We fix $n \geq 1$. Let $f$ be a continuous function $f : \mathbb R_+^* \to \mathbb R^n$. Suppose that we have $n$ polynomials in $n+1$ variables

$$\begin{aligned} P_1(Y, X_1, \dots, X_n)\\ P_2(Y, X_1, \dots, X_n)\\ \vdots \qquad\\ P_n(Y,X_1,\dots,X_n)\end{aligned}$$

such that for $a \in \mathbb R_+^*, f(a)$ is an $\textbf{isolated}$ solution of the following system $(S_a)$ with unknown $(x_1,\dots,x_n)$

\begin{equation} \left\{ \begin{aligned} P_1(a,x_1,\dots,x_n) &= 0\\ P_2(a,x_1,\dots,x_n) &= 0\\ &\vdots\\ P_n(a,x_1,\dots,x_n) &= 0 \end{aligned} \right. \end{equation}

Finally we suppose that we have $p \geq 1$, such that for all $i$, $\mbox{deg} (P_i) \leq p$. I would like to:

show that $f$ has a limit in $0$.

show $f$ admits a Pusieux expansion near $0$.

have information about the first term of this Puiseux series (depending on $n$ and $p$) to be able to give a majorant of the rate of convergence of $f$ to its limit in $0$.

I know that this question is really specific, and I don't know how hard it is since I don't know a lot about systems of polynomials equations.

Solving systems of polynomial equations. It might be of some help. $\endgroup$ – Igor Khavkine Jun 16 '18 at 18:11