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definition. monomial [cox1997ideals, 1.1.1] [ag-0003]

A monomial in \(n\) formal variables \(x_1, \ldots , x_n\), denoted \(x^\alpha \), is a formal expression of the form \[ x_1^{\alpha _1} x_2^{\alpha _2} \cdots x_n^{\alpha _n} \] where \(n \in \mathbb N\), \(\alpha =\left (\alpha _1, \ldots , \alpha _n\right )\) is an \(n\)-tuple of nonnegative integers.

The total degree of the monomial is denoted \(|\alpha |=\alpha _1+\cdots +\alpha _n\).