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Section: New Results

Reasoning on polynomial expressions

Participants : José Grimm, Julianna Zsido, Yves Bertot.

Continuing previous work by Bertot, we showed that if $p$ is a polynomial on any ordered ring, that has $n$ positive roots, the list of its coefficients has at least $n$ sign changes. If there is exactly one sign change, and the ring is an Archimedian field, there is a number $a$ such that the polynomial is negative on $[0,a]$ and strictly increasing after $a$; thus it has at most one positive root, and there is a Cauchy sequence $x_i$ such that $p\left(x_i\right)<0$ but $p(x_i+c/2^n)>0$.

The publication by Bertot, Mahboubi, and Guilhot in 2011 on Bernstein polynomials describes a procedure that works only for polynomials with simple roots. We added the proofs that describe how to obtain such polynomials, starting from arbitrary ones. In other words, we proved the following statement: for every polynomial $p$, $p$ divided by the greatest common divisor of $p$ and its derivative has the same roots as $p$ and all the roots are simple.

We started working on a proof that the dichotomy process based on Bernstein polynomials is bound to terminate, concentrating on a theorem known as the theorem of three circles.