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Approximate roots of a valuation and the Pierce-Birkhoff conjecture

F. Lucas, J. Madden, D. Schaub, M. Spivakovsky (2012)

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

Reza Naghipour (2006)

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 ( , N ) : = n 1 A s s R R / ( ) a ( N ) of associated primes is finite. The purpose of this paper is to show that the topology on N defined by ( ) a ( N ) : R n 1 is finer than the topology defined by ( ) a ( N ) n 1 if and only if A * a ( , N ) is disjoint from the quintasymptotic primes of with respect to N. Moreover, we show...

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

Charles N. Delzell (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

Let h : n 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 i j , for some finite collection of polynomials f i j [ x 1 , ... , x n ] . (A simple example is h ( x 1 ) = | x 1 | = sup { x 1 , - x 1 } .) In 1984, L. Mahé and, independently, G. Efroymson, proved this for n 2 ; it remains open for n 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;...

Positive polynomials and hyperdeterminants

Fernando Cukierman (2007)

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.

Real holomorphy rings and the complete real spectrum

D. Gondard, M. Marshall (2010)

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 α = ( 𝔭 , v , P ) , where 𝔭 is a real prime of A , v is a real valuation of the field k ( 𝔭 ) : = qf ( A / 𝔭 ) 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 [5]. 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

Murray Marshall (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

Consider a compact subset K of real n -space defined by polynomial inequalities g 1 0 , , g s 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 + + 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 [11] as well as arguments from quadratic form theory...

Separating ideals in dimension 2.

James J. Madden, Niels Schwartz (1997)

Revista Matemática de la Universidad Complutense de Madrid

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

M. Dickmann, A. Petrovich (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

The notion of a real semigroup was introduced in [8] 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

Katarzyna Osiak (2007)

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...

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