Roots of polynomials

Polynomials are, on the one hand, a privileged tool in algebra and, on the other, a convenient and powerful means of investigation in analysis. In both cases, the roots of one-indeterminate polynomials play a fundamental role, either in the arithmetic-algebraic framework of field extensions, or in the many numerical problems linked to approximation by polynomials: interpolation, solving numerical equations, for example. Of course, many other fields are also concerned: finding the eigenvalues of a matrix and, consequently, studying discrete or continuous dynamic systems, linear or non-linear; traditional arithmetic, complex geometry, real algebraic geometry are all examples.

The aim of this article is to give some fairly general tools for locating, separating or estimating the roots of polynomials, mainly with real or complex coefficients. Only methods specific to polynomials will be studied, those applicable in more general situations being the subject of another article.