Related papers: Eine Bemerkung zu einem Satz von E. Becker und D. …

A famous theorem of Hilbert from 1888 states that a positive semidefinite (psd) real form is a sum of squares (sos) of real forms if and only if $n=2$ or $d=1$ or $(n,2d)=(3,4)$, where $n$ is the number of variables and $2d$ the degree of…

We prove that the number of connected components of a smooth hypersurface in the positive orthant of $\mathbb{R}^n$ defined by a real polynomial with $d + k + 1$ monomials, where $d$ is the dimension of the affine span of the exponent…

For a fixed rational number g and integers a and d the sets N_g(a,d), respectively R_g(a,d), of primes p for which the order, respectively the index of g(mod p) is congruent to a(mod d), are considered. Under the Generalized Riemann…

Exhibiting a deep connection between purely geometric problems and real algebra, the complexity class $\exists \mathbb{R}$ plays a crucial role in the study of geometric problems. Sometimes $\exists \mathbb{R}$ is referred to as the 'real…

Experimental evidence, the heuristics of indistinguishability, and its logical inconsistency with quantum formalism all argue against the existence of a quantum mixture uncorrelated with the exterior, that is, argue for the postulate "The…

In [8], P. Lecomte conjectured the existence of a natural and projectively equivariant quantization. In [1], M. Bordemann proved this existence using the framework of Thomas-Whitehead connections. In [9], we gave a new proof of the same…

Let f1, ..., fs be a polynomial family in Q[X1,..., Xn] (with s less than n) of degree bounded by D. Suppose that f1, ..., fs generates a radical ideal, and defines a smooth algebraic variety V. Consider a projection P. We prove that the…

It is proven that for any topological or analytical types of isolated singular points of plane curves, there exists a non-real irreducible plane algebraic curve of degree $d$ which goes through $d^2$ real distinct points and has imaginary…

Section 10.4 of the 1998 Springer-Verlag book {\em Complexity and Real Computation}, by Blum, Cucker, Shub, and Smale, contains a particularly elegant proof of the Fundamental Theorem of Algebra: The central idea of the proof naturally…

We study the Euler characteristic of the real Milnor fibres of a real analytic map, using a relation between complex monodromy and complex conjugation. We deduce the result of Coste and Kurdyka that the Euler characteristic of the link of…

The results presented in this paper are refinements of some results presented in a previous paper. Three such refined results are presented. The first one relaxes one of the basic hypotheses assumed in the previous paper, and thus extends…

Consider an $F$-crystal over a noetherian scheme $S$. De Jong--Oort's purity theorem states that the associated Newton polygons over all points of $S$ are constant if this is true outside a subset of codimension bigger than 1. In this paper…

Let $S_0$ be a smooth and compact real variety given by a reduced regular sequence of polynomials $f_1, ..., f_p$. This paper is devoted to the algorithmic problem of finding {\em efficiently} a representative point for each connected…

A very small amount of K\"ahler algebra (i.e. Clifford algebra of differential forms) in the real plane makes x + ydxdy emerge as a factor between the differentials of the Cartesian and polar coordinates, largely replacing the concept of…

In the present article, real number representations, that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are…

Given any polynomial with real coefficients, the existence of a real quadratic polynomial factor is proven using only basic real analysis. The aim is to provide an approachable proof to anybody who is familiar with the least upper bound…

We consider the problem of determining the number of distinct distances between two point sets in $\mathbb{R}^2$ where one point set $\mathcal{P}_1$ of size $m$ lies on a real algebraic curve of fixed degree $r$, and the other point set…

Zaremba's conjecture (1971) states that every positive integer number $d$ can be represented as a denominator (continuant) of a finite continued fraction $\frac{b}{d}=[d_1,d_2,...,d_{k}],$ with all partial quotients $d_1,d_2,...,d_{k}$…

In this paper we give a formula for the classes (in the Grothendieck ring of complex quasi-projective varieties) of irreducible components of $(1,k)$-quasi-homogeneous Hilbert schemes of points on the plane. We find a new simple geometric…

We investigate the relations between the rings ${\bf E}$, ${\bf G}$ and ${\bf D}$ of values taken at algebraic points by arithmetic Gevrey series of order either $-1$ ($E$-functions), $0$ (analytic continuations of $G$-functions) or $1$…