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 $P=K[x_{1},\ldots ,x_{n}]$ be the polynomial ring over K in n indeterminates and let $\mathbb {T} (P)=\{x_{1}^{\alpha _{1}}\cdots x_{n}^{\alpha _{n}}\mid \alpha _{1},\ldots ,\alpha _{n}\in \mathbb {N} _{0}\}$ 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 $t_{1}\leq _{\sigma }t_{2}</math.Then<math>\sigma$ is called a term ordering on T(P) iff it is multiplicative, i.e. $t_{1}\leq _{\sigma }t_{2}$ implies $st_{1}\leq _{\sigma }st_{2}$ for each $t_{1},t_{2},s\in \mathbb {T} (P)$, and we have $1\leq _{\sigma }t$ 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

• ${\rm {LT}}_{\sigma }(f)=t_{1}$ is the leading terms of f,
• ${\rm {LC}}_{\sigma }(f)=c_{1}$ is the leading monomial of f and
• ${\rm {LM}}_{\sigma }(f)=c_{1}t_{1}$ is the leading coefficient of f.