Exercise 11, § 6

In this exercise we assume that A is a field, $f(0) \neq 0$ and $\deg g < \deg f = p$. Let $\sum \alpha_n X^n$ be the expansion in a formal power series of the rational fraction $g(X)/f(X)$; for each integer $k \geq 0$ let $H_n^{(k)}$ be the Hankel determinant defined in Exercise 1 of IV, p. 90. a) Show that $H_n^{(k)} = 0$ for $k > p$ and calculate $H_0^{(p)}$ as function of $\operatorname{res}(f, g)$. b) Under what conditions do we have $H_0^{(p)} = H_0^{(p-1)} = \ldots = H_0^{(p-r+1)} = 0$?