The nearest polynomial to multiple given polynomials with a given zero

Abstract

The following type of problems have been well-studied in the area of symbolic-numeric computation for about twenty years: Given a polynomial f ∈ C[x] and a point z ∈ C, find the nearest polynomial f̃ ∈ C[x] to f with f̃(z) = 0. A common framework for such problems is described in [7]. In previous works, for example [3, 4, 7, 6], problems for one given polynomial were considered. Here, we consider a problem for multiple given polynomials. Through observation or by using different numerical algorithms for a given input data, we may obtain multiple polynomials being equal in theory but being slightly different each other. Thus, it is worth considering the problem for multiple polynomials. In this abstract, after the preliminaries, we define the nearest polynomial to multiple given polynomials. In the definition, we use a pair of norms to measure the nearness between polynomials. We remark the difficulty of the problem of finding the nearest polynomial depends on the norm pair. Finally, we describe an algorithm for the problem when both of the norms are the ∞-norm.