Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the degree two entry of a Gorenstein $ h$-vector and a conjecture of Stanley

Authors: Juan Migliore, Uwe Nagel and Fabrizio Zanello
Journal: Proc. Amer. Math. Soc. 136 (2008), 2755-2762
MSC (2000): Primary 13E10; Secondary 13H10, 13D40
Published electronically: April 10, 2008
MathSciNet review: 2399039
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this short paper we establish a (non-trivial) lower bound on the degree two entry $ h_2$ of a Gorenstein $ h$-vector of any given socle degree $ e$ and any codimension $ r$.

In particular, when $ e=4$, that is, for Gorenstein $ h$-vectors of the form $ h=(1,r,h_2,r,1)$, our lower bound allows us to prove a conjecture of Stanley on the order of magnitude of the minimum value, say $ f(r)$, that $ h_2$ may assume. In fact, we show that

$\displaystyle \lim_{r\rightarrow \infty} \frac{f(r)}{ r^{2/3}}= 6^{2/3}.$

In general, we wonder whether our lower bound is sharp for all integers $ e\geq 4$ and $ r\geq 2$.

References [Enhancements On Off] (What's this?)

  • [At] C.A. Athanasiadis: Ehrhart polynomials, simplicial polytopes, magic squares and a conjecture of Stanley, J. Reine Angew. Math. 583 (2005), 163-174. MR 2146855 (2006a:05171)
  • [BI] D. Bernstein and A. Iarrobino: A nonunimodal graded Gorenstein Artin algebra in codimension five, Comm. Algebra 20 (1992), No. 8, 2323-2336. MR 1172667 (93i:13012)
  • [BG] A.M. Bigatti and A.V. Geramita: Level algebras, lex segments and minimal Hilbert functions, Comm. Algebra 31 (2003), 1427-1451. MR 1971070 (2004f:13020)
  • [Bo] M. Boij: Graded Gorenstein Artin algebras whose Hilbert functions have a large number of valleys, Comm. Algebra 23 (1995), No. 1, 97-103. MR 1311776 (96h:13040)
  • [BL] M. Boij and D. Laksov: Nonunimodality of graded Gorenstein Artin algebras, Proc. Amer. Math. Soc. 120 (1994), 1083-1092. MR 1227512 (94g:13008)
  • [BH] W. Bruns and J. Herzog: Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics 39, Revised edition, Cambridge, U.K. (1998). MR 1251956 (95h:13020)
  • [Ge] A.V. Geramita: Inverse Systems of Fat Points: Waring's Problem, Secant Varieties of Veronese Varieties and Parameter Spaces for Gorenstein Ideals, Queen's Papers in Pure and Applied Mathematics 102, The Curves Seminar at Queen's (1996), Vol. X, Queen's University, Kingston, ON, 2-114. MR 1381732 (97h:13012)
  • [Gr] M. Green: Restrictions of linear series to hyperplanes, and some results of Macaulay and Gotzmann, Algebraic Curves and Projective Geometry (1988), 76-86, Trento; Lecture Notes in Math. 1389, Springer, Berlin (1989). MR 1023391 (90k:13021)
  • [Hu] C. Huneke: Hyman Bass and Ubiquity: Gorenstein Rings, Contemp. Math. 243, Amer. Math. Soc., Providence, RI (1999), 55-78. MR 1732040 (2001m:13001)
  • [IK] A. Iarrobino and V. Kanev: Power Sums, Gorenstein Algebras, and Determinantal Loci, Lecture Notes in Mathematics 1721, Springer-Verlag, Berlin-Heidelberg (1999). MR 1735271 (2001d:14056)
  • [IS] A. Iarrobino and H. Srinivasan: Some Gorenstein Artin algebras of embedding dimension four, I: Components of $ \mathbb{P}{\rm Gor}(H)$ for $ H=(1,4,7,...,1)$, J. Pure Appl. Algebra 201 (2005), 62-96. MR 2158748 (2006g:13033)
  • [Kl] P. Kleinschmidt: Über Hilbert-Funktionen graduierter Gorenstein-Algebren, Arch. Math. (Basel) 43 (1984), 501-506. MR 775736 (86c:13020)
  • [KS] A.R. Klivans and A. Shpilka: Learning arithmetic circuits via partial derivatives, in: Proc. 16th Annual Conference on Computational Learning Theory, Morgan Kaufmann Publishers (2003), 463-476.
  • [Na] U. Nagel: Empty simplices of polytopes and graded Betti numbers, Discrete Comput. Geom. 39 (2008), 389-410.
  • [Pa] R. Pandharipande: Three questions in Gromov-Witten theory, in: Proceedings of the International Congress of Mathematics, Vol. II, Higher Ed. Press, Beijing (2002), 503-512. MR 1957060 (2003k:14069)
  • [Re] I. Reiten: The converse to a theorem of Sharp on Gorenstein modules, Proc. Amer. Math. Soc. 32 (1972), 417-420. MR 0296067 (45:5128)
  • [St1] R. Stanley: Hilbert functions of graded algebras, Adv. Math. 28 (1978), 57-83. MR 0485835 (58:5637)
  • [St2] R. Stanley: Combinatorics and Commutative Algebra, First Ed., Progress in Mathematics 41, Birkhäuser, Boston, MA (1983). MR 725505 (85b:05002)
  • [St3] R. Stanley: Combinatorics and Commutative Algebra, Second Ed., Progress in Mathematics 41, Birkhäuser, Boston, MA (1996). MR 1453579 (98h:05001)
  • [St4] R. Stanley: A monotonicity property of $ h$-vectors and $ h^{*}$-vectors, European J. Combin. 14 (1993), 251-258. MR 1215335 (94f:52016)
  • [St5] R. Stanley: The number of faces of a simplicial convex polytope, Adv. Math. 35 (1980), 236-238. MR 563925 (81f:52014)
  • [Za1] F. Zanello: Stanley's theorem on codimension $ 3$ Gorenstein $ h$-vectors, Proc. Amer. Math. Soc. 134 (2006), No. 1, 5-8. MR 2170536 (2006c:13037)
  • [Za2] F. Zanello: When is there a unique socle-vector associated to a given $ h$-vector?, Comm. in Algebra 34 (2006), No. 5, 1847-1860. MR 2229494 (2007a:13021)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13E10, 13H10, 13D40

Retrieve articles in all journals with MSC (2000): 13E10, 13H10, 13D40

Additional Information

Juan Migliore
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

Uwe Nagel
Affiliation: Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, Kentucky 40506-0027

Fabrizio Zanello
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Address at time of publication: Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan 49931-1295

Keywords: Artinian algebra, Gorenstein $h$-vector, unimodality, Green's theorem.
Received by editor(s): May 7, 2007
Received by editor(s) in revised form: December 1, 2007
Published electronically: April 10, 2008
Additional Notes: The second author gratefully acknowledges partial support from and the hospitality of the Institute for Mathematics and its Applications at the University of Minnesota
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society