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FOOTPRINT AND MINIMUM DISTANCE FUNCTIONS

  • Nunez-Betancourt, Luis (Centro de Investigacion en Matematicas) ;
  • Pitones, Yuriko (Departamento de Matematicas Centro de Investigacion y de Estudios Avanzados del IPN) ;
  • Villarreal, Rafael H. (Departamento de Matematicas Centro de Investigacion y de Estudios Avanzados del IPN)
  • Received : 2017.04.04
  • Accepted : 2017.06.21
  • Published : 2018.01.31

Abstract

Let S be a polynomial ring over a field K, with a monomial order ${\prec}$, and let I be an unmixed graded ideal of S. In this paper we study two functions associated to I: The minimum distance function ${\delta}_I$ and the footprint function $fp_I$. It is shown that ${\delta}_I$ is positive and that $fp_I$ is positive if the initial ideal of I is unmixed. Then we show that if I is radical and its associated primes are generated by linear forms, then ${\delta}_I$ is strictly decreasing until it reaches the asymptotic value 1. If I is the edge ideal of a Cohen-Macaulay bipartite graph, we show that ${\delta}_I(d)=1$ for d greater than or equal to the regularity of S/I. For a graded ideal of dimension ${\geq}1$, whose initial ideal is a complete intersection, we give an exact sharp lower bound for the corresponding minimum distance function.

Keywords

Acknowledgement

Supported by : SNI, CONACYT

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