Effective Khovanskii, Ehrhart Polytopes, and the Erd\H{o}s Multiplication Table Problem

Let $P(k,n)$ be the set of products of $k$ factors from the set $\{1,\ldots , n\}.$ In 1955, Erd\H{o}s posed the problem of determining the order of magnitude of $|P (2, n)|$ and proved that $|P (2, n)| = o(n^2 )$ for $n \to\infty$. In 2015, Darda and Hujdurovi\'c asked whether, for each fixed $n$, $|P (k, n)|$ is a polynomial in $k$ of degree $\pi(n)$ - the number of primes not larger than $n$. Recently, Granville, Smith and Walker published an effective version of Khovanskii's Theorem. We apply this new result to show, that for each integer $n$, there is a polynomial $q_n$ of degree $\pi(n)$ such that $|P (k, n)|=q_n(k)$ for each $k\geq n^2\cdot\left(\prod_{m=1}^{\pi(n)} \log_{p_m}(n)\right)-n+1.$ Moreover, we give an upper estimate of the leading coefficient of $q_n$.