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Theorem 3 (Division Algorithm in $k[x_1, ldots, x_n]$)

[theorem] Let $>$ be a monomial order on $\ZZ_{\ge 0}^n$ 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 $q_i, r\in k[x_1, \ldots, x_n]$, and either $r = 0$ or $r$ is a linear combination, with coefficients of $k$, of monomials, none of which is divisible by any of $\operatorname{LT}(f_1)$, $\ldots$, $\operatorname{LT}(f_s)$. We call $r$ a remainder of $f$ on division by $F$. Furthermore, if $q_i f_i \neq 0$, then

[/theorem]