Minimal lattice points in the Newton polyhedron and application to normal ideals (Q6561500)

Let \(S\) be a ring and \(I\) be an ideal in \(S\). An element \(f\in S\) is \textit{integral} over \(I\), if there exists an equation \N\[\Nf^k+c_1f^{k-1}+\cdots +c_{k-1}f+c_k=0\text{ with }c_i\in I^i.\N\]\NThe set of elements \(\overline{I}\) in \(S\) which are integral over \(I\) is the \textit{integral closure} of \(I\). The ideal \(I\) is called \textit{integrally closed}, if \(I=\overline{I}\), and \(I\) is said to be \textit{normal} if all powers of \(I\) are integrally closed. Now, consider a monomial ideal \(I=(x_1^{a_1}, \ldots, x_n^{a_n}) \subset R=K[x_1, \ldots, x_n]\) with each \(a_i\) is a positive integer and \(K\) is a field. Let \(\mathbf{I}(a_1, \ldots, a_n)\) denote the integral closure of the ideal \(I\). The main aim of this paper is to present an elementary and simpler proof of Theorem 5.1 in [\textit{L. Reid} et al., Commun. Algebra 31, No. 9, 4485--4506 (2003; Zbl 1021.13008)]. To do this, the author uses the elementary definition of convex sets, and in particular, a simple characterization of the exponents of the minimal generators of \(\mathbf{I}(a_1, \ldots, a_n)\). In fact, let \(L= \mathbf{I}(a_1, \ldots, a_n, a_{n+1} +l)\) and \(J=\mathbf{I}(a_1, \ldots, a_n, a_{n+1})\), where \(l=\mathrm{lcm}(a_1, \ldots, a_n)\). Then the author proves that:\N\N{Corollary 1.} If \(L\) is normal, then \(J\) is so. \N\N{Corollary 2.} If \(J\) is normal and \(a_{n+1} \geq l\), then \(L\) is so.