On Brown’s constant associated with irreducible polynomials over henselian valued fields

Let v be a henselian valuation of arbitrary rank of a field K and be the prolongation of v to the algebraic closure of K with value group . In 2008, Ron Brown gave a class P of monic irreducible polynomials over K such that to each g(x) belonging to P, there corresponds a smallest constant λ_g belonging to (referred to as Brown’s constant) with the property that whenever is more than λ_g with K(β) a tamely ramified extension of (K,v), then K(β) contains a root of g(x). In this paper, we determine explicitly this constant besides giving an important property of λ_g without assuming that K(β)/K is tamely ramified.     

Fisher information for generalised linear mixed models

The Fisher information for the canonical link exponential family generalised linear mixed model is derived. The contribution from the fixed effects parameters is shown to have a particularly simple form.     

Noether’s problem for GL(2, 3)

We prove that Noether’s problem has an affirmative answer for the group GL(2, 3), over every field K. In particular, the group admits a generic polynomial over . As a consequence, so does the group . Research was partially supported by MCYT grant BFM2003-01898.     

Corrigendum and addendum to my paper concerning Kummer extensions with few roots of unity

This note presents corrections and additions to my paper (J. Number Theory 41 (1992) 322-358).     

Efficient Software-Implementation of Finite Fields with Applications to Cryptography

In this work, we present a survey of efficient techniques for software implementation of finite field arithmetic especially suitable for cryptographic applications. We discuss different algorithms for three types of finite fields and their special versions popularly used in cryptography: Binary fields, prime fields and extension fields. Implementation details of the algorithms for field addition/subtraction, field multiplication, field reduction and field inversion for each of these fields are discussed in detail. The efficiency of these different algorithms depends largely on the underlying micro-processor architecture. Therefore, a careful choice of the appropriate set of algorithms has to be made for a software implementation depending on the performance requirements and available resources.     

A generalization of a theorem of Ore

Let (K,v)$(K,v)$ be a discrete rank one valued field with valuation ring _Rv$R_v$. Let L/K$L/K$ be a finite extension such that the integral closure S   of _Rv$R_v$ in L   is a finitely generated _Rv$R_v$-module. Under a certain condition of v  -regularity, we obtain some results regarding the explicit computation of _Rv$R_v$-bases of S, thereby generalizing similar results that had been obtained for algebraic number fields in El Fadil et al. (2012) [7]. The classical Theorem of Index of Ore is also extended to arbitrary discrete valued fields. We give a simple counter example to point out an error in the main result of Montes and Nart (1992) [12] related to the Theorem of Index and give an additional necessary and sufficient condition for this result to be valid.     

Topological differential fields

We consider first-order theories of topological fields admitting a model-completion and their expansion to differential fields (requiring no interaction between the derivation and the other primitives of the language). We give a criterion under which the expansion still admits a model-completion which we axiomatize. It generalizes previous results due to M. Singer for ordered differential fields and of C. Michaux for valued differential fields. As a corollary, we show a transfer result for the NIP property. We also give a geometrical axiomatization of that model-completion. Then, for certain differential valued fields, we extend the positive answer of Hilbert’s seventeenth problem and we prove an Ax-Kochen-Ershov theorem. Similarly, we consider first-order theories of topological fields admitting a model-companion and their expansion to differential fields, and under a similar criterion as before, we show that the expansion still admits a model-companion. This last result can be compared with those of M. Tressl: on one hand we are only dealing with a single derivation whereas he is dealing with several, on the other hand we are not restricting ourselves to definable expansions of the ring language, taking advantage of our topological context. We apply our results to fields endowed with several valuations (respectively several orders).     

Pro-p galois groups of rank ≤4

Letp be a prime >2, letF be a field of characteristic ≠p containing a primitivep-th root of unity and letG _F (p) be the Galois group of the maximal Galois-p-extension ofF. Ifrk G _F (p)≤4 thenG _F (p) is a free pro-p product of metabelian groups orG _F (p) is a Demuškin group of rank 4.     

MOMENT ESTIMATION FOR MULTIVARIATE EXTREME VALUE DISTRIBUTION

Moment estimation for multivariate extreme value distribution is described in this paper. Asymptotic covariance matrix of the estimators is given. The relative efficiencies of moment estimators as compared with the maximum likelihood and the stepwise estimators are computed. We show that when there is strong dependence between the variates, the generalized variance of moment estimators is much lower than the stepwise estimators. It becomes more obvious when the dimension increases.     

Sample covariance shrinkage for high dimensional dependent data

For high dimensional data sets the sample covariance matrix is usually unbiased but noisy if the sample is not large enough. Shrinking the sample covariance towards a constrained, low dimensional estimator can be used to mitigate the sample variability. By doing so, we introduce bias, but reduce variance. In this paper, we give details on feasible optimal shrinkage allowing for time series dependent observations.     

A complex version of the baer-krull theorems

In the setting of the complex spectrum of a field of characteristic zero we give complex analogs of the celebrated (real) Baer-Krull theorems relat-ing the orderings of a field compatible with a given valuation ring with the orderings of the residue class field of that valuation ring.     

GAR-Realisierungen klassischer Gruppen

Ohne Zusammenfassung

---

Der Autor dankt der Deutschen Forschungsgemeinschaft für finanzielle Unterstützung     

Generic polynomials are descent-generic

Let be a generic polynomial for a group G in the sense that every Galois extension N/L of infinite fields with group G and K≤L is given by a specialization of g(X). We prove that then also every Galois extension whose group is a subgroup of G is given in this way. Received: 15 January 2001     

Polynomials and finite sets connected with an identity

It is shown that the polynomials satisfying the identityf(x) f(x + 1) = f(x ² +x – a), wherea either belongs to a field of characteristic zero or is transcendental over a prime field of characteristic exceeding 2, are precisely those of the form(x ² –a) ⁿ ; thus extending a result proved by Nathanson in the complex case. The result is not, in general, true in characteristic 2. Additionally, a class of finite sets, considered by Nathanson in connection with the identity, is completely determined.