Solving systems of polynomial inequalities over a real closed field in subexponential time

Let _m= (₁,..., _m) denote an ordered field, where _i+1>0 is infinitesimal relative to the elements of _i, 0 < –i < m (by definition, ₀= ). Given a system of inequalities f₁ > 0, ..., f_s > 0, f_s+1 0, ..., f_k 0, where f_{j m} [X₁,..., X_n] are polynomials such that, and the absolute value of any integer occurring in the coefficients of the f_js is at most 2^M. An algorithm is constructed which tests the above system of inequalities for solvability over the real closure of _m in polynomial time with respect to M, ((d)ⁿd₀)^n+m. In the case _m=, the algorithm explicitly constructs a family of real solutions of the system (provided the latter is consistent). Previously known algorithms for this problem had complexity of the order ofM(d d ₀ ^{m 2U(n)} .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 174, pp. 3–36, 1988.     

Sur l'homotopie rationnelle des espaces fonctionnels

Let X be a nilpotent space such that it exists k1 with H^p (X,) = 0 p > k and H^k (X,) 0, let Y be a (m–1)-connected space with mk+2, then the rational homotopy Lie algebra of Y^X (resp. is isomorphic as Lie algebra, to H^* (X,) (_* (Y) ) (resp.⁺ (X,) (_* (Y) )). If X is formal and Y -formal, then the spaces Y^X and are -formal. Furthermore, if dim _* (Y) is infinite and dim H^* (Y,Q) is finite, then the sequence of Betti numbers of grows exponentially.     

Hecke algebras,type III factors and phase transitions with spontaneous symmetry breaking in number theory

In this paper, we construct a naturalC*-dynamical system whose partition function is the Riemann function. Our construction is general and associates to an inclusion of rings (under a suitable finiteness assumption) an inclusion of discrete groups (the associated ax+b groups) and the corresponding Hecke algebras of bi-invariant functions. The latter algebra is endowed with a canonical one parameter group of automorphisms measuring the lack of normality of the subgroup. The inclusion of rings provides the desiredC*-dynamical system, which admits the function as partition function and the Galois group Gal(^cycl/) of the cyclotomic extension ^cycl of as symmetry group. Moreover, it exhibits a phase transition with spontaneous symmetry breaking at inverse temperature =1 (cf. [Bos-C]). The original motivation for these results comes from the work of B. Julia [J] (cf. also [Spe]).     

Canonical Equivalence Relations on ℚ n

We show that for each positive integer n there is a finite list of equivalence relations on [] ⁿ with the property that for every other equivalence relation E on [] ⁿ there is X of order type equal to the order type of , such that E[X] ⁿ is equal to one of the equivalence relations from the list.     

Complexity of deciding the first-order theory of real closed fields

Let denote an ordered field, where _i+1 is infinitesimal relative to the elements of the field _i 0 i < m (by definition, ₀=). Given a formula of the first-order theory of the real closed field _m, of the following form:, where P is a quantifier-free formula containing k atomic subformulas of the form (f_j 0), where are polynomials such that, and the absolute value of any integer occurring in the coefficients of the polynomials f_j is at most 2_M. Let =S₁+...+S_a denote the number of variables anda n the number of quantifiers in the formula. An algorithm is constructed which decides the truth of formulas of the above form in polynomial time with respect to M,. Previously known algorithms for this problem had complexity of the order of.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 174, pp. 53–100, 1988.     

Sur la construction d'un corps de Hilbert

We are concerned with the following: If k is a quadratic field and N a cyclic unramified extension of degree qⁿ over k, q a prime number, determine N explicitely via a primitive element , i.e., N=k(), in the spirit of Helmut Hasse [3]. We propose a method which determines these extensions, once we are able to specify the arithmetic of a certain field . To explicit our method, we construct the Hilbert fields of (226) and (646).     

Über lineare Unabhängigkeit

It is the aim of the present work to prove, under appropriate conditions, lower estimates for the dimension of w ₁ + ... + w _m over , wherew ₁,...,w _m are given real numbers. In particular, if this dimension ism, i.e. ifw ₁,...,w _m are linearly independent over , we are also interested in a quantitative version of this fact. Our qualitative theorems generalize a result of Nesterenko. Its formulation is quite similar to the axiomatization of methods for algebraic independence, as it became usual during the last decade.

     

A topological characterization for closed sets under polar duality in ℚ n

A topological characterization is given for closed sets in ⁿ under the restriction of (cone) polar duality to ⁿ .     

Unique <Emphasis Type="Italic">a</Emphasis>-closure for some ℓ-groups of rational valued functions

Usually, an abelian -group, even an archimedean -group, has a relatively large infinity of distinct a-closures. Here, we find a reasonably large class with unique and perfectly describable a-closure, the class of archimedean -groups with weak unit which are -convex. ( is the group of rationals.) Any C(X, ) is -convex and its unique a-closure is the Alexandroff algebra of functions on X defined from the clopen sets; this is sometimes C(X).     

Compressing Coefficients While Preserving Ideals in K-Theory for C*-Algebras

An invariant based on orderedK-theory with coefficients in _n>1 /n and an infinite number of natural transformations has proved to be necessary and sufficient to classify a large class of nonsimple C* -algebras. In this paper, we expose and explain the relations between the order structure and the ideals of the C* -algebras in question.As an application, we give a new complete invariant for a large class of approximately subhomogeneous C*-algebras. The invariant is based on ordered K-theory with coefficients in /. This invariant is more compact (hence, easier to compute) than the invariant mentioned above, and its use requires computation of only four natural transformations.     

Die stabile Reduktion der Fermatkurve über einem Zahlkörper

Let p be an odd prime and F the Fermat curve of degree p, defined by x^p+y^p=1 over . Although the curve F has bad reduction at the prime (p), the stable reduction theorem assures that over some number field K/ we can get stable reduction of the curve F at the primes lying above p. We have determined it in this paper. See Abb.1.     

Impossibility of a Certain Cyclotomic Equation with Applicationsto Difference Sets

Under certain conditions, we show the nonexistence ofan element in the p-th cyclotomicfield over , that satisfies . As applications, we establish the nonexistence ofsome difference sets and affine difference sets.     

Categories equivalent to the category of rational H-spaces

The category of rationalH-spaces is shown to be equivalent to the category of commutative Hopf algebras over , the category of cocommutative Hopf algebras over , and the categoryL of graded Lie algebras over by the rational cohomology, homology, and homotopy functors, respectively. Several consequences of these equivalences are derived. It is also proved that the loop-space functor is an equivalence from the category of coformal rational spaces to . Dually, the category of rationalcoH-spaces is shown to be equivalent to the comonoid category ofL and to the category of cocommutative coalgebras over . The suspension functor is an equivalence from the category of formal, rational spaces to the category of 2-connected, rationalcoH-spaces.     

Exact bounds on the order of torsion points of curves of genus 1

Let k be an algebraic number field of degree n on ₂; and , respectively, the curves on k; let, and _m, '_m be the bases of groups of all points of order m on and g, respectively. A proof of the following theorem is sketched: let p>3 be prime; if, then (p^t)6n; if k, then (p^t)4n. The resulting bounds are unimprovable.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 151, pp. 57–65, 1986.     

A characterization of Gaussian measures on locally compact Abelian groups

Let and be independent random variables having equal variance. In order that + and – be independent, it is necessary and sufficient that and have normal distributions. This result of Bernshtein [1] is carried over in [7] to the case when and take values in a locally compact Abelian group. In the present note, a characterization of Gaussian measures on locally compact Abelian groups is given in which in place of + and –, functions of and are considered which satisfy the associativity equation.Translated from Matematicheskie Zametki, Vol. 22, No. 5, pp. 759–762, November, 1977.     

Über Hyperflächen aus Quadriken und quadratischen Kegeln im reellenm-dimensionalen projektiven Raum

Zusammenfassung Die vorgestellte Kennzeichnung für eine reguläreC ²(U)-Hyperflächef(U) mit zusammenhängender offener ParametermengeU ^m–1 aus einer Quadrik im reellenm-dimensionalen projektiven Raum ^m ,m 3 über dem (m+l)-dimensionalen reellen Vektorraum ^m+1 stützt sich auf die ebenen Schnitte vonf(U) durch eine im Punktf(a) der Hyperfläche nichttangentiale Geradeg. Als Anwendung ergeben sich einerseits eine Charakterisierung von Hyperflächen aus einer Quadrik unter Benützung gewisser scheinbarer Umrisse und andererseits eine Verallgemeinerung eines Ergebnisses über die zentralaffine Metrik nach D. Laugwitz. Weiters werden Hyperflächen aus quadratischen Kegeln in ^m mit Hilfe ihrer Schnitte mit Unterräumen geeigneter fester Dimension durch eine Gerade gekennzeichnet.Wir verwenden wie in [6] einen Flächenbegriff unter Berücksichtigung der Quotientenstruktur von ^m im Sinne differenzierbarer Mannigfaltigkeiten. Unter Benützung von Aussagen aus [6] wird für reguläre Hyperflächen in ^m die Existenz entweder einer oskulierenden Quadrik oder eines oskulierenden quadratischen Kegels oder einer oskulierenden Hyperebene in jedem Flächenpunkt nachgewiesen.     

Krümmungskennzeichnung der Ketten einer allgemeinen Geometrie Σ(ℜ,\mathfrak{L})

Lie theory renders a concept of curvature for the geometries (,) introduced by W. Benz [1]. Among regular curves in the chains of (, ) are characterized by =0.     

Limiting behavior of weighted central paths in linear programming

We study the limiting behavior of the weighted central paths{(x(), s())} > 0 in linear programming at both = 0 and = . We establish the existence of a partition (B ,N ) of the index set { 1, ,n } such thatx _i() ands _j () as fori B , andj N , andx _N (),s _B () converge to weighted analytic centers of certain polytopes. For allk 1, we show that thekth order derivativesx ^(k) () ands ^(k) () converge when 0 and . Consequently, the derivatives of each order are bounded in the interval (0, ). We calculate the limiting derivatives explicitly, and establish the surprising result that all higher order derivatives (k 2) converge to zero when .     

Note on initial topologies on rational vector spaces induced by realvalued linear mappings

LetG be a vector space over the field of rational numbers andf, g:G -linear mappings. equipped with the usual norm topology. Denote by _f , _g the initial topologies onG induced byf respectivelyg.Then the following result holds: If there is a nonvoid open setU whose complement contains at least one inner point such thatf ^–1 U _g , then there is ac withf=cg. In particular, iff0, the topologies coincide.Furthermore, a -linear mappingh: (G, _f )(G, _g ) is continuous if and only if there is a real constantc withg ^o h=cf.Dedicated to Professor János Aczél on his 60th birthday