Majoration de la norme des facteurs d'un polynôme : cas où toutes les racines du polynôme sont réelles
Mémoire sur la résolution des équations algébriques les unes par les autres.
Méthode de l'élimination des intervalles pour servir à la résolution des équations algébriques et transcendantes
Minimal degree solutions for the Bezout equation
Multivariate Sturm-Habicht sequences: real root counting on n-rectangles and triangles.
The main purpose of this note is to show how Sturm-Habicht Sequence can be generalized to the multivariate case and used to compute the number of real solutions of a polynomial system of equations with a finite number of complex solutions. Using the same techniques, some formulae counting the number of real solutions of such polynomial systems of equations inside n-dimensional rectangles or triangles in the plane are presented.
New inequalities on polynomial divisors.
Niven’s Theorem
This article formalizes the proof of Niven’s theorem [12] which states that if x/π and sin(x) are both rational, then the sine takes values 0, ±1/2, and ±1. The main part of the formalization follows the informal proof presented at Pr∞fWiki (https://proofwiki.org/wiki/Niven’s_Theorem#Source_of_Name). For this proof, we have also formalized the rational and integral root theorems setting constraints on solutions of polynomial equations with integer coefficients [8, 9].
Norms of certain rational functions of a matrix and Schur's criterion for polynomials
Norms of matrix powers
Note sur la continuité des racines des équations algébriques
Note sur la règle des signes de Descartes
O rovnicích kvadratických
On a Stratification Defined by Real Roots of Polynomials
2000 Mathematics Subject Classification: 12D10We prove smoothness of the strata and a transversality property of their tangent spaces.
On Arrangements of Real Roots of a Real Polynomial and Its Derivatives
2000 Mathematics Subject Classification: 12D10.We prove that all arrangements (consistent with the Rolle theorem and some other natural restrictions) of the real roots of a real polynomial and of its s-th derivative are realized by real polynomials.
On cubic factors of certain trinominals.
On equations and properties concerning some classes of chordal polygons.
On generalized “ham sandwich” theorems
In this short note we utilize the Borsuk-Ulam Anitpodal Theorem to present a simple proof of the following generalization of the “Ham Sandwich Theorem”: Let be subsets with finite Lebesgue measure. Then, for any sequence of -linearly independent polynomials in the polynomial ring there are real numbers , not all zero, such that the real affine variety simultaneously bisects each of subsets , . Then some its applications are studied.
On intervals containing full sets of conjugates of algebraic integers
On multiple zeros of Bernoulli polynomials
On multivariate Descartes' rule – a counterexample.