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### Approximate roots of a valuation and the Pierce-Birkhoff conjecture

Annales de la faculté des sciences de Toulouse Mathématiques

In this paper, we construct an object, called a system of approximate roots of a valuation, centered in a regular local ring, which describes the fine structure of the valuation (namely, its valuation ideals and the graded algebra). We apply this construction to valuations associated to a point of the real spectrum of a regular local ring $A$. We give two versions of the construction: the first, much simpler, in a special case (roughly speaking, that of rank 1 valuations), the second – in the case...

### Associated primes, integral closures and ideal topologies

Colloquium Mathematicae

Let ⊆ be ideals of a Noetherian ring R, and let N be a non-zero finitely generated R-module. The set Q̅*(,N) of quintasymptotic primes of with respect to N was originally introduced by McAdam. Also, it has been shown by Naghipour and Schenzel that the set $A{*}_{a}\left(,N\right):={\bigcup }_{n\ge 1}As{s}_{R}R/{\left(ⁿ\right)}_{a}^{\left(N\right)}$ of associated primes is finite. The purpose of this paper is to show that the topology on N defined by ${{\left(ⁿ\right)}_{a}^{\left(N\right)}{:}_{R}⟨⟩}_{n\ge 1}$ is finer than the topology defined by ${{\left(ⁿ\right)}_{a}^{\left(N\right)}}_{n\ge 1}$ if and only if $A{*}_{a}\left(,N\right)$ is disjoint from the quintasymptotic primes of with respect to N. Moreover, we show...

### Convexity, valuations and Prüfer extensions in real algebra.

Documenta Mathematica

### Extension of the Two-Variable Pierce-Birkhoff conjecture to generalized polynomials

Annales de la faculté des sciences de Toulouse Mathématiques

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 ${f}_{ij}\in ℝ\left[{x}_{1},...,{x}_{n}\right]$. (A simple example is $h\left({x}_{1}\right)=|{x}_{1}|=sup\left\{{x}_{1},-{x}_{1}\right\}$.) 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;...

### Forms Derived from the Arithmetic-Geometric Inequality.

Mathematische Annalen

### Positive polynomials and hyperdeterminants

Collectanea Mathematica

Let F be a homogeneous real polynomial of even degree in any number of variables. We consider the problem of giving explicit conditions on the coefficients so that F is positive definite or positive semi-definite. In this note we produce a necessary condition for positivity, and a sufficient condition for non-negativity, in terms of positivity or semi-positivity of a one-variable characteristic polynomial of F. Also, we revisit the known sufficient condition in terms of Hankel matrices.

### Positive polynomials and polynomial inequalities.

Boletín de la Asociación Matemática Venezolana

### Real holomorphy rings and the complete real spectrum

Annales de la faculté des sciences de Toulouse Mathématiques

The complete real spectrum of a commutative ring $A$ with $1$ is introduced. Points of the complete real spectrum ${Sper}^{c}A$ are triples $\alpha =\left(𝔭,v,P\right)$, where $𝔭$ is a real prime of $A$, $v$ is a real valuation of the field $k\left(𝔭\right):=qf\left(A/𝔭\right)$ and $P$ is an ordering of the residue field of $v$. ${Sper}^{c}A$ is shown to have the structure of a spectral space in the sense of Hochster . The specialization relation on ${Sper}^{c}A$ is considered. Special attention is paid to the case where the ring $A$ in question is a real holomorphy ring.

### Representations of non-negative polynomials having finitely many zeros

Annales de la faculté des sciences de Toulouse Mathématiques

Consider a compact subset $K$ of real $n$-space defined by polynomial inequalities ${g}_{1}\ge 0,\cdots ,{g}_{s}\ge 0$. For a polynomial $f$ non-negative on $K$, natural sufficient conditions are given (in terms of first and second derivatives at the zeros of $f$ in $K$) for $f$ to have a presentation of the form $f={t}_{0}+{t}_{1}{g}_{1}+\cdots +{t}_{s}{g}_{s}$, ${t}_{i}$ a sum of squares of polynomials. The conditions are much less restrictive than the conditions given by Scheiderer in [11, Cor. 2.6]. The proof uses Scheiderer’s main theorem in  as well as arguments from quadratic form theory...

### Separating ideals in dimension 2.

Experience shows that in geometric situations the separating ideal associated with two orderings of a ring measures the degree of tangency of the corresponding ultrafilters of semialgebraic sets. A related notion of separating ideals is introduced for pairs of valuations of a ring. The comparison of both types of separating ideals helps to understand how a point on a surface is approached by different half-branches of curves.

### Spectral Real Semigroups

Annales de la faculté des sciences de Toulouse Mathématiques

The notion of a real semigroup was introduced in  to provide a framework for the investigation of the theory of (diagonal) quadratic forms over commutative, unitary, semi-real rings. In this paper we introduce and study an outstanding class of such structures, that we call spectral real semigroups (SRS). Our main results are: (i) The existence of a natural functorial duality between the category of SRSs and that of hereditarily normal spectral spaces; (ii) Characterization of the SRSs as the...

### The Boolean space of higher level orderings

Fundamenta Mathematicae

Let K be an ordered field. The set X(K) of its orderings can be topologized to make it a Boolean space. Moreover, it has been shown by Craven that for any Boolean space Y there exists a field K such that X(K) is homeomorphic to Y. Becker's higher level ordering is a generalization of the usual concept of ordering. In a similar way to the case of ordinary orderings one can define a topology on the space of orderings of fixed exact level. We show that it need not be Boolean. However, our main theorem...