Section: New Results

Isolation of polynomial roots

Participants : Yves Bertot [correspondant] , Julianna Zsidó.

Together with techniques to produce square-free polynomials (polynomials whose roots are all simple), Bernstein polynomials provide a way to decide whether a polynomial has roots in a given interval. Together with a dichotomy procedure, this makes it possible to isolate all the roots of a polynomial, or to show that no root of a given polynomial occur in a given interval. At the end of 2012, Julianna Zsidó started to study this procedure: she showed the properties of the procedure to obtain square-free polynomials and she then formalized a proof for a theorem known as the theorem of three circles which plays a rôle in proving that dichotomy will terminate. This work has been published as an article in the Journal of Automated Reasoning.

We expect to wrap up all this work by producing easy-to-use tactics to prove properties of polynomial formulas and generalizing it to polynomials in several variables.

During a summer internship, Konstantinos Lentzos worked on the representation of algebraic numbers (which can always be represented as roots of polynomials in a given interval) and the question of finding polynomials for algebraic numbers obtained through simple operations (like addition, multiplication, opposite, and inversion). However, this work was made extremely difficult by the problem of finding morphisms between various fields definable on top of a polynomial ring.