Bounds for the minimum distance function

Luis Núñez-Betancourt; Yuriko Pitones; Rafael H. Villarreal

doi:10.2478/auom-2021-0042

Abstract

Let I be a homogeneous ideal in a polynomial ring S. In this paper, we extend the study of the asymptotic behavior of the minimum distance function δ_I of I and give bounds for its stabilization point, r_I, when I is an F -pure or a square-free monomial ideal. These bounds are related with the dimension and the Castelnuovo–Mumford regularity of I.

References

[AE05] Ian M. Aberbach and Florian Enescu. The structure of F-pure rings. Math. Z., 250(4):791–806, 2005.10.1007/s00209-005-0776-y
Search in Google Scholar Back to article
[DSNnB18] Alessandro De Stefani and Luis Núñez Betancourt. F -thresholds of graded rings. Nagoya Math. J., 229:141–168, 2018.10.1017/nmj.2016.65
Search in Google Scholar Back to article
[GSRTR02] M. González-Sarabia, C. Rentería, and H. Tapia-Recillas. Reed-Muller-type codes over the Segre variety. Finite Fields Appl., 8(4):511–518, 2002.10.1006/ffta.2002.0360
Search in Google Scholar Back to article
[GW78] Shiro Goto and Keiichi Watanabe. On graded rings. I. J. Math. Soc. Japan, 30(2):179–213, 1978.10.2969/jmsj/03020179
Search in Google Scholar Back to article
[HH] Melvin Hochster and Craig Huneke. Tight closure in equal characteristic zero. Unpublished manuscript available at http://www.math.lsa.umich.edu/hochster/tcz.ps.
Search in Google Scholar Back to article
[HR76] Melvin Hochster and Joel L. Roberts. The purity of the Frobenius and local cohomology. Adv. Math., 21(2):117–172, 1976.10.1016/0001-8708(76)90073-6
Search in Google Scholar Back to article
[HS06] Craig Huneke and Irena Swanson. Integral closure of ideals, rings, and modules, volume 336 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2006.
Search in Google Scholar Back to article
[JV21] Delio Jaramillo and Rafael H. Villarreal. The v-number of edge ideals. J. Combin. Theory Ser. A, 177:105310, 35, 2021.10.1016/j.jcta.2020.105310
Search in Google Scholar Back to article
[MBPV17] José Martínez-Bernal, Yuriko Pitones, and Rafael H. Villarreal. Minimum distance functions of graded ideals and Reed-Muller-type codes. J. Pure Appl. Algebra, 221(2):251–275, 2017.10.1016/j.jpaa.2016.06.006
Search in Google Scholar Back to article
[NnBPV18] Luis Núñez Betancourt, Yuriko Pitones, and Rafael H. Villarreal. Footprint and minimum distance functions. Commun. Korean Math. Soc., 33(1):85–101, 2018.
Search in Google Scholar Back to article
[RMSV11] Carlos Rentería-Márquez, Aron Simis, and Rafael H. Villarreal. Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields. Finite Fields Appl., 17(1):81–104, 2011.10.1016/j.ffa.2010.09.007
Search in Google Scholar Back to article
[Sch10] Karl Schwede. Centers of F -purity. Math. Z., 265(3):687–714, 2010.10.1007/s00209-009-0536-5
Search in Google Scholar Back to article
[TW04] Shunsuke Takagi and Kei-ichi Watanabe. On F-pure thresholds. J. Algebra, 282(1):278–297, 2004.10.1016/j.jalgebra.2004.07.011
Search in Google Scholar Back to article
[Vil15] R. Villarreal. Monomial Algebras. Monographs and Research Notes in Mathematics. Chapman and Hall/CRC, second edition, 2015.
Search in Google Scholar Back to article

Bounds for the minimum distance function

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