Let $P=k[x_1...x_n]$ be a poly over a field. Suppose that $R=k[x_1...x_n]/I$. The canonical module of $R$ is $\omega_R=Ext^{n-dim(R)}_P(R,P)$.
The question is that is there any upper bound for the minimum number of generators of $\omega_R$ in terms of $r=\text{the minimum number of generators of}$ $I$? and\or $\text{the embedding dim of}$ $R$?
One may add more reasonable conditions to $R$, e.g. assume that $R$ is Cohen-Macaulay?
and\or
Assume that $I$ is almost complete intersection i.e. $r=codim(I)+1$.
A desired upper bound would be $r\choose 2$ i.e. $\mu(\omega_R)\leq {r\choose 2}$.
Recall that min num gen $\omega_R$ denoted by $\mu(\omega_R)$ is also called the type of the ring $R$ denoted by $r(R)$. And the latter is one iff $R$ is Gorenstein , this is a result of P.Roberts.
Let's also remind that in the CM case the question is tantamount to ask an upper bound for the last betti number in the min free resolution of $R$ over $P$.
