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?
Is there a reason why the $S$-polynomial is defined in this way?
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$\begingroup$
abstract-algebra
polynomials
groebner-basis
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0I 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? – 2012-11-12
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0We are using a pdf that was typed by the instructor. This is a general abstract algebra class. – 2012-11-12
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0Ah, 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).