arXiv subjects

Lindsay C. Piechnik

Publications and source records attributed to Lindsay C. Piechnik.

3 recordsLinked to original sources

arXiv2017

Existence of unimodular triangulations - positive results

Unimodular triangulations of lattice polytopes arise in algebraic geometry, commutative algebra, integer programming and, of course, combinatorics. In this article, we review several classes of polytopes that do have unimodular triangulations and constructions that preserve their existence. We include, in particular, the first effective proof of the classical result by Knudsen-Mumford-Waterman stating that every lattice polytope has a dilation that admits a unimodular triangulation. Our proof yields an explicit (although doubly exponential) bound for the dilation factor.

math.CO↗

arXiv2015

Constructing monomial ideals with a given minimal resolution

This paper gives a description of various recent results which construct monomial ideals with a given minimal free resolution. We show that these are all instances of coordinatizing a finite atomic lattice as defined by Mapes. Subsequently, we explain how in some of these cases (in one case work by Faridi, and in another work by Fl\oystad), where questions still remain, this point of view can be applied. We also prove an equivalence in the case of trees between the notion of maximal defined by Fl\oystad and a notion of being maximal in a Betti stratum.

math.AC↗

arXiv2010

Non-Generic Tropical Hyperplane Arrangements and the Secondary polytope of $Δ_{n-1} \times Δ_{d-1}$

Ardila and Develin's paper on tropical oriented hyperplane arrangements and tropical oriented matroids defines tropical oriented matroids and conjectures a bijection between them and triangulations of products of simplices $Δ_{n-1} \times Δ_{d-1}$. Oh and Yoo recently confirmed this conjecture; however, neither group addressed the case of hyperplanes that are not in generic position. These non-generic arrangements do not correspond to tropical oriented matroids, but they encode information about subdivisions of $Δ_{n-1} \times Δ_{d-1}$. This note considers the non-generic case and presents some preliminary results in the area.

math.CO↗