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Given a vector space $V$ of homogeneous polynomials of the same degree overan infinite field, consider a generic subspace $W$ of $V$. The main result ofthis paper is a lower-bound (in general sharp) for the dimensions of the spacesspanned 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 ofthe forms generating the original space $V$. Rephrasing our result in thelanguage of commutative algebra (where this result finds its most importantapplications), 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 ofthe 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 $,$$H_u^{c,gen}\geq {1\over t^2-1}((t-c)h_{e-u}+(ct-1)h_u).$$ This resultgeneralizes a recent theorem of Iarrobino (which treats the case $t=2$).Finally, we begin to obtain, as a consequence, some structure theorems forlevel $h$-vectors of type bigger than 2, which is, at this time, a very littleexplored topic.

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