Exercise 3, § 1

a) Let $(\alpha_i, \beta_i)$ be $n$ distinct pairs of real numbers $\geqslant 0$, distinct from $(0, 0)$, and such that $\inf_i \alpha_i = \inf_i \beta_i = 0$. Consider the equation

where the $a_i$ are real numbers $\neq 0$ and the $\varphi_i$ are continuous functions on a square $0 \leqslant x \leqslant a, 0 \leqslant y \leqslant a$, tending to 0 as $(x, y)$ tends to $(0, 0)$. Suppose that there exists a function $g$ which is positive and continuous on an interval $[0, b]$, tending to 0 with $x$ and such that $f(x, g(x)) = 0$ for all $x \in [0, b]$. For every real number $\mu > 0$ show that $g(x)/x^\mu$ tends to a limit, finite or infinite, as $x$ tends to 0 (use V, p. 217, prop. 9, considering, for every number $t \geqslant 0$, the equation $f(x, tx^\mu) = 0$).

b) For $g(x)/x^\mu$ to tend to a finite limit $\neq 0$ it is necessary that $\mu$ should be such that there exist at least two distinct pairs $(\alpha_h, \beta_h)$ and $(\alpha_k, \beta_k)$ such that $\alpha_h + \mu \beta_h = \alpha_k + \mu \beta_k$ and such that, for every other pair $(\alpha_i, \beta_i)$, one has $\alpha_i + \mu \beta_i \geqslant \alpha_h + \mu \beta_h$. One thus obtains a finite number of possible values $\mu_j$ ($1 \leqslant j \leqslant r$); the numbers $-1/\mu_j$ are the gradients of the affine lines in the plane $\mathbf{R}^2$ which contain at least two of the set of points $(\alpha_i, \beta_i)$ and such that all the other points of this set lie above the line considered ("Newton polygon").

c) Let $\mu_1$ be the smallest of the numbers $\mu_j$. Show that $g(x)/x^{\mu_1}$ tends to a finite limit (possibly zero). (Show first that one may always assume that if $i$ and $j$ are two distinct indices one cannot have both $\alpha_i \leqslant \alpha_j$ and $\beta_i \leqslant \beta_j$; deduce that one can assume $\alpha_1 = 0$, $\alpha_i > 0$ and $\beta_i < \beta_1$ for $i \neq 1$; now, putting $g(x) = t(x) x^{\mu_1}$, show that $t(x)$ cannot tend to $+\infty$ as $x$ tends to 0.)

d) Deduce from c), by recursion on $r$, that there exists an index $j$ such that $g(x)/x^{\mu_j}$ tends to a finite limit $\neq 0$ as $x$ tends to 0.