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
Abstract: Given a vector space of homogeneous polynomials of the same degree over an infinite field, consider a generic subspace of . 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 , in terms of the dimensions of the spaces spanned by the partial derivatives of the forms generating the original space .
Rephrasing our result in the language of commutative algebra (where this result finds its most important applications), we have: let be a type artinian level algebra with -vector , and let, for , be the -vector of the generic type level quotient of having the same socle degree . Then we supply a lower-bound (in general sharp) for the -vector . Explicitly, we will show that, for any ,
Finally, we begin to obtain, as a consequence, some structure theorems for level -vectors of type bigger than 2, which is, at this time, a very little explored topic.
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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.