I started to study properties of Artinian and Gorestein Rings, trying to approach the Fröberg conjecture, but I note that I am having trouble computing some examples. For instance, I would like to compute by hand, the socle degrees of a ring of the form $R/I$, where $R = k[x,y,z,w]$ and $I = \langle x^a,y^b,z^c,w^d\rangle $.
I understand that the socle degree is the highest degree of the polynomials in $R/I$, but I am not sure how can I be able to compute it in terms of $a,b,c,d$. Any help will be welcome, and references too.
Thank you.
Perhaps I'm overlooking something, but it seems like an example of an polynomial with maximal degree in the quotient is exhibited by $x^{a-1}y^{b-1}z^{c-1}w^{d-1}$. So would the answer in your case not simply be $a+b+c+d-4$?