Face Module for Realizable Z-matroids

  • Ivan Martino Northeastern University

Abstract

In this work, we define the face ring for a matroid over Z.  Its Hilbert series is, indeed, the expected specialization of the Grothendieck-Tutte polynomial defined by Fink and Moci in [10].

Author Biography

Ivan Martino, Northeastern University
Zelevinsky Research Institute and Wallenberg Post-Doctoral FellowF

References

1. Petter Brändén and Luca Moci, The multivariate arithmetic Tutte polynomial, Trans. Amer. Math. Soc. 366 (2014), no. 10, 5523-5540. MR 3240933

2. Filippo Callegaro and Emanuele Delucchi, The integer cohomology algebra of toric arrangements, Advances in Mathematics 313 (2017), 746 - 802.

3. Corrado De Concini and Giovanni Gaiffi, Projective wonderful models for toric arrangements, Advances in Mathematics (2017).

4. Michele D'Adderio and Luca Moci, Arithmetic matroids, the Tutte polynomial and toric arrangements, Adv. Math. 232 (2013), 335-367. MR 2989987

5. C. De Concini and C. Procesi, Hyperplane arrangements and box splines, Michigan Math. J. 57 (2008), 201-225, With an appendix by A. Björner, Special volume in honor of Melvin Hochster. MR 2492449

6. Torsten Ekedahl, A geometric invariant of a finite group, arXiv:0903.3148v1, 2009.

7. Torsten Ekedahl, The Grothendieck group of algebraic stacks, arXiv:0903.3143v2, 2009.

8. Alex Fink and Ivan Martino, Realizable Z-matroids, In preparation, 2018.

9. Alex Fink and Ivan Martino, Toric arrangements are shellable, In preparation, 2018.

10. Alex Fink and Luca Moci, Matroids over a ring, J. Eur. Math. Soc. (JEMS) 18 (2016), no. 4, 681-731. MR 3474454

11. Alex Fink and Luca Moci, Polyhedra and Parameter Spaces for Matroids over Valuation Rings, ArXiv:1707.01026v2, 2017.

12. Adriano M. Garsia, Combinatorial methods in the theory of Cohen-Macaulay rings, Adv. in Math. 38 (1980), no. 3, 229-266. MR 597728

13. Daniel R. Grayson and Michael E. Stillman, Macaulay2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/.

14. Bernd Kind and Peter Kleinschmidt, Schälbare Cohen-Macauley-Komplexe und ihre Parametrisierung, Math. Z. 167 (1979), no. 2, 173-179. MR 534824

15. Ivan Martino, The Ekedahl invariants forfinite groups, J. Pure Appl. Algebra 220 (2016), no. 4, 1294-1309. MR 3423448

16. Ivan Martino, Introduction to the Ekedahl Invariants, MATH. SCAND. 120 (2017), 211-224a.

17. Luca Moci, A Tutte polynomial for toric arrangements, Trans. Amer. Math. Soc. 364 (2012), no. 2, 1067-1088. MR 2846363

18. Luca Moci, Wonderful models for toric arrangements, Int. Math. Res. Not. IMRN (2012), no. 1, 213-238. MR 2874932

19. Richard P. Stanley, f-vectors and h-vectors of simplicial posets, J. Pure Appl. Algebra 71 (1991), no. 2-3, 319-331. MR 1117642

20. Richard P. Stanley, Combinatorics and commutative algebra, second ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1453579
Published
2018-12-31
Section
Articles