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- W2951416441 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 $uin lbrace 1,...,erbrace $, $$H_u^{c,gen}geq {1over t^2-1}((t-c)h_{e-u}+(ct-1)h_u).$$ 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." @default.
- W2951416441 created "2019-06-27" @default.
- W2951416441 creator A5022091558 @default.
- W2951416441 date "2005-02-22" @default.
- W2951416441 modified "2023-09-27" @default.
- W2951416441 title "Partial derivatives of a generic subspace of a vector space of forms: quotients of level algebras of arbitrary type" @default.
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