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A conic and an $M$-quintic with a point at infinity.

Journal of Mathematical Sciences (New York)

A continuous, constructive solution to Hilbert's 17th problem.

Inventiones mathematicae

A first-order version of Pfaffian closure

Fundamenta Mathematicae

The purpose of this paper is to extend a theorem of Speissegger [J. Reine Angew. Math. 508 (1999)], which states that the Pfaffian closure of an o-minimal expansion of the real field is o-minimal. Specifically, we display a collection of properties possessed by the real numbers that suffices for a version of the proof of this theorem to go through. The degree of flexibility revealed in this study permits the use of certain model-theoretic arguments for the first time, e.g. the compactness theorem....

A Formula for the Euler characteristic of a real algebraic manifold.

Manuscripta mathematica

A geometric point of view on mean-variance models

Applicationes Mathematicae

This paper deals with the mathematics of the Markowitz theory of portfolio management. Let E and V be two homogeneous functions defined on ℝⁿ, the first linear, the other positive definite quadratic. Furthermore let Δ be a simplex contained in ℝⁿ (the set of admissible portfolios), for example Δ : x₁+ ... + xₙ = 1, ${x}_{i}\ge 0$. Our goal is to investigate the properties of the restricted mappings (V,E):Δ → ℝ² (the so called Markowitz mappings) and to classify them. We introduce the notion of a generic model...

A group law on smooth real quartics having at least $3$ real branches

Journal de théorie des nombres de Bordeaux

Let $C$ be a smooth real quartic curve in ${ℙ}^{2}$. Suppose that $C$ has at least $3$ real branches ${B}_{1},{B}_{2},{B}_{3}$. Let $B={B}_{1}×{B}_{2}×{B}_{3}$ and let $O\in B$. Let ${\tau }_{O}$ be the map from $B$ into the neutral component Jac$\left(C\right){\left(ℝ\right)}^{0}$ of the set of real points of the jacobian of $C$, defined by letting ${\tau }_{O}\left(P\right)$ be the divisor class of the divisor $\sum {P}_{i}-{O}_{i}$. Then, ${\tau }_{O}$ is a bijection. We show that this allows an explicit geometric description of the group law on Jac$\left(C\right){\left(ℝ\right)}^{0}$. It generalizes the classical geometric description of the group law on the neutral component of the set of real points of...

A linear extension operator for Whitney fields on closed o-minimal sets

Annales de l’institut Fourier

A continuous linear extension operator, different from Whitney’s, for ${𝒞}^{p}$-Whitney fields (p finite) on a closed o-minimal subset of ${ℝ}^{n}$ is constructed. The construction is based on special geometrical properties of o-minimal sets earlier studied by K. Kurdyka with the author.

A local Bernstein inequality on real algebraic varieties.

Mathematische Zeitschrift

A Measure for Semialgebraic Sets Related to Boolean Complexity.

Mathematische Annalen

A Note on a Theorem of Lion

Bulletin of the Polish Academy of Sciences. Mathematics

In this note we bind together Wilkie's complement theorem with Lion's theorem on geometric, regular and 0-regular families of functions.

A note on Bierstone-Milman-Pawłucki's paper "Composite differentiable functions"

Annales Polonici Mathematici

We demonstrate that the composite function theorems of Bierstone-Milman-Pawłucki and of Glaeser carry over to any polynomially bounded, o-minimal structure which admits smooth cell decomposition. Moreover, the assumptions of the o-minimal versions can be considerably relaxed compared with the classical analytic ones.

A note on boundaries of open polynomial images of $mathbb{R}^{2}$.

Revista Matemática Iberoamericana

A note on global Nash subvarieties and Artin-Mazur theorem

Bollettino dell'Unione Matematica Italiana

It is shown that every connected global Nash subvariety of ${\mathbb{R}}^{n}$ is Nash isomorphic to a connected component of an algebraic variety that, in the compact case, can be chosen with only two connected components arbitrarily near each other. Some examples which state the limits of the given results and of the used tools are provided.

A note on Robinson's non-negativity criterion

Fundamenta Mathematicae

A proof of the valuation property and preparation theorem

Annales Polonici Mathematici

The purpose of this article is to present a short model-theoretic proof of the valuation property for a polynomially bounded o-minimal theory T. The valuation property was conjectured by van den Dries, and proved for the polynomially bounded case by van den Dries-Speissegger and for the power bounded case by Tyne. Our proof uses the transfer principle for the theory ${T}_{conv}$ (i.e. T with an extra unary symbol denoting a proper convex subring), which-together with quantifier elimination-is due to van den...

A real nullstellensatz and positivstellensatz for the semipolynomials over an ordered field.

Extracta Mathematicae

Let K be an ordered field and R its real closure. A semipolynomial will be defined as a function from Rn to R obtained by composition of polynomial functions and the absolute value. Every semipolynomial can be defined as a straight-line program containing only instructions with the following type: polynomial, absolute value, sup and inf and such a program will be called a semipolynomial expression. It will be proved, using the ordinary real positivstellensatz, a general real positivstellensatz concerning...

A really elementary proof of real Lüroth's theorem.

Classical Lüroth theorem states that every subfield K of K(t), where t is a transcendental element over K, such that K strictly contains K, must be K = K(h(t)), for some non constant element h(t) in K(t). Therefore, K is K-isomorphic to K(t). This result can be proved with elementary algebraic techniques, and therefore it is usually included in basic courses on field theory or algebraic curves. In this paper we study the validity of this result under weaker assumptions: namely, if K is a subfield...

A resolution theorem for homology cycles of real algebraic varieties.

Inventiones mathematicae

A set on which the Łojasiewicz exponent at infinity is attained

Annales Polonici Mathematici

We show that for a polynomial mapping $F=\left(f₁,...,fₘ\right):{ℂ}^{n}\to {ℂ}^{m}$ the Łojasiewicz exponent ${}_{\infty }\left(F\right)$ of F is attained on the set $z\in {ℂ}^{n}:f₁\left(z\right)·...·fₘ\left(z\right)=0$.

A theorem on generic intersections in an o-minimal structure

Fundamenta Mathematicae

Consider a transitive definable action of a Lie group G on a definable manifold M. Given two (locally) definable subsets A and B of M, we prove that the dimension of the intersection σ(A) ∩ B is not greater than the expected one for a generic σ ∈ G.

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