Exercise 13, § 6

Let $n \geq 1$ be an integer, and order the elements of $\mathbf{N}^n$ lexicographically (notation $a \leq \beta$). For each $a \in \mathbf{N}^n$ let $T_a$ be the set of elements of $\mathbf{Z}[X_1, \ldots, X_n]$ of the form $X^\alpha + \sum u_{\alpha \beta} X^\beta$ (with $u_{\alpha \beta} \in \mathbf{Z}$).

a) Show that the product of an element of $T_a$ by an element of $T_\beta$ lies in $T_{\alpha + \beta}$.

b) Let $\varepsilon_k$ be the element $(1, 1, \ldots, 1, 0, \ldots, 0)$ of $\mathbf{N}^n$. Show that $s_k \in T_{\varepsilon_k}$. Deduce that

belongs to $T_a$, for $a \in \mathbf{N}^n$ such that $a_1 \geq a_2 \geq \ldots \geq a_n$. (Set $a_{n+1} = 0$.)

c) With the notation of IV, p. 66, deduce that $c_s = d_s = 1$ for $a \in D_k$ and $c_{\alpha \beta} = d_{\alpha \beta} = 0$ if $\alpha < \beta$.