Do most polynomials generate a prime ideal? (Q288524)

Let \(\mathbb{K}\) be an algebraically closed field and \(R=\mathbb{K}[x_1^{\pm1},\ldots,x_n^{\pm1}]\) be the Laurent polynomial ring in \(n\) variables over \(\mathbb{K}\). In this article the author studies the problem for which monomial support do most sets of polynomials generate an ideal whose radical is prime in \(R\).NEWLINENEWLINEIn more details, the author in the main theorem proves the following:\newline Suppose \(1\leq k\leq n,\;A_1,\ldots,A_k\subset \mathbb{Z}^n\) and \(0\in A_j,\;\forall j\). General polynomials \(f_1,\ldots,f_k\) with supports \(A_1,\ldots,A_k\) respectively, generate a proper ideal whose radical is prime in \(R\) if and only if for every non-empty subset \(J\subseteq [k]\) one of the following holds: {\parindent=0.7cm\begin{itemize}\item[(1)] \(\text{dim span}_{\mathbb{Q}}\bigcup_{j\in J}A_j\geq |J|+1\) \item[(2)] \(\text{dim span}_{\mathbb{Q}}\bigcup_{j\in J}A_j= |J|\) and the mixed volume of \((\text{conv}(A_j))_{j\in J}\) is 1. NEWLINENEWLINE\end{itemize}} For the case of characteristic zero, the same conditions give primeness.NEWLINENEWLINEApplying the above theorem the author is able to show an interaction with the connectivity of stable intersection of tropical hypersurfaces.