Descartes' rule of signs, Newton polygons, and polynomials over hyperfields

Descartes' Rule of Signs

In this work, we formally proved Descartes Rule of Signs, which relates the number of positive real roots of a polynomial with the number of sign changes in its coefficient list. Our proof follows the simple inductive proof given by Arthan [1], which was also used by John Harrison in his HOL Light formalisation. We proved most of the lemmas for arbitrary linearly-ordered integrity domains (e.g....

Newton Polygons and Families of Polynomials

We consider a continuous family (fs), s ∈ [0, 1] of complex polynomials in two variables with isolated singularities, that are Newton non-degenerate. We suppose that the Euler characteristic of a generic fiber is constant. We firstly prove that the set of critical values at infinity depends continuously on s, and secondly that the degree of the fs is constant (up to an algebraic automorphism of...

Counting zeros of generalized polynomials: Descartes’ rule of signs and Laguerre’s extensions

A slightly different question is how many positive zeros a polynomial has. Here the basic result is known as “Descartes’ rule of signs”. It says that the number of positive zeros is no more than the number of sign changes in the sequence of coefficients. Descartes included it in his treatise La Géométrie, which appeared in 1637. It can be proved by a method based on factorization, but, again, j...

Real root isolation for exact and approximate polynomials using Descartes' rule of signs

Collins und Akritas (1976) have described the Descartes method for isolating the real roots of an integer polynomial in one variable. This method recursively subdivides an initial interval until Descartes’ Rule of Signs indicates that all roots have been isolated. The partial converse of Descartes’ Rule by Obreshkoff (1952) in conjunction with the bound of Mahler (1964) and Davenport (1985) lea...

Higher Order Bernoulli Polynomials and Newton Polygons