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

• Published in: Research in the mathematical sciences 2022-03, Vol.9 (1), Article 13
• Main Authors: Grisalde, Gonzalo, Seceleanu, Alexandra, Villarreal, Rafael H.
• Format: Article
• Language: English
• Subjects: Algorithms; Applications of Mathematics; Combinatorial analysis; Computational Mathematics and Numerical Analysis; Developments in Commutative Algebra: In honor of Jürgen Herzog on the occasion of his 80th Birthday; Integer programming; Linear programming; Lower bounds; Mathematics; Mathematics and Statistics; Optimization; Polyhedra
• ISSN: 2522-0144; 2197-9847
• DOI: 10.1007/s40687-021-00310-2

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.