2 Basic monomial operations and representations

Given an integer n > 0 we define the set of exponent vectors M_n by $\{\alpha = (\alpha_1, \ldots, \alpha_n) \vert\alpha \in {\bf N}^n\}$. Notice that monomials usually denote terms of the form $c \, x_1^{\alpha_1} \ldots x_n^{\alpha_n}$. However, in this paper we do only consider the exponent vector of a monomial and shall therefore use the words exponent vector and monomial interchangeably (i.e., we identify a monomial with its exponent vector).

We furthermore use Greek letters to denote monomials and the letter n to denote the a-priory given length of monomials (which is the number of variables in the corresponding polynomial ring).

Monomials play a central role in GB computations. In this section, we describe the basic monomial operations and discuss basic facts about monomial (resp. polynomial) representations for GB computations.