Combinatorics and geometry of power ideals
HTML articles powered by AMS MathViewer
- by Federico Ardila and Alexander Postnikov PDF
- Trans. Amer. Math. Soc. 362 (2010), 4357-4384 Request permission
Erratum: Trans. Amer. Math. Soc. 367 (2015), 3759-3762.
Abstract:
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.
References
- Federico Ardila, Computing the Tutte polynomial of a hyperplane arrangement, Pacific J. Math. 230 (2007), no. 1, 1–26. MR 2318445, DOI 10.2140/pjm.2007.230.1
- F. Ardila. Enumerative and algebraic aspects of matroids and hyperplane arrangements. Ph.D. thesis, Massachusetts Institute of Technology, 2003.
- 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. MR 2029863, DOI 10.1007/978-0-8176-8170-8_{5}
- A. Berget. Products of linear forms and Tutte polynomials. Preprint, 2008.
- 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). MR 1710758, DOI 10.1016/S0012-9593(01)80005-7
- 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. MR 825481, DOI 10.4064/sm-82-3-243-263
- C. De Concini and C. Procesi, Wonderful models of subspace arrangements, Selecta Math. (N.S.) 1 (1995), no. 3, 459–494. MR 1366622, DOI 10.1007/BF01589496
- C. De Concini and C. Procesi. The algebra of the box spline. Preprint, 2006. arXiv:math/0602019v1.
- J. Emsalem and A. Iarrobino, Inverse system of a symbolic power. I, J. Algebra 174 (1995), no. 3, 1080–1090. MR 1337186, DOI 10.1006/jabr.1995.1168
- Anthony V. Geramita and Henry K. Schenck, Fat points, inverse systems, and piecewise polynomial functions, J. Algebra 204 (1998), no. 1, 116–128. MR 1623949, DOI 10.1006/jabr.1997.7361
- O. Holtz and A. Ron. Zonotopal algebra. Preprint, 2007. arXiv:0708.2632.
- Brian Harbourne, Problems and progress: a survey on fat points in $\mathbf P^2$, 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. MR 1898832
- Peter Orlik and Hiroaki Terao, Commutative algebras for arrangements, Nagoya Math. J. 134 (1994), 65–73. MR 1280653, DOI 10.1017/S0027763000004852
- James G. Oxley, Matroid theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992. MR 1207587
- Alexander Postnikov and Boris Shapiro, Trees, parking functions, syzygies, and deformations of monomial ideals, Trans. Amer. Math. Soc. 356 (2004), no. 8, 3109–3142. MR 2052943, DOI 10.1090/S0002-9947-04-03547-0
- 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. MR 1729365, DOI 10.1090/trans2/194/10
- Nicholas Proudfoot and David Speyer, A broken circuit ring, Beiträge Algebra Geom. 47 (2006), no. 1, 161–166. MR 2246531
- Steven M. Roman and Gian-Carlo Rota, The umbral calculus, Advances in Math. 27 (1978), no. 2, 95–188. MR 485417, DOI 10.1016/0001-8708(78)90087-7
- Henry K. Schenck, Linear systems on a special rational surface, Math. Res. Lett. 11 (2004), no. 5-6, 697–713. MR 2106236, DOI 10.4310/MRL.2004.v11.n5.a12
- 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. MR 2187739, DOI 10.1017/CBO9780511734885.009
- 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. MR 2383131, DOI 10.1090/pcms/013/08
- B. Sturmfels and Z. Xu. Sagbi bases of Cox-Nagata rings. Preprint, 2008. arXiv:0803.0892.
- Hiroaki Terao, Algebras generated by reciprocals of linear forms, J. Algebra 250 (2002), no. 2, 549–558. MR 1899865, DOI 10.1006/jabr.2001.9121
- David G. Wagner, Algebras related to matroids represented in characteristic zero, European J. Combin. 20 (1999), no. 7, 701–711. MR 1721927, DOI 10.1006/eujc.1999.0316
- Neil White (ed.), Matroid applications, Encyclopedia of Mathematics and its Applications, vol. 40, Cambridge University Press, Cambridge, 1992. MR 1165537, DOI 10.1017/CBO9780511662041
Additional Information
- Federico Ardila
- Affiliation: Department of Mathematics, San Francisco State University, 1600 Holloway Avenue, San Francisco, California 94110
- MR Author ID: 725066
- Email: federico@math.sfsu.edu
- Alexander Postnikov
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachu- setts Avenue, Cambridge, Massachusetts 02139
- Email: apost@math.mit.edu
- Received by editor(s): October 31, 2008
- Received by editor(s) in revised form: February 11, 2009
- Published electronically: April 1, 2010
- Additional Notes: The first author was supported in part by NSF Award DMS-0801075.
The second author was supported in part by NSF CAREER Award DMS-0504629. - © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 4357-4384
- MSC (2000): Primary 05A15, 05B35, 13P99, 41A15, 52C35
- DOI: https://doi.org/10.1090/S0002-9947-10-05018-X
- MathSciNet review: 2608410