Definition b4. Let \(\alpha = (\alpha_1, \ldots, \alpha_n) \in \mathbb{N}_0^n\) = ( 1, , \alpha n) \in \mathbbN_0^n be an \(n\)n-tuple of non-negative integers. A monomial in \(x_1, \ldots, x_n\)x 1, , x n is a product \[\prod_{i=1}^n x_i^{\alpha_i} = x_1^{\alpha_1} \cdot x_2^{\alpha_2} \cdots x_n^{\alpha_n}.\] We write \(x^\alpha = \prod_{i=1}^n x_i^{\alpha_i}\)x = i=1^n x_i^_i for short. The total degree of \(x^\alpha\)x is \(\lvert x^\alpha \rvert = \sum_{i=1}^n \alpha_i\) x = \sum i=1^n \alpha_i. Note that \(x^\alpha = 1\)x = 1 when \(\alpha = (0, \ldots, 0)\) = (0, , 0) and that any monomial is fully determined by \(\alpha\).
Definition b5. Let \(x^\alpha\)x be a monomial and let \(\mathbb{F}\)F be a field. A term with a non-zero coefficient \(c_\alpha \in \mathbb{F}\)c F is the product \(c_\alpha x^\alpha\)c x .
Definition bh. A polynomial \(f\)f with coefficients in a field \(\mathbb{F}\)F is a finite sum of terms: \[f = \sum_\alpha c_\alpha \cdot x^\alpha, \quad c_\alpha \in \mathbb{F}.\] The zero polynomial is denoted \(0\)0. Polynomials of one variable are called univariate polynomials and are denoted \(f(x) \in \mathbb{F}[x]\)f(x) F[x].
Definition b7. Let \(f = \sum_{\alpha \in \mathbb{N}_0^n} c_\alpha x^\alpha\)f = \mathbbN 0^n c_\alpha x^\alpha and \(g = \sum_{\alpha \in \mathbb{N}_0^n} d_\alpha x^\alpha\)g = \mathbbN 0^n d_\alpha x^\alpha be polynomials.
- Their sum is \(f + g = \sum_{\alpha \in \mathbb{N}_0^n} (c_\alpha + d_\alpha) x^\alpha\)f + g = \mathbbN 0^n (c_\alpha + d_\alpha) x^\alpha.
- Their product is \(f \cdot g = \sum_{\gamma \in \mathbb{N}_0^n} \left( \sum_{\alpha + \beta = \gamma} c_\alpha d_\beta \right) x^\gamma\)f g = \in \mathbbN 0^n \left( \sum_{\alpha + \beta = \gamma} c_\alpha d_\beta \right) x^\gamma.
Definition b6. Let \(f = \sum c_\alpha x^\alpha \neq 0 \in \mathbb{F}[\mathbf{x}]\)f = c x \neq 0 \in \mathbbF[\mathbfx] be a non-zero polynomial. The total degree of \(f\)f, denoted \(\deg(f)\)(f), is the maximum \(\lvert x^\alpha \rvert\) x such that the corresponding coefficient \(c_\alpha\)c is nonzero. The degree of \(0\)0 is undefined.
Remark b9. The set of all polynomials in \(x_1, \ldots, x_n\)x 1, , x n with coefficients in a field \(\mathbb{F}\)F is denoted \(\mathbb{F}[x_1, \ldots, x_n]\)F[x 1, , x n] or \(\mathbb{F}[\mathbf{x}]\)F[x] for short. This set forms a ring under standard polynomial addition and multiplication and is called the polynomial ring (Becker 1993). Not all polynomials have multiplicative inverses and so \(\mathbb{F}[\mathbf{x}]\)F[x] does not form a field. We say that \(f\)f divides \(g\)g and write \(f \mid g\)f g if \(g = fh\)g = fh for some polynomial \(h \in \mathbb{F}[\mathbf{x}]\)h F[x].
Definition ny. Let \(f(x) \in \mathbb{F}[x]\)f(x) F[x] be a univariate polynomial of positive degree. The polynomial \(f(x)\)f(x) is irreducible over \(\mathbb{F}[x]\)F[x] if there is no factorization \(f(x) = p(x)q(x)\)f(x) = p(x)q(x), where \(p(x)\)p(x) and \(q(x)\)q(x) are also univariate polynomials of positive degree in \(\mathbb{F}[x]\)F[x].