for $m(k,n;s)$, including a gcd$(k,n)=1$ case where the value becomes independent of the chosen $s$ and a concrete coset-based construction, thereby delivering a finite-field analogue of Verstraëte's conjecture in this setting and clarifying when a main-term linear in $q$ appears. Together, these results connect polynomial-value-set constraints to the additive structure of ${\mathbb Z}_n$ and yield a clear structural picture of extremal sets in the finite-field setting.
Abstract
Erdős, Sárközy, and Sós studied the asymptotics of the maximum size of a subset of $\{1,2,\ldots, N\}$ such that it does not contain $k$ distinct elements whose product is a perfect square. More generally, Verstraëte proposed a conjecture regarding the asymptotic behavior of the same quantity with the set of perfect squares replaced by the value set of a polynomial in $\mathbb{Z}[x]$. In this paper, we study a finite field analogue of Verstraëte's conjecture.