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Rees algebras of filtrations of covering polyhedra and integral closure of powers of monomial ideals

Rees algebras of filtrations of covering polyhedra and integral closure of powers of monomial ideals The aims of this work are to study Rees algebras of filtrations of monomial ideals associated with covering polyhedra of rational matrices with nonnegative entries and nonzero columns using combinatorial optimization and integer programming and to study powers of monomial ideals and their integral closures using irreducible decompositions and polyhedral geometry. We study the Waldschmidt constant and the ic-resurgence of the filtration associated with a covering polyhedron and show how to compute these constants using linear programming. Then, we show a lower bound for the ic-resurgence of the ideal of covers of a graph and prove that the lower bound is attained when the graph is perfect. We also show lower bounds for the ic-resurgence of the edge ideal of a graph and give an algorithm to compute the asymptotic resurgence of squarefree monomial ideals. A classification of when Newton’s polyhedron is the irreducible polyhedron is presented using integral closure. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

Rees algebras of filtrations of covering polyhedra and integral closure of powers of monomial ideals

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Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022
eISSN
2197-9847
DOI
10.1007/s40687-021-00310-2
Publisher site
See Article on Publisher Site

Abstract

The aims of this work are to study Rees algebras of filtrations of monomial ideals associated with covering polyhedra of rational matrices with nonnegative entries and nonzero columns using combinatorial optimization and integer programming and to study powers of monomial ideals and their integral closures using irreducible decompositions and polyhedral geometry. We study the Waldschmidt constant and the ic-resurgence of the filtration associated with a covering polyhedron and show how to compute these constants using linear programming. Then, we show a lower bound for the ic-resurgence of the ideal of covers of a graph and prove that the lower bound is attained when the graph is perfect. We also show lower bounds for the ic-resurgence of the edge ideal of a graph and give an algorithm to compute the asymptotic resurgence of squarefree monomial ideals. A classification of when Newton’s polyhedron is the irreducible polyhedron is presented using integral closure.

Journal

Research in the Mathematical SciencesSpringer Journals

Published: Mar 1, 2022

Keywords: Monomial ideal; Rees algebra; Filtration; Integral closure; Symbolic powers; Linear programming; Resurgence; Waldschmidt constant; Irreducible representation; Covering polyhedra; Normal ideal; Primary 13C70; Secondary 13F20; 13F55; 05E40; 13A30; 13B22

References