# HowTo:Term Orderings

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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 ${\displaystyle P=K[x_{1},\ldots ,x_{n}]}$ be the polynomial ring over K in n indeterminates and let ${\displaystyle \mathbb {T} (P)=\{x_{1}^{\alpha _{1}}\cdots x_{n}^{\alpha _{n}}\mid \alpha _{1},\ldots ,\alpha _{n}\in \mathbb {N} _{0}\}}$ be the set of all power products in P.

Let ${\displaystyle \sigma }$ be a total order on T(P). For a pair ${\displaystyle (t_{1},t_{2})\in \sigma }$, we write ${\displaystyle t_{1}\leq _{\sigma }t_{2} is called a term ordering on T(P) iff it is multiplicative, i.e. ${\displaystyle t_{1}\leq _{\sigma }t_{2}}$ implies ${\displaystyle st_{1}\leq _{\sigma }st_{2}}$ for each ${\displaystyle t_{1},t_{2},s\in \mathbb {T} (P)}$, and we have ${\displaystyle 1\leq _{\sigma }t}$ for all terms ${\displaystyle t\in \mathbb {T} (P)}$.

For a polynomial ${\displaystyle f\in P}$, we can write ${\displaystyle f=c_{1}t_{1}+\cdots +c_{s}t_{s}}$ where ${\displaystyle t_{i}\in \mathbb {T} (P)}$ and ${\displaystyle c_{i}\in K^{\times }}$ for each i such that ${\displaystyle t_{1}>_{\sigma }\cdots >_{\sigma }t_{s}}$. Then we use the following notation

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