definition. polynomial [cox1997ideals, 1.1.2, 1.1.3] [ag-0004]

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A polynomial \(f\) over a ring \(R\) in \(n\) variables is a finite linear combination (with coefficients \(a_\alpha \) in \(R\) ) of monomials, i.e. the formal expression of the form \[ f=\sum _\alpha a_\alpha x^\alpha , \quad a_\alpha \in R, \]

The set of all polynomials in \(x_1, \ldots , x_n\) with coefficients in \(R\) is denoted \(R\left [x_1, \ldots , x_n\right ]\).

\(a_\alpha x^\alpha \) is called a term of \(f\) if \(a_\alpha \neq 0\).

The total degree of \(f \neq 0\), denoted \(\operatorname {deg}(f)\), is the maximum \(|\alpha |\) such that the coefficient \(a_\alpha \) is nonzero. The total degree of the zero polynomial is undefined.