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Locally smooth solutions for monic polynomial equations

Source: Admission to SNSB, 2002

October 4, 2019
differential equationsfunctionalgebrapolynomial

Problem Statement

Provided that the roots of the polynom Xn+a1Xn−1+a2Xn−2+⋯+an−1X+an:∈R[X], X^n+a_1X^{n-1} +a_2X^{n-2} +\cdots +a_{n-1}X +a_n:\in\mathbb{R}[X] , of degree n≥2, n\ge 2, are all real and pairwise distinct, prove that there exists is a neighbourhood V \mathcal{V} of (a1,a2,…,an) \left( a_1,a_2,\ldots ,a_n \right) in Rn \mathbb{R}^n and n n functions x1,x2,…,xn∈C∞(V) x_1,x_2,\ldots ,x_n\in\mathcal{C}^{\infty } \left( \mathcal{V} \right) whose values at (a1,a2,…,an) \left( a_1,a_2,\ldots ,a_n \right) are roots of the mentioned polynom.
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