# Questions about polynomial systems with parameter

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 $(x_1,\dots,x_n)$

\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.

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.

• This previous question discusses systems over complex numbers mathoverflow.net/questions/88867/… . For related algorithms, see Maurer, J. Manuscripta Math (1980) 32: 91. doi.org/10.1007/BF01298184 and Jensen, A.N., Markwig, H. & Markwig, T. Collect. Math. (2008) 59: 129. doi.org/10.1007/BF03191365 – j.c. Jun 16 '18 at 16:17
• There's a book by Sturmfels, Solving systems of polynomial equations. It might be of some help. – Igor Khavkine Jun 16 '18 at 18:11
• Thanks a lot for your answers. In the answer by samuele in topic "Puiseux series expansion for space curves?", do you know if can we give a minoration of the integer $p$ he introduced ? – John Jun 16 '18 at 22:27
• I meant a majoration of the integer $p$ Samuele introduced, not a minoration sorry – John Jun 16 '18 at 22:32