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On the normality of a class of monomial ideals via the Newton polyhedron. (English) Zbl 1415.13008
Summary: Let $$I=\left\langle x_{1}^{a_{1}},\ldots,x_{n}^{a_{n}}\right\rangle \subset R=K[x_{1},\ldots,x_{n}]$$ with $$a_{1},\ldots,a_{n}$$ positive integers and $$K$$ a field, and let $$J$$ be the integral closure of $$I$$. A criterion for the normality of $$J$$ is developed. This criterion is used to show that $$J$$ is normal if and only if the integral closure of the ideal $$\langle x_{1}^{b_{1}},\ldots,x_{n}^{b_{n}},\ldots,x_{r}^{b_{r}}\rangle \subset R[x_{n+1},\ldots,x_{r}]$$ is normal, where $$b_{i}\in \left\{ a_{1},\ldots,a_{n}\right\}$$ for all $$i$$, this generalizes the work of the first author [Rocky Mt. J. Math. 39, No. 1, 1–9 (2009; Zbl 1166.13023)]. If $$l=lcm (a_{1},\ldots,a_{n})$$ and the integral closure of $$\left\langle x_{1}^{a_{1}},\ldots,x_{n}^{a_{n}},x_{n+1}^{l}\right\rangle \subset R[x_{n+1}]$$ is not normal, then we show that the integral closure of $$\left\langle x_{1}^{a_{1}},\ldots,x_{n}^{a_{n}},x_{n+1}^{s}\right\rangle$$ is not normal for any $$s>l$$. Also, we give a shorter proof of a main result of H. Coughlin [Classes of normal monomial ideals. Eugene, OR: University of Oregon (PhD Thesis) (2004)].

##### MSC:
 13B22 Integral closure of commutative rings and ideals 06B10 Lattice ideals, congruence relations 52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
##### Software:
Normaliz; SINGULAR
Full Text:
##### References:
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