Real Closed Research Articles

Radicals of binomial ideals

In this paper we investigate radical operations on binomial ideals, i.e. ideals generated by sums of at most two terms, especially the L-radical, α-radical and τ-radical for an arbitrary extension field L of the base field K resp. an arbitrary ordering α resp. preordering τ on K. This is the vanishing ideal of the set of L-rational points of the ideal resp. the R-radical for an arbitrary real closure R of α resp. the intersection of the α-radicals for all orders α on K containing τ. We derive necessary and sufficient conditions on L resp. τ for these radicals of arbitrary binomial ideals to be again binomial and find several cases (incl. L = K and L a real or separable closure of K) where this is true. There are counterexamples for the ordinary radical. Further we describe algorithms for radical computations and root counting which are designed for the special structure of binomial ideals, and we give Bezout-type bounds for the number of L-rational points in the case that their number is finite.

More on imaginaries in p-adic fields

According to [4, p. 1154], a complete L-theory T eliminates imaginaries just in case for every L-formula φ(x1,… , xm, y1, …, yn), every model M of T, and every ā Є Mn, there is a subset A of M's domain with the following property: if N ≽ M and f is an automorphism of N, thenif and only ifAmong the several equivalent conditions discussed in [4, p. 1155], one may single out the following: if T is a complete theory in which two distinct objects are definable, T eliminates imaginaries just in case every T-definable n-ary equivalence relation may be defined by a formulawhere g is a T-definable n-ary function taking k-tuples as values (for some natural number k).Say that an L-structure M eliminates imaginaries just in case Th(M) does. If L is the language of rings with unit, [4, p. 1158] shows that any algebraically closed field eliminates imaginaries, and [2, p. 629] points out that any real-closed field eliminates imaginaries.

Classification of hermitian forms and semisimple groups over fields of virtual cohomological dimension one

Letk be a perfect field with cdk(i)≤1. It has recently been proved by the author that homogeneous spaces under connected linear groups overk satisfy a Hasse principle with respect to the real closures ofk. Using this result we classify the semisimple algebraic groups overk and, in particular, characterize the anisotropic ones. Similarly we classify the various types of hermitian forms over skew fields overk and exhibit to what extent weak or strong local-global principles hold. In the case wherek is the function field of a smooth projective curveX over ℝ, we also cover the local-global questions vis-a-vis the completions ofk at the points ofX.

Principe de Hasse pour les espaces principaux homogènes sous les groupes classiques sur un corps de dimension cohomologique virtuelle au plus 1

Letk be a perfect field of virtual cohomological dimension at most 1. LetG be a semi-simple, simply connected classical group. In this paper it is proved that a principal homogeneous space underG which is trivial over every real closure ofk is trivial.

Dynamic evaluation and real closure

Abstract The aim of this paper is to present how the dynamic evaluation method can be used to deal with the real closure of an ordered field. Two kinds of questions, or tests, may be asked in an ordered field: equality tests (a = b?) and sign tests (a > b?). Equality tests are handled through splittings, exactly as in the algebraic closure of a field. Sign tests are handled through a structure called “Tarski data type”.

Generic computation of the real closure of an ordered field

Abstract This paper describes a generalization of the real closure computation of an ordered field (Rioboo, 1991) enabling to use different techniques to code a single real algebraic number.

A constructive predictor-corrector algorithm for the direct position kinematics problem for a general 6-6 Stewart platform

This paper deals with the direct position kinematics of a general 6-6 Stewart platform and presents a new and efficient algorithm to solve the problem. Unlike the convention of developing a set of kinematic equations and then solving them, an alternative approach is proposed which involves just a search and verification purely from geometric considerations. An attempt at conceptual construction of the mechanism under given constraints results in the prediction of tentative solutions, which are subsequently corrected. The algorithm is implemented and two case studies are reported. The proposed algorithm is simple to implement, finds all the real closures and is applicable to the most general case of the Stewart platform.

Are fifth-degree equations over GF(5 m ) solvable by radicals?

Irreducible quintics over finite fields are solvable in closed form with the possible exception of characteristic 5 fields. It is shown that this is equally true for fifth-degree equations overGF(5m). The result follows from an Artin-Schreier theorem that yields explicit expressions for the roots ofx5−x−a. In addition to what at present is known for all other finite fields, any quintic can be solved in closed form.

Formally real rings and their real closure

Formally real rings and their real closure

Orders and Order Closures for Not Necessarily Formally Real Fields

"Closures" and "orders" of fields which are not necessarily formally real are introduced here. The closed fields include real closed fields, algebraically closed fields, and the p-adically closed fields of arbitrary p-rank. The theorem of Artin and Schreier on the bijective correspondence between orderings and isomorphism classes of real closures is generalized. An isomorphism theorem for Henselian extensions is proved which generalizes theorems of Becker and of Prestel and Roquette for generalized real closed fields and Henselian p-adic fields, respectively. A basic tool is the theory of Henselizations of fields with respect to "extended" absolute values, i.e., ones which can take the value ∞.

A class of connected theories of order

In 1962 Jan Mycielski proposed a very general notion of interpretability [M1]. This led to the question whether a given theory could be interpreted in the disjoint union of two theories, without being interpretable in any of them. He argued that in such a case it would be presumably simpler to study each of these theories separately, and hence conjectured that this situation can never occur for any of the well-known theories of mathematics. This conjecture has now been verified for the following theories (see [MPS], [P], [S1, 2]): ELO (endless, i.e., without maximal element, linear order), Th(〈ℚ, ≤〉), Th(〈ω, ≤〉) and all sequential theories (those which can code finite sequences of elements of their models). The latter include PA, ZF, GB and Th(〈ω,+,·〉). In view of these confirmations it became ever more plausible that the conjecture is valid also for RCF (real closed fields), i.e., for Th(〈ℝ,≤,+,·,0,1〉). In the present paper we show that Mycielski's conjecture is valid for a class of theories which includes RCF and OF (ordered fields).We consider only theories with equality and without function symbols. Interpretations will be meant local, multidimensional, and with parameters, as defined in [M1], [M2] and surveyed in [MPS] (for a recent definition see also [S2]). We shall write T0 ≪ T1 to say that T0 is interpretable in T1 (or that T1 interprets T0), and this will mean that for every theorem α of T0 there is an interpretation of α in T1.

A real nullstellensatz and positivstellensatz for the semi- polynomials over an ordered field

Let K be an ordered field and R its real closure. A semipolynomial will be defined as a function from R n 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”, “max” and “min” and such a program will be called a semipolynomial expression. It will be proved, using the ordinary Real Positivstellensatz, a general Real Positivstellensatz concerning the semipolynomial expressions. Using this semipolynomial version for the Real Positivstellensatz we shall get as consequences a continuous and rational solution for the 17th Hilbert problem, rational and continuous versions for several cases in the Real Positivstellensatz and constructive proofs for several theorems concerning the algebra over the real numbers.

Some examples of lattice-orders in real closed fields

Some examples of lattice-orders in real closed fields

Quantifier Elimination in p-adic Fields

We present a tutorial survey of quantifier elimination and decision procedures in p-adic fields. The p-adic fields are studied in the (so-called) Pn-formalism of Angus Macintyre, for which motivation is provided through a rich body of analogies with real-closed fields. Quantifier elimination and decision procedures are described proceeding via a Cylindrical Algebraic Decomposition of affine p-adic space. Effective complexity analyses are also provided.

Topologische Divisionsalgebren ohne zugehörige topologische affine Ebene

We prove that many (non-associative) topological division algebrasD of dimensionn ∈ N over the centreK do not yield topological affine or projective planes (of Lenz-Barlotti type V) in contrast to the results of SKORNJAKOV [20], SALZMANN [18] and [19], GRUNDHOFER [7], HARTMANN [11] and RINK [17] concerning projective planes coordinatized by compact or special topological ternary fields. In particular, this holds for every non-trivial and non-archimedian valuation topology ofK distinct from the order topology ifK is a real-closed field, and if the division algebraD =Kn carries the product topology.

�ber die Fortsetzungen der Anordnung eines K�rpers auf eine einfach transzendente Erweiterung

Let an ordered field (K, <) be given. It is well-known that the orderings 〈 on the rational function field K(t) extending the ordering < on K correspond in a unique way to the cuts in the real closure R of K. Here we show that for two of these orderings, (K(t),〈1) is K-isomorphic to (K(t), 〈2) as ordered fields iff there exists a fractional linear function on K transforming the corresponding cuts in R into each other. As a corollary we can determine the group of all K-automorphisms on (K(t), 〈).

Topologische Affine Ebenen Mit Nichtstetigem Parallelismus

In this paper, we present a general method of constructing topological affine planes having non-continuous parallelism. We prove that a topological affine plane E with point set Lk×Lk, and with a special ‘K-algebraic slope’ has a topological affine subplane with non-continuous parallelism (Satz 4.6). Here, K is a real-closed subfield of a real-closed field L. The crucial tools needed to make our method work are the notion of a slope and the notion of K-algebraicity, a concept which is introduced and intensively studied here. As an application of our general method, we obtain in Section 5 affine Salzmann planes with lines being bent countably infinitely often admitting a subplane with non-continuous parallelism. This provides a negative answer to a question posed by H. Salzmann [13, p. 52].

Inverse real closed spaces

Inverse real closed spaces

Plongement dense d'un corps ordonné dans sa clôture réelle

AbstractWe study the structures (K ⊂ Kr), where K is an ordered field and Kr its real closure, in the language of ordered fields with an additional unary predicate for the subfield K. Two such structures (K ⊂ Kr) and (L ⊂ Lr) are not necessarily elementary equivalent when K and L are. But with some saturation assumption on K and L, then the two structures become equivalent, and we give a description of the complete theory.

Existence and uniqueness of the real closure of an ordered field without Zorn's Lemma

In this paper the existence of a real closure of an ordered field is given without the use of the axiom of choice. The possibility of such a proof and related results demonstrate fundamental differences between the concepts of real and algebraic closures of fields.