# Minimal syzygies for polynomial ideals

Let $I$ be an ideal of $S=k[x_1,\dots,x_n]$. I am asked to find a minimal free graded resolution of $I$, by means of syzygy matrices. I suppose there has to be an algoritmic approach to it, provided that the ideal is not very complicated and the number of variables is decent.

I found in Ralf Froeberg - An Introduction to Groebner Bases (section 3.10) that for monomial ideals, there's a simple approach. What if the ideal is not monomial?

I'll describe further the approach that I've been taught to have.

For example, take $I=(x^2,y^2,xy+yz) \trianglelefteq k[x,y,z]=S$. I can find the syzygies on its generators, using Buchberger's algorithm, but before that I found a Groebner basis for $I$ being $G=\{f_1, \dots,f_4\}=I \cup (yz^2)$.

Now, if I compute the S-polynomials, I get 6 relations (arising from $S(f_i,f_j)$, with $1 \leq i < j \leq 6$). How can I tell which of these relations (syzygies) are minimal? Of course, if I could find some linear combinations of some of them being equal to some other, it'd be great, but they don't seem obvious for me.

So, how can I decide which syzygies are minimal? Both in this case and in general.

I then can form the resolution like this: $0 \rightarrow X \rightarrow S(-2)^3 \oplus S(-3) \rightarrow S \rightarrow S/I \rightarrow 0$, where $X$ is about to be determined. The shifts in the third nonzero term from the right correspond to the degrees of the polynomials in $G$.

Provided that I find the minimal syzygies, I determine the degrees of the polynomials that appear in its columns and those will be the shifts in $X$.

Another question: could it happen that in a column of the above matrix there are polynomials of different degrees? If so, what do I do?