CASTELNUOVO-MUMFORD REGULARITY OF THE INTEGRAL CLOSURE OF POWERS OF THE EDGE IDEAL OF THE PETERSEN GRAPH

Abstract

For a monomial ideal, the function ????????????(????????̅) being linear for sufficiently large ???? is a well-known property of the regularity index function. The problem of determining the stabilization index ???????? such that ????????????(????????̅) becomes linear, and finding the coefficients ????,???? in the expression ????????????(????????̅)=????????+???? for all ????≥????????, has attracted much attention from researchers. This problem can be approached through the concept of minimal free resolutions, the properties of generators of the ideal ????, or linear programming methods. In this paper, I show that the regularity index function of the integral closure of powers of the edge ideal ????????????(????????̅) is a linear function with slope 2 for sufficiently large ????, and determine the stabilization index ????????????−????????????????̅(????) for the Petersen graph via Newton polyhedra and linear programming.