Buchberger's algorithm requires computing S-pairs (more on page 83 of Ideals, Varities and Algorithm by Cox et all 2008, 3rd edition)
S(f,g)=LCM(LT(f),LT(g))/LT(f) *f - LCM(LT(f),LT(g))/LT(g) * g
where LCM is the least common multiple (equivalent to x^\gamma in the book notation) and LT is the leading term.
How can you compute S-pairs in Macaulay2 or otherwise programmatically?
Example: S-pair in Graded Lexicographic order with g1=x^2-y and g2=x^3-z where S(g1,g2)=xz-xy.
S-pair command for Macaulay2 as a function by the definition
and also the code in text for easy copying
Spair = (g1,g2) -> lcm(leadTerm(g1),leadTerm(g2))/leadTerm(g1)*g1-lcm(leadTerm(g1),leadTerm(g2))/leadTerm(g2)*g2;
that works such that Spair(polynomial1, polynomial2) computes the S-pair polynomial for polynomial1 and polynomial2.
In contrast, alternative method for Spairs can apparently be deduced in terms of syzygies and generators by the Theorem 9 of the book (1)
S * G = \sum_{i=1}^t h_i g_i \rightarrow_G 0
where the mapping is modulo G, S-pairs are somehow related to the expression in terms of syzygies and generators, some examples the below. Perhaps syz and gb are useful to compute S-pairs.
Relationship of S-pairs to generating sets in terms of Gröbner basis and syzygies
"Let us now take a look at the first syzygies (or minimal S-pairs [1, §2.9]) among the sixteen minimal generators. " (Ideals, Varieties and Macaulay 2, p9)
"The matrix spairs contains all the S-pairs between generators of J corresponding to the minimal first syzygies of M" (p.195 here)
References
(1) Ideals, Varities, and Algorithms by Cox et all (2008, 3rd ed)