# Clarification about a proof regarding a sum of polynomials being expressed as a linear combination of S-polynomials

I'm reading this proof from Ideals, Varieties, and Algorithms by David A. Cox, Donal O'Shea, and John Little. You can find an online version here.

This is Lemma 5 of Chapter 2, page 85. From my understanding, what it is saying is that if you have some polynomials with the same multidegree and you sum them together and get a polynomial with a smaller multidegree, then this sum can be expressed as some linear combination of S-polynomials. The 'cancellation' of terms of the highest multidegree in a sense happens in the S-polynomial. Correct me if I'm wrong.

I don't have a problem with the proof per se, but I have some questions that need clarification:

1) The proof seems to imply that the second S-polynomial has to be fixed in the summation. In $$\sum_{i=0}^{s-1}d_iS(p_i,p_s)$$ on the last line of page 85, the summation is summed over $$i$$, so the $$p_s$$ term is unvaried. So when the proof says expressed as a linear combination of S-polynomials, do they mean any S-polynomial generated by $$2$$ completely different S-polynomials, or does one of the polynomials generating the S-polynomial has to stay constant? If the former case is true, how can we prove it?

2) As an extension: would the following lemma still apply if the polynomials, $$p_i$$ have different multidegrees? I feel like it should, but I have no idea how to start the proof.

• What is an $S$-polynomial? – Servaes Apr 12 at 15:29
• An S-polynomial is defined here: maths.qmul.ac.uk/~whitty/LSBU/MathsStudyGroup/Buchberger.pdf or you can look inside the book given in my post. The intuition is that the leading monomials inside 2 given polynomials will cancel out. Hope it helps. Unfortunately, the actual definition is quite hard to write out in this comment without reference to prior terminology. – Yip Jung Hon Apr 12 at 15:32