TOPICS
Search

S-Polynomial

For nonzero multivariate polynomials f and g with leading monomials LM(f) and LM(g) and leading terms LT(f) and LT(g), their S-polynomial is

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

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, Monomial Order

Explore with Wolfram|Alpha

References

Cox, D.; Little, J.; and O'Shea, D. Ideals, Varieties, and Algorithms: An Introduction to Algebraic Geometry and Commutative Algebra, 2nd ed. New York: Springer-Verlag, 1996.

Cite this as:

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

Subject classifications