Gröbner bases

Let $G=\{g_1,...,g_k\}\subset F[x_1,...,x_n]$ be a finite collection of polynomials with coefficients in some field $F$. Let $I=\langle G\rangle$ denote the ideal in $F[x_1,...,x_n]$ generated by $G$.

Definition 2.8.7   We say that $G$ is a Gröbner basis for $I$ if each $f\in I$ has a leading term (with respect to some fixed ordering on the set of monomials) which is divisible by the leading term of some element of $G$.

A construction Gröbner bases (due to Buchburger) will be given later.

Definition 2.8.8   Let $f,g\in F[x_1,...,x_n]$ be non-zero polynomials. The $S$-polynomial of $f,g$ is

$$S(f,g)= \frac{lcm(lt(f),lt(g)}{lt(f)}\cdot f -\frac{lcm(lt(f),lt(g)}{lt(g)}\cdot g.$$