Eine Bemerkung über die Wurzelpolynome Galoisscher Gleichungen.
Le but de cet article, à travers l’étude des travaux en analyse algébrique finie d’Étienne Bézout (1730-1783), est de mieux faire connaître ses résultats, tels qu’il les a effectivement trouvés, et de mettre en valeur aussi bien les points de vue novateurs que les méthodes originales, mis en œuvre à cet effet. L’idée de ramener le problème de l’élimination d’une ou plusieurs inconnues à l’étude d’un système d’équations du premier degré, son utilisation inhabituelle des coefficients indéterminés...
2000 Mathematics Subject Classification: 12D10.In the present paper we consider degree 6 hyperbolic polynomials (HPs) in one variable (i.e. real and with all roots real). We are interested in such HPs whose number of equalities between roots of the polynomial and/or its derivatives is higher than expected. We give the complete study of the four families of such degree 6 even HPs and also of HPs which are primitives of degree 5 HPs.Research partially supported by research project 20682 for cooperation...
Let be a continuous, piecewise-polynomial function. The Pierce-Birkhoff conjecture (1956) is that any such is representable in the form , for some finite collection of polynomials . (A simple example is .) In 1984, L. Mahé and, independently, G. Efroymson, proved this for ; it remains open for . In this paper we prove an analogous result for “generalized polynomials” (also known as signomials), i.e., where the exponents are allowed to be arbitrary real numbers, and not just natural numbers;...
We prove that there are absolute constants and such that for everythere aresuch thathas at least distinct sign changes in . This improves and extends earlier results of Bloch and Pólya.