We say that a field K has the Extension Property if every automorphism of K(X) extends to an automorphism of K. J.M. Gamboa and T. Recio [2] have introduced this concept, naive in appearance, because of its crucial role in the study of homogeneity conditions in spaces of orderings of functions fields. Gamboa [1] has studied several classes of fields with this property: Algebraic extensions of the field Q of rational numbers; euclidean, algebraically closed and pythagorean fields; fields with an...

For a large class of Hardy fields their extensions containing non-${}^{\infty }$-germs are constructed. Hardy fields composed of only non-${}^{\infty }$-germs, apart from constants, are also considered.

Let $h:{ℝ}^n\to ℝ$ be a continuous, piecewise-polynomial function. The Pierce-Birkhoff conjecture (1956) is that any such $h$ is representable in the form ${sup}_i{inf}_j{f}_{ij}$, for some finite collection of polynomials $f_{ij}\in ℝ[x_1,...,x_n]$. (A simple example is $h\left(x_1\right)=\left|x_1\right|=sup\{x_1,-x_1\}$.) In 1984, L. Mahé and, independently, G. Efroymson, proved this for $n\le 2$; it remains open for $n\ge 3$. 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;...

Soient $(K,\nu )$ un corps valué et $L$ est une extension monogène finie de $K$ définie par $L=K\left[x\right]/\left(P\right)$, alors toute valuation de $L$ qui prolonge $\nu$ définit une pseudo-valuation $\zeta$ de $K\left[x\right]$ de noyau l’idéal $\left(P\right)$. Nous savons associer à $\zeta$ une famille de valuations de $K\left[x\right]$, appelée famille admissible, construite de façon explicite à partir de valuations augmentées et de valuations augmentées limites.Nous donnons une condition nécessaire et suffisante pour qu’une valuation $\mu$ de $K\left[x\right]$ appartienne à la famille admissible associée à une pseudo-valuation...

We generalize a question of Büchi: Let R be an integral domain, C a subring and k ≥ 2 an integer. Is there an algorithm to decide the solvability in R of any given system of polynomial equations, each of which is linear in the kth powers of the unknowns, with coefficients in C? We state a number-theoretical problem, depending on k, a positive answer to which would imply a negative answer to the question for R = C = ℤ. We reduce a negative answer for k = 2 and for...

We prove that there are absolute constants $c_1\>0$ and $c_2\>0$ such that for every$$\{a_0,a_1,...,a_n\}\subset [1,M],1\le M\le exp\left(c_1{n}^{1/4}\right),$$there are$$b_0,b_1,...,b_n\in \{-1,0,1\}$$such that$$P\left(z\right)=\sum _{j=0}^n{b}_j{a}_j{z}^j$$has at least $c_2{n}^{1/4}$ distinct sign changes in $(0,1)$. This improves and extends earlier results of Bloch and Pólya.

Dans [Swe], Sweedler a caractérisé les extensions purement inséparables $K/k$ d’exposant fini qui sont produit tensoriel d’extensions simples. En vue d’étendre ce résultat aux extensions d’exposants non bornés, L. Kime dans [Kim] propose les extensions $k\left(x^{p^{-\infty }}\right)=k(x^{p^{-1}},x^{p^{-2}},\cdots )$ comme généralisation d’extensions simples. Dans ce travail, on propose d’autres généralisations naturelles. Ceci nous a permis de décrire explicitement toutes les extensions purement inséparables $K/k$ lorsque le degré d’imperfection de $k$ est $\le 2$. Dans [Dev2]...