×

zbMATH — the first resource for mathematics

Combinatorics and geometry of power ideals. (English) Zbl 1226.05019
Trans. Am. Math. Soc. 362, No. 8, 4357-4384 (2010); corrigendum ibid. 367, No. 5, 3759-3762 (2015).
Summary: We investigate ideals in a polynomial ring which are generated by powers of linear forms. Such ideals are closely related to the theories of fat point ideals, Cox rings, and box splines.
We pay special attention to a family of power ideals that arises naturally from a hyperplane arrangement \( \mathcal{A}\). We prove that their Hilbert series are determined by the combinatorics of \( \mathcal{A}\) and can be computed from its Tutte polynomial. We also obtain formulas for the Hilbert series of certain closely related fat point ideals and zonotopal Cox rings.
Our work unifies and generalizes results due to Dahmen-Micchelli, Holtz-Ron, Postnikov-Shapiro-Shapiro, and Sturmfels-Xu, among others. It also settles a conjecture of Holtz-Ron on the spline interpolation of functions on the lattice points of a zonotope.

MSC:
05A15 Exact enumeration problems, generating functions
05B35 Combinatorial aspects of matroids and geometric lattices
13P99 Computational aspects and applications of commutative rings
41A15 Spline approximation
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Federico Ardila, Computing the Tutte polynomial of a hyperplane arrangement, Pacific J. Math. 230 (2007), no. 1, 1 – 26. · Zbl 1152.52011 · doi:10.2140/pjm.2007.230.1 · doi.org
[2] F. Ardila. Enumerative and algebraic aspects of matroids and hyperplane arrangements. Ph.D. thesis, Massachusetts Institute of Technology, 2003.
[3] Victor V. Batyrev and Oleg N. Popov, The Cox ring of a del Pezzo surface, Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002) Progr. Math., vol. 226, Birkhäuser Boston, Boston, MA, 2004, pp. 85 – 103. · Zbl 1075.14035 · doi:10.1007/978-0-8176-8170-8_5 · doi.org
[4] A. Berget. Products of linear forms and Tutte polynomials. Preprint, 2008. · Zbl 1219.05032
[5] Michel Brion and Michèle Vergne, Arrangement of hyperplanes. I. Rational functions and Jeffrey-Kirwan residue, Ann. Sci. École Norm. Sup. (4) 32 (1999), no. 5, 715 – 741 (English, with English and French summaries). · Zbl 0945.32003 · doi:10.1016/S0012-9593(01)80005-7 · doi.org
[6] Wolfgang Dahmen and Charles A. Micchelli, On the local linear independence of translates of a box spline, Studia Math. 82 (1985), no. 3, 243 – 263. · Zbl 0545.41018
[7] C. De Concini and C. Procesi, Wonderful models of subspace arrangements, Selecta Math. (N.S.) 1 (1995), no. 3, 459 – 494. · Zbl 0842.14038 · doi:10.1007/BF01589496 · doi.org
[8] C. De Concini and C. Procesi. The algebra of the box spline. Preprint, 2006. arXiv:math/0602019v1. · Zbl 1176.20040
[9] J. Emsalem and A. Iarrobino, Inverse system of a symbolic power. I, J. Algebra 174 (1995), no. 3, 1080 – 1090. · Zbl 0842.14002 · doi:10.1006/jabr.1995.1168 · doi.org
[10] Anthony V. Geramita and Henry K. Schenck, Fat points, inverse systems, and piecewise polynomial functions, J. Algebra 204 (1998), no. 1, 116 – 128. · Zbl 0934.13013 · doi:10.1006/jabr.1997.7361 · doi.org
[11] O. Holtz and A. Ron. Zonotopal algebra. Preprint, 2007. arXiv:0708.2632. · Zbl 1223.13010
[12] Brian Harbourne, Problems and progress: a survey on fat points in \?², Zero-dimensional schemes and applications (Naples, 2000) Queen’s Papers in Pure and Appl. Math., vol. 123, Queen’s Univ., Kingston, ON, 2002, pp. 85 – 132. · Zbl 1052.14052
[13] Peter Orlik and Hiroaki Terao, Commutative algebras for arrangements, Nagoya Math. J. 134 (1994), 65 – 73. · Zbl 0801.05019
[14] James G. Oxley, Matroid theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992. · Zbl 0784.05002
[15] Alexander Postnikov and Boris Shapiro, Trees, parking functions, syzygies, and deformations of monomial ideals, Trans. Amer. Math. Soc. 356 (2004), no. 8, 3109 – 3142. · Zbl 1043.05038
[16] Alexander Postnikov, Boris Shapiro, and Mikhail Shapiro, Algebras of curvature forms on homogeneous manifolds, Differential topology, infinite-dimensional Lie algebras, and applications, Amer. Math. Soc. Transl. Ser. 2, vol. 194, Amer. Math. Soc., Providence, RI, 1999, pp. 227 – 235. · Zbl 0971.53034 · doi:10.1090/trans2/194/10 · doi.org
[17] Nicholas Proudfoot and David Speyer, A broken circuit ring, Beiträge Algebra Geom. 47 (2006), no. 1, 161 – 166. · Zbl 1095.13024
[18] Steven M. Roman and Gian-Carlo Rota, The umbral calculus, Advances in Math. 27 (1978), no. 2, 95 – 188. · Zbl 0375.05007 · doi:10.1016/0001-8708(78)90087-7 · doi.org
[19] Henry K. Schenck, Linear systems on a special rational surface, Math. Res. Lett. 11 (2004), no. 5-6, 697 – 713. · Zbl 1077.14013 · doi:10.4310/MRL.2004.v11.n5.a12 · doi.org
[20] Alan D. Sokal, The multivariate Tutte polynomial (alias Potts model) for graphs and matroids, Surveys in combinatorics 2005, London Math. Soc. Lecture Note Ser., vol. 327, Cambridge Univ. Press, Cambridge, 2005, pp. 173 – 226. · Zbl 1110.05020 · doi:10.1017/CBO9780511734885.009 · doi.org
[21] Richard P. Stanley, An introduction to hyperplane arrangements, Geometric combinatorics, IAS/Park City Math. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2007, pp. 389 – 496. · Zbl 1136.52009
[22] B. Sturmfels and Z. Xu. Sagbi bases of Cox-Nagata rings. Preprint, 2008. arXiv:0803.0892. · Zbl 1202.14053
[23] Hiroaki Terao, Algebras generated by reciprocals of linear forms, J. Algebra 250 (2002), no. 2, 549 – 558. · Zbl 1049.13011 · doi:10.1006/jabr.2001.9121 · doi.org
[24] David G. Wagner, Algebras related to matroids represented in characteristic zero, European J. Combin. 20 (1999), no. 7, 701 – 711. · Zbl 0996.16027 · doi:10.1006/eujc.1999.0316 · doi.org
[25] Neil White , Matroid applications, Encyclopedia of Mathematics and its Applications, vol. 40, Cambridge University Press, Cambridge, 1992. · Zbl 0742.00052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.