Is there a reason why the $S$-polynomial is defined in this way?

Question

In my book the $S$-polynomial of two nonzero polynomials $f$ and $g$ is defined as $$S(f,g) = \displaystyle\frac{x^w}{LT(f)} \cdot f - \frac{x^w}{LT(g)} \cdot g$$ where $\displaystyle x^w$ is the least common multiple of $LT(f)$ and $LT(g)$. My question is where did this come from?

Answer 1

The $S$-polynomials come from Buchberger's criterion, which is a necessary and sufficient condition for a set of polynomials to be a Grobner basis. Here is a nice brief explanation of what a Grobner basis is, and Buchberger's algorithm for finding them. It requires a bit of basic background in algebra (multivariate polynomials and ideals, mostly).