Monomials, coefficients, degree in each variable and total degree in $F[x_1,\dots ,x_n]$

Let $F$ be a field. Every polynomial in the iterated ring $F[x_1,\dots ,x_n]$ has a unique finite expansion

$f={\sum }_{\mathbf{t}}{c}_{\mathbf{t}}{x}^{\mathbf{t}},x^{\mathbf{t}}:=x_1^{t_1}\cdots x_n^{t_n},$

over finitely many multi-indices $\mathbf{t}=(t_1,\dots ,t_n)\in {\mathbb{N}}^n$.

For such an index, the scalar $c_{\mathbf{t}}$ is the coefficient $[x^{\mathbf{t}}]f$ of the monomial $x^{\mathbf{t}}$.

The degree in $x_i$ is the largest $t_i$ with $c_{\mathbf{t}}\mathrm{\ne }0$, and the total degree is the largest value of $t_1+\cdots +t_n$ with $c_{\mathbf{t}}\mathrm{\ne }0$. For the zero polynomial, every degree is left undefined.

Evaluation at a point $(a_1,\dots ,a_n)\in F^n$ is the iterated substitution supplied by the universal property of the polynomial ring.

• The page uses this expansion constantly. The Nullstellensatz and the grid reduction need both the separate variable degrees and the total degree, and neither is visible in the bare iterated-ring construction unless it is named here.