Identity theorem for bounded p-adic meromorphic functions

Let K be a complete ultrametric algebraically closed field. We investigate several properties of sequences (an)_n∈N in a disk d(0,R⁻) with regards to bounded analytic functions in that disk: sequences of uniqueness (when f(an)=0∀n∈N implies f=0), identity sequences (when limn→+∞f(an)=0 implies f=0) and analytic boundaries (when lim supn→∞|f(an)|=‖f‖). Particularly, we show that identity sequences and analytic boundary sequences are two equivalent properties. For certain sequences, sequences of uniqueness and identity sequences are two equivalent properties. A connection with Blaschke sequences is made. Most of the properties shown on analytic functions have continuation to meromorphic functions.     

About the ultrametric corona problem

Let K be a complete ultrametric algebraically closed field and let A be the K-Banach algebra of bounded analytic functions in the disk . Let Mult(A,‖⋅‖) be the set of continuous multiplicative semi-norms of A, let Multm(A,‖⋅‖) be the subset of the ?∈Mult(A,‖⋅‖) whose kernel is a maximal ideal and let Multa(A,‖⋅‖) be the subset of the ?∈Multm(A,‖⋅‖) whose kernel is of the form (if ?∈Multm(A,‖⋅‖)?Multa(A,‖⋅‖), the kernel of ? is then of infinite codimension). The main problem we examine is whether Multa(A,‖⋅‖) is dense inside Multm(A,‖⋅‖) with respect to the topology of simple convergence. This a first step to the conjecture of density of Multa(A,‖⋅‖) in the whole set Mult(A,‖⋅‖): this is the corresponding problem to the well-known complex corona problem. We notice that if ?∈Multm(A,‖⋅‖) is defined by an ultrafilter on D, ? lies in the closure of Multa(A,‖⋅‖). Particularly, we shaw that this is case when a maximal ideal is the kernel of a unique ?∈Multm(A,‖⋅‖). Thus, if every maximal ideal is the kernel of a unique ?∈Multm(A,‖⋅‖), Multa(A,‖⋅‖) is dense in Multm(A,‖⋅‖). And particularly, this is the case when K is strongly valued. In the general context, we find a subset of Multm(A,‖⋅‖)?Multa(A,‖⋅‖) which is included in the closure of Multa(A,‖⋅‖). More generally, we show that if ψ∈Mult(A,‖⋅‖) does not define the Gauss norm on polynomials (‖⋅‖), then it is characterized by a circular filter, like on rational functions and analytic elements. As a consequence, if ψ does not lie in the closure of Multa(A,‖⋅‖), then its restriction to polynomials is the Gauss norm.     

Some results of functional analysis for ultrametric spaces and valued vector spaces

We prove a Hahn-Banach type theorem and a generalization of Baire's theorem for ultrametric spaces (with totally ordered value sets). Some applications to valued vector spaces with value groups of arbitrary rank are given (Principle of Uniform Boundedness, Open Mapping Theorem).Dedicated to Prof. H. Salzmann on the occasion of his 65th birthday     

A Mazur–Ulam theorem in non-Archimedean normed spaces

The classical Mazur–Ulam theorem which states that every surjective isometry between real normed spaces is affine is not valid for non-Archimedean normed spaces. In this paper, we establish a Mazur–Ulam theorem in the non-Archimedean strictly convex normed spaces.     

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.     

Existence and regularity of the solution of a mixed boundary value problem for the Keldysh equation with a nonlinear absorption term

The Keldysh equation is a more general form of the classic Tricomi equation from fluid dynamics. Its well-posedness and the regularity of its solution are interesting and important. The Keldysh equation is elliptic in y>0 and is degenerate at the line y=0 in R². Adding a special nonlinear absorption term, we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain—similar to the potential fluid shock reflection problem. By means of an elliptic regularization technique, a delicate a priori estimate and compact argument, we show that the solution of a mixed boundary value problem of the Keldysh equation is smooth in the interior and Lipschitz continuous up to the degenerate boundary under some conditions. We believe that this kind of regularity result for the solution will be rather useful.     

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.     

Some Problems on Homomorphisms and Real Function Algebras

 In this paper we solve a problem about the representation of all homomorphisms on a real function algebra as point evaluations and another two about function algebras in which homomorphisms are point evaluations on sequences in the algebra. (Received 4 December 2000; in revised form 2 April 2001)     

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).     

On integrally closed simple extensions of valuation rings

Let v be a Krull valuation of a field with valuation ring $R_v$. Let θ be a root of an irreducible trinomial $F(x)=x^n+a{x}^m+b$ belonging to $R_v[x]$. In this paper, we give necessary and sufficient conditions involving only $a,b,m,n$ for $R_v[\theta ]$ to be integrally closed. In the particular case when v is the p-adic valuation of the field $\mathbb{Q}$ of rational numbers, $F(x)\in \mathbb{Z}[x]$ and $K=\mathbb{Q}(\theta )$, then it is shown that these conditions lead to the characterization of primes which divide the index of the subgroup $\mathbb{Z}[\theta ]$ in $A_K$, where $A_K$ is the ring of algebraic integers of K. As an application, it is deduced that for any algebraic number field K and any quadratic field L not contained in K, we have $A_{KL}=A_K{A}_L$ if and only if the discriminants of K and L are coprime.     

Positive elimination in valued fields

Let K be an algebraically closed field with a valuation ring or a real closed field with a convex valuation ring . We show that the projection of a basic (see “Introduction”) subset of to K ⁿ is again basic.     

Spectral synthesis and other results in some topological algebras of vector-valued functions

Using results from theory of bundles of topological vector spaces, we prove some spectral synthesis and other results in certain topological algebras of vector-valued functions. In particular, we extend and generalize results of J.W. Kitchen and the second-named author (1996). We obtain as a corollary a recent (2008) result of Arizmendi-Peimbert/Carillo-Hoyo/García on the spectral synthesis property in (C _b (X),β), the space of bounded continuous functions on the completely regular Hausdorff space X under the strict topology β which arises from weighting by the non-negative upper semicontinuous functions on X which disappear at infinity.     

Linear mixed models and penalized least squares

Linear mixed-effects models are an important class of statistical models that are used directly in many fields of applications and also are used as iterative steps in fitting other types of mixed-effects models, such as generalized linear mixed models. The parameters in these models are typically estimated by maximum likelihood or restricted maximum likelihood. In general, there is no closed-form solution for these estimates and they must be determined by iterative algorithms such as EM iterations or general nonlinear optimization. Many of the intermediate calculations for such iterations have been expressed as generalized least squares problems. We show that an alternative representation as a penalized least squares problem has many advantageous computational properties including the ability to evaluate explicitly a profiled log-likelihood or log-restricted likelihood, the gradient and Hessian of this profiled objective, and an ECME update to refine this objective.     

Numerically positive divisors on algebraic surfaces

Numerically positive line bundles on a complex projective smooth algebraic surfaceS are studied. In particular for any such line bundleL Pic(S) we prove the following facts: (i)g(L) 0 and (ii)L is ample ifg(L) 1,g standing for the arithmetic genus. Some applications are discussed. We also investigate numerically positive non-ample line bundlesL withg(L)=2.     

<Emphasis Type="Italic">P</Emphasis>-adic sets of range uniqueness

We study sets of range uniqueness (SRU’s) in a complete, ultrametric, algebraically closed fieldK for analytic elements. We find monotonic distances sequences which appear to be SRU’s completely different from those known in ©. On the other hand, most of open closed sets cannot be SRU’s.     

Representation theorems for Banach algebras

For a space X   denote by _Cb(X)$C_b(X)$ the Banach algebra of all continuous bounded scalar-valued functions on X   and denote by _C0(X)$C_0(X)$ the set of all elements in _Cb(X)$C_b(X)$ which vanish at infinity.     

The n-generator property in rings of integer-valued polynomials determined by finite sets

Let D be an integral domain and E a non-empty finite subset of D. For n ≧ 2, we show that D has the n-generator property if and only if Int(E, D) has the n-generator property if and only if Int(E, D) has the strong (n + 1)-generator property. Thus, iterating the Int(E, D) construction cannot produce Prüfer domains whose finitely generated ideals require an ever larger number of generators. We also show that, for n ≧ 2, a non-zero polynomial f ∈Int(E, D) is a strong n-generator in Int(E, D) if and only if f (a) is a strong n-generator in D for all a ∈E. Received: 15 July 2004