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S-Polynomial

The S-polynomial of two nonzero polynomials f,g in K[x_1,...,x_n] over a field K, relative to a fixed monomial order, is

S(f,g)=(lcm(LM(f),LM(g)))/(LT(f))f-(lcm(LM(f),LM(g)))/(LT(g))g.

Here, LM denotes the leading monomial and LT the leading term. The two multiples have the same leading monomial, so their difference cancels it. S-polynomials are used by Buchberger's algorithm to determine whether a set of polynomials is a Gröbner basis.

See also

Buchberger's Algorithm, Gröbner Basis, Leading Monomial, Leading Term, Monomial Order, Polynomial Reduction

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References

Cox, D.; Little, J.; and O'Shea, D. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 2nd ed. New York: Springer-Verlag, 1997.

Cite this as:

Weisstein, Eric W. "S-Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/S-Polynomial.html

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