Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

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

Download PDF
33 pages

Loading next page...
 
/lp/springer-journals/rees-algebras-of-filtrations-of-covering-polyhedra-and-integral-k72Q5h8teY
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 Sciences – Springer 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

  • Packing properties of cubic square-free monomial ideals
    Alilooee, A; Banerjee, A
  • The Waldschmidt constant for squarefree monomial ideals
    Bocci, C; Cooper, S; Guardo, E; Harbourne, B; Janssen, M; Nagel, U; Seceleanu, A; Van Tuyl, A; Vu, T
  • Comparing powers and symbolic powers of ideals
    Bocci, C; Harbourne, B
  • Consequences of the packing problem
    Bodas, H; Drabkin, B; Fong, C; Jin, S; Kim, J; Li, W; Seceleanu, A; Tang, T; Williams, B
  • Symbolic powers of monomial ideals
    Cooper, SM; Embree, RJD; Hà, HT; Hoefel, AH
  • On resurgence via asymptotic resurgence
    DiPasquale, M; Drabkin, B
  • Asymptotic resurgence via integral closures
    DiPasquale, M; Francisco, CA; Mermin, J; Schweig, J
  • Algebraic and combinatorial properties of ideals and algebras of uniform clutters of TDI systems
    Dupont, LA; Villarreal, RH
  • Blowup algebras of square-free monomial ideals and some links to combinatorial optimization problems
    Gitler, I; Reyes, E; Villarreal, RH
  • A note on Rees algebras and the MFMC property
    Gitler, I; Valencia, C; Villarreal, RH
  • Symbolic powers of monomial ideals and Cohen-Macaulay vertex-weighted digraphs
    Gimenez, P; Martínez-Bernal, J; Simis, A; Villarreal, RH; Vivares, CE; Greuel, GM
  • The Cohen-Macaulay and Gorenstein Rees algebras associated to filtrations
    Goto, S; Nishida, K
  • Asymptotic resurgences for ideals of positive dimensional subschemes of projective space
    Guardo, E; Harbourne, B; Van Tuyl, A
  • Embedded associated primes of powers of square-free monomial ideals
    Hà, HT; Morey, S
  • Membership criteria and containments of powers of monomial ideals
    Hà, HT; Trung, NV
  • Symbolic powers of monomial ideals and vertex cover algebras
    Herzog, J; Hibi, T; Trung, NV
  • Standard graded vertex cover algebras, cycles and leaves
    Herzog, J; Hibi, T; Trung, NV; Zheng, X
  • Splittings and symbolic powers of square-free monomial ideals
    Montaño, J; Núñez-Betancourt, L
  • Monomial ideals of weighted oriented graphs
    Pitones, Y; Reyes, E; Toledo, J
  • Filtrations and Noetherian symbolic blow-up rings
    Schenzel, P
  • On total dual integrality
    Schrijver, A
  • Integral closures of monomial ideals and Fulkersonian hypergraphs
    Trung, NV
  • Rees algebras and polyhedral cones of ideals of vertex covers of perfect graphs
    Villarreal, RH