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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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I should have asked to clarify first: what exactly is your book? Is this a class in general abstract algebra or are you talking about ideals and varieties? –  icurays1 Nov 12 '12 at 8:29
    
We are using a pdf that was typed by the instructor. This is a general abstract algebra class. –  Low Scores Nov 12 '12 at 8:43
    
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. –  icurays1 Nov 12 '12 at 8:46

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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).

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