Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Partial derivatives of a generic subspace of a vector space of forms: Quotients of level algebras of arbitrary type

Author: Fabrizio Zanello
Journal: Trans. Amer. Math. Soc. 359 (2007), 2675-2686
MSC (2000): Primary 13E10; Secondary 13H10
Published electronically: January 4, 2007
MathSciNet review: 2286051
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Given a vector space $ V$ of homogeneous polynomials of the same degree over an infinite field, consider a generic subspace $ W$ of $ V$. The main result of this paper is a lower-bound (in general sharp) for the dimensions of the spaces spanned in each degree by the partial derivatives of the forms generating $ W$, in terms of the dimensions of the spaces spanned by the partial derivatives of the forms generating the original space $ V$.

Rephrasing our result in the language of commutative algebra (where this result finds its most important applications), we have: let $ A$ be a type $ t$ artinian level algebra with $ h$-vector $ h=(1,h_1,h_2,...,h_e)$, and let, for $ c=1,2,...,t-1$, $ H^{c,gen}=(1,H_1^{c,gen},H_2^{c,gen},...,H_e^{c,gen})$ be the $ h$-vector of the generic type $ c$ level quotient of $ A$ having the same socle degree $ e$. Then we supply a lower-bound (in general sharp) for the $ h$-vector $ H^{c,gen}$. Explicitly, we will show that, for any $ u\in \lbrace 1,...,e\rbrace $,

$\displaystyle H_u^{c,gen}\geq {1\over t^2-1}\left((t-c)h_{e-u}+(ct-1)h_u\right).$

This result generalizes a recent theorem of Iarrobino (which treats the case $ t=2$).

Finally, we begin to obtain, as a consequence, some structure theorems for level $ h$-vectors of type bigger than 2, which is, at this time, a very little explored topic.

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

  • [BH] W. Bruns and J. Herzog: Cohen-Macaulay rings, Cambridge studies in advanced mathematics, No. 39, Revised edition (1998), Cambridge, U.K. MR 1251956 (95h:13020)
  • [Ge] A.V. Geramita: Inverse Systems of Fat Points: Waring's Problem, Secant Varieties and Veronese Varieties and Parametric Spaces of Gorenstein Ideals, Queen's Papers in Pure and Applied Mathematics, No. 102, The Curves Seminar at Queen's (1996), Vol. X, 3-114. MR 1381732 (97h:13012)
  • [GHMS] A.V. Geramita, T. Harima, J. Migliore and Y.S. Shin: The Hilbert Function of a Level Algebra, Memoirs of the Amer. Math. Soc., to appear.
  • [Ia] A. Iarrobino: Hilbert functions of Gorenstein algebras associated to a pencil of forms, Proc. of the Conference on Projective Varieties with Unexpected Properties (Siena 2004), C. Ciliberto et al. eds. (de Gruyter), 2005, pp. 273-286, MR 2202259
  • [Ia2] A. Iarrobino: Compressed Algebras: Artin algebras having given socle degrees and maximal length, Trans. Amer. Math. Soc. 285 (1984), 337-378. MR 0748843 (85j:13030)
  • [IK] A. Iarrobino and V. Kanev: Power Sums, Gorenstein Algebras, and Determinantal Loci, Springer Lecture Notes in Mathematics (1999), No. 1721, Springer, Heidelberg. MR 1735271 (2001d:14056)
  • [NW] N. Nisam and A. Wigderson: Lower bounds on arithmetic circuits via partial derivatives, Comput. Complexity 6 (1996/1997), No. 3, 217-234. MR 1486927 (99f:68107)
  • [Za] F. Zanello: Level algebras of type 2, Comm. Algebra 34 (2006), No. 2, 691-714. MR 2211949

Similar Articles

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

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

Additional Information

Fabrizio Zanello
Affiliation: Dipartimento di Matematica, Università di Genova, Genova, Italy
Address at time of publication: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

Keywords: Artinian algebra, level algebra, $h$-vector, generic quotient, dimension, partial derivatives.
Received by editor(s): February 22, 2005
Received by editor(s) in revised form: March 17, 2005
Published electronically: January 4, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society