Deformation of Minimal Polynomials and Approximation of Several Intervals by an Inverse Polynomial Mapping

In this paper we show that for a given set of l real disjoint intervals E_l=^l_j=1 [a_2j−1, a_2j] and given >0 there exists a real polynomial and a set of l disjoint intervals E_l=^l_j=1 [ã_2j−1, ã_2j] with E_lE_l and (ã₁, …, ã_2l)−(a₁, …, a_2l)_max<, such that ⁻¹([−1, 1])=E_l. The statement follows by showing how to get in a constructive way by a continuous deformation procedure from a minimal polynomial on E_l with respect to the maximum norm a polynomial mapping of E_l.