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On the Normality of a Class of Monomial Ideals via the Newton Polyhedron

  • Ibrahim Al-AyyoubEmail author
  • Imad Jaradat
  • Khaldoun Al-Zoubi
Article
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Abstract

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 Al-Ayyoub (Rocky Mt Math 39(1):1–9, 2009). 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 Coughlin (Classes of Normal Monomial Ideals. Ph.D. thesis, 2004).

Keywords

Newton polyhedron lattice points integral closure normal ideals convex hull 

Mathematics Subject Classification

13B22 06B10 52B20 

Notes

Acknowledgements

The authors are grateful to professor Irena Swanson for the advice throughout the course of this research and for the Macaulay2 code.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsJordan University of Science and TechnologyIrbidJordan

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