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
DOI:
https://doi.org/10.1090/S0002-9947-07-04015-9
Published electronically:
January 4, 2007
MathSciNet review:
2286051
Full-text PDF
Abstract | References | Similar Articles | Additional Information
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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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
Email:
zanello@kth.se
DOI:
https://doi.org/10.1090/S0002-9947-07-04015-9
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.