Algebraic matroids are almost entropic

Matúš, František

doi:10.1090/proc/13846

Algebraic matroids are almost entropic
HTML articles powered by AMS MathViewer

by František Matúš

Proc. Amer. Math. Soc. 152 (2024), 1-6

DOI: https://doi.org/10.1090/proc/13846

Published electronically: October 6, 2023

PDF | Request permission

Abstract:

Algebraic matroids capture properties of the algebraic dependence among elements of extension fields. Almost entropic matroids have the rank functions arbitrarily well approximated by the entropies of subvectors of random vectors. The former class of matroids is included in the latter. A key argument in the proof is the Lang-Weil bound on the number of points in algebraic varieties.

References

Aner Ben-Efraim, Secret-sharing matroids need not be algebraic, Discrete Math. 339 (2016), no. 8, 2136–2145. MR 3500143, DOI 10.1016/j.disc.2016.02.012
Antonio Cafure and Guillermo Matera, Improved explicit estimates on the number of solutions of equations over a finite field, Finite Fields Appl. 12 (2006), no. 2, 155–185. MR 2206396, DOI 10.1016/j.ffa.2005.03.003
Mahdi Cheraghchi and Amin Shokrollahi, Almost-uniform sampling of points on high-dimensional algebraic varieties, STACS 2009: 26th International Symposium on Theoretical Aspects of Computer Science, LIPIcs. Leibniz Int. Proc. Inform., vol. 3, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2009, pp. 277–288. MR 2870659
A. Dress and L. Lovász, On some combinatorial properties of algebraic matroids, Combinatorica 7 (1987), no. 1, 39–48. MR 905150, DOI 10.1007/BF02579199
Satoru Fujishige, Polymatroidal dependence structure of a set of random variables, Inform. and Control 39 (1978), no. 1, 55–72. MR 514262, DOI 10.1016/S0019-9958(78)91063-X
A. W. Ingleton and R. A. Main, Non-algebraic matroids exist, Bull. London Math. Soc. 7 (1975), 144–146. MR 369110, DOI 10.1112/blms/7.2.144
Serge Lang and André Weil, Number of points of varieties in finite fields, Amer. J. Math. 76 (1954), 819–827. MR 65218, DOI 10.2307/2372655
Bernt Lindström, A reduction of algebraic representations of matroids, Proc. Amer. Math. Soc. 100 (1987), no. 2, 388–389. MR 884485, DOI 10.1090/S0002-9939-1987-0884485-6
František Matúš, Matroid representations by partitions, Discrete Math. 203 (1999), no. 1-3, 169–194. MR 1696241, DOI 10.1016/S0012-365X(99)00004-7
František Matúš, Two constructions on limits of entropy functions, IEEE Trans. Inform. Theory 53 (2007), no. 1, 320–330. MR 2292891, DOI 10.1109/TIT.2006.887090
František Matúš, Adhesivity of polymatroids, Discrete Math. 307 (2007), no. 21, 2464–2477. MR 2359593, DOI 10.1016/j.disc.2006.11.013
František Matúš and Lászlo Csirmaz, Entropy region and convolution, IEEE Trans. Inform. Theory 62 (2016), no. 11, 6007–6018. MR 3565097, DOI 10.1109/TIT.2016.2601598
F. Matúš, Classes of matroids closed under the minors and principal extensions (to appear in Combinatorica, DOI: 10.1007/s00493-016-3534-y)
James Oxley, Matroid theory, 2nd ed., Oxford Graduate Texts in Mathematics, vol. 21, Oxford University Press, Oxford, 2011. MR 2849819, DOI 10.1093/acprof:oso/9780198566946.001.0001