I am trying to work through the text Ideals, Varieties, and Algorithms and I came across this lemma in the section discussing S-Polynomials & Buchberger's Criterion.
It is stated:
Suppose we have a sum $\sum_{i=1}^{s} {c_i f_i}$, where $c_i \in k$ and multideg($f_i)$ = $\delta \in Z^{n}$ $|$ $n \geq 0$ for all i. If multideg ($\sum_{i=1}^{s} {c_i f_i}$) < $\delta, $ then $\sum_{i=1}^{s} {c_i f_i}$ is a linear combination, with coefficients in K, of the S-polynomials $S(f_j,\,f_k) \; \text{for} \;1 \leq j,k \leq s$. Furthermore each $S(f_i,\:f_k)$ has multidegree $< \delta$.
In particular my specific questions are:
1) When it says "multideg($f_i)$ = $\delta \in Z^{n}$ $|$ $n \geq 0$ for all i" does this mean every $f_i$ in the ideal has the same multideg? or does it mean multideg$(f_i) = \delta_i$. If it is the former, why does every polynomial have to have the same multideg?
2) "If multideg ($\sum_{i=1}^{s} {c_i f_i}$) < $\delta$". Related to question 1, what does this $\delta$ refer to?
I don't have the book with me, but I don't think there are any typos. 1. It means each of (only) f_1 to f_s has multidegree $\delta$, not that every polynomial in the ideal has the same degree (only every polynomial in the list). Why? Because we want to say that in the sum $\Sigma c_if_i$ Has smaller degree because we are saying Some cancellation occurs and that's how we end up with something of a different degree. Look at a concrete example involving the s-polynomial and you will see what I mean.