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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.

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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 $a+b+c+d-4$?

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    I agree that a element of the form $x^{a-1} y^{b-1}z^{c-1}w^{d-1} $ will be not zero in the algebra, but the point is that I guess it is possible to find an element of the form $x^{i} y^{j}z^{k}w^{l}$ not zero, such that $i+j+k+l = a+b+c+d-1$, but I don't really know how to do it. even if the right answer is $a+b+c+d-4$, the only way of showing this is showing that any other polynomial with greater degree is zero right? – 2017-02-09
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    @user123456 What you are describing is patently not possible. If $i+j+k+l > a+b+c+d-4$, then that means at least one of the following is true: $i > a-1$, $j> b-1$, $k > c-1$, $l > d-1$. It follows immediately that $x^iy^jz^kw^l=0$ in $R/I$. – 2017-02-09
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    dear @rschwieb, now I see, in fact your answer is correct, sorry for the delay on understanding it. – 2017-02-09