Exercise 8, § 5

Suppose that $K = \mathbf{Z}$ and that X is finite. For every integer $k \geq 0$, let $(e_{k,\alpha})$ denote the basis of $A_k(X)$.

(a) Let $(w_n)_{n \in \mathbf{N}}$ be a sequence of elements of F. Let $w_{k,\alpha}(n)$ be the coefficient of $e_{k,\alpha}$ in the term of $g(w_n) \in \hat{A}(X)$ of degree k. Then:

The sequence $(w_n)$ is called typical if, for every ordered pair $(k, \alpha)$ the function $w_{k,\alpha}: n \mapsto w_{k,\alpha}(n)$ is a binomial polynomial of degree $\leq k$, cf. Exercise 4. This condition is independent of the choice of the bases $(e_{r,\alpha})$. Show that it is equivalent to the existence of a sequence $(a_k)_{k \in \mathbf{N}}$ of elements of $\hat{A}_q(X)$ such that $\omega(a_k) \geq k$ for all $k$ and

(b) Show that, if $(w_n)$ and $(w'_n)$ are two typical sequences, so are $(w_n^{-1})$ and $(w_n w'_n)$.

(c) Let $w$ be an element of $\mathrm{C}^k\mathbf{F}$ for $k \geq 1$ and let $f$ be a binomial polynomial of degree $\leq k$. Show that $(w^{f(n)})$ is a typical sequence. In particular if $z \in \mathbf{F}$, the sequence $(z^n)$ of powers of $z$ is typical.

(d) Let $(w_n)$ be a sequence of elements of $\mathbf{F}$. Show that $(w_n)$ is typical if and only if there exist $y_0, y_1, \ldots, y_n, \ldots$ in $\mathbf{F}$, where $y_i \in \mathrm{C}^i\mathbf{F}$ for all $i \geq 1$ and

for all $n \in \mathbf{Z}$ and all $k \geq 1$. (Determine the $y_i$ from $(w_n)$ by setting successively $n = 0, 1, \ldots$. For example $w_0 = y_0, w_1 = y_0 y_1, \ldots$.)

(e) Let $\mathbf{H}$ be a Hall set relative to $X$ and let $(w_n)$ be a sequence of elements of $\mathbf{F}$. Let $(\alpha(m, n))_{m \in \mathbf{H}, n \in \mathbf{Z}}$ be the family of integers such that

for all $n \in \mathbf{Z}$ and all $c \geq 1$ (cf. no. 4, Remark). Show that $(w_n)$ is typical if and only if, for all $m \in \mathbf{H}$, the function $n \mapsto \alpha(m, n)$ is a binomial polynomial of degree $\leq l(m)$, where $l(m)$ is the length of $m.$†

(f) A sequence $(w_n)$ is called 1-typical if the corresponding functions $w_{k,\alpha}$ are binomial polynomials of degree $\leq k - 1$. Show that, if $(w_n)$ is 1-typical, there exist $y_i \in \mathrm{C}^{i+1}\mathbf{F}$ for $i = 0, 1, \ldots$, such that

for all $n \in \mathbf{Z}$ and all $k \geq 1$.