Proposition 1

Let $u$ and $v$ be two polynomials.

(i) If $\deg u \neq \deg v$, we have

If $\deg u = \deg v$ we have $\deg(u + v) \leq \deg u$.

(ii) We have $\deg(uv) \leq \deg u + \deg v$. The proof is immediate.

Let $J \subset I$ and $B = A[(X_i)_{i \in I - J}]$; we shall identify $A[(X_i)_{i \in I}]$ with $B[(X_i)_{i \in J}]$ (III, p. 453 f.). The degree of $u \in A[(X_i)_{i \in I}]$, qua element of $B[(X_i)_{i \in J}]$, is called the degree of $u$ with respect to the $X_i$ of index $i \in J$ (III, p. 454).

Let $u = \sum_{k=0}^n a_k X^k \in A[X]$ be a non-zero polynomial of degree $n$ in a single indeterminate. The coefficient $a_n$, which is $\neq 0$ by hypothesis, is called the leading coefficient of $u$. A polynomial $u \neq 0$ whose leading coefficient is equal to 1 is called a monic polynomial.

In $A[X_1, X_2, \ldots, X_q]$ the number of monomials of total degree $p$ is equal to the number of elements $(n_k)_{1 \leq k \leq q}$ of $\mathbf{N}^q$ such that $\sum_{k=1}^q n_k = p$, that is $\binom{q+p-1}{p}$ (Sets III, Prop. 15, p. 182)

More generally, let A be a commutative monoid and $(\delta_i)_{i \in I}$, a family of elements of A. There exists a unique graduation of type A of the algebra $A[(X_i)_{i \in I}]$ such that each monomial $X^\nu$ is of degree $\sum_{i \in I} \nu_i \delta_i$ (III, p. 458, example 3). The case considered above is that where $A = \mathbf{N}$ and $\delta_i = 1$. In the general case, to avoid confusion, we shall use the word « weight » instead of « degree » and « isobaric » instead of « homogeneous ». For example, there exists a unique graduation of type N of the algebra $A[(X_i)_{i \geq 1}]$ such that $X_i$ is of weight $i$ for each integer $i \geq 1$. The isobaric elements of weight $n$ are the polynomials of the form $\sum a_{\nu} X^{\nu}$, where $a_{\nu} = 0$ for $\sum_{i \geq 1} i \cdot \nu_i \neq n$.