HowTo:Term Orderings

This page is an introduction into term orderings in CoCoA-5 or ApCoCoA-2. In order to understand this topic, we assume that the reader is familiar with the concept of term orderings on polynomial rings.

Mathematical definition

Let K be a field, let $P=K[x_1,\dots ,x_n]$ be the polynomial ring over K in n indeterminates and let $\mathbb{𝕋}(P)=\{x_1^{{\alpha }_1}\cdots x_n^{{\alpha }_n}\mid {\alpha }_1,\dots ,{\alpha }_n\in {\mathbb{\mathbb{N}}}_0\}$ be the set of all power products in P.

Let $\sigma$ be a total order on T(P). For a pair $(t_1,t_2)\in \sigma$, we write $t_1{\le }_{\sigma }{t}_2</math.Then<math>\sigma$ is called a term ordering on T(P) iff it is multiplicative, i.e. $t_1{\le }_{\sigma }{t}_2$ implies $s{t}_1{\le }_{\sigma }s{t}_2$ for each $t_1,t_2,s\in \mathbb{𝕋}(P)$, and we have $1{\le }_{\sigma }t$ for all terms $t\in \mathbb{𝕋}(P)$.

For a polynomial $f\in P$, we can write $f=c_1{t}_1+\cdots +c_s{t}_s$ where $t_i\in \mathbb{𝕋}(P)$ and $c_i\in K^{\times }$ for each i such that $t_1{>}_{\sigma }\cdots {>}_{\sigma }{t}_s$. Then we use the following notation

• ${\mathrm{L}\mathrm{T}}_{\sigma }(f)=t_1$ is the leading terms of f,
• ${\mathrm{L}\mathrm{C}}_{\sigma }(f)=c_1$ is the leading monomial of f and
• ${\mathrm{L}\mathrm{M}}_{\sigma }(f)=c_1{t}_1$ is the leading coefficient of f.