Given a nonzero polynomial \(f \in k[x]\), let

where \(c_i \in k\) and \(c_0 \neq 0\) [thus, \(m = \deg (f)\)]. Then we say that \(c_0 x^m\) is the leading term of \(f\), written

Fix a monomial order \({\gt}\) on \(\mathbb {Z}_{\geq 0}^n\), and let \(F = (f_1, \ldots , f_s)\) be an ordered \(s\)-tuple of polynomials in \(k[x_1, \ldots , x_n]\). Then every \(f \in k[x_1, \ldots , x_n]\) can be written as

where \(a_i, r \in k[x_1, \ldots , x_n]\), and either \(r = 0\) or \(r\) is a linear combination, with coefficients in \(k\), of monomials, none of which is divisible by any of \(\mathrm{LT}(f_1), \ldots , \mathrm{LT}(f_s)\). We will call \(r\) a remainder of \(f\) on division by \(F\).

Fix a monomial order on the polynomial ring \(k[x_1, \ldots , x_n]\).A finite subset \(G = \{ g_1, \ldots , g_t\} \) of an ideal \(I \subseteq k[x_1, \ldots , x_n]\), with \(I \ne \{ 0\} \), is said to be a Gröbner basis (or standard basis) if

Using the convention that \(\langle \emptyset \rangle = \{ 0\} \), we define the empty set \(\emptyset \) to be the Gröbner basis of the zero ideal \(\{ 0\} \).

The \(S\)-polynomial of \(f\) and \(g\) is the combination