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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?

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    Ah, okay. My answer stands - the S-polynomials were defined this way because they satisfy certain properties that are needed to construct Groebner bases. The process is somewhat similar to Gaussian elimination from linear algebra, but with polynomials (usually multivariate ones, at that). I recommend the book "Ideals, Varieties, and Algorithms" if you want a gentle introduction to this sort of thing.2012-11-12

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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).