non-Archimedean Valuations Research Articles

Implicit linear difference equations over a non-Archi-medean ring

Over any field an implicit linear difference equation one can reduce to the usual explicit one, which has infinitely many solutions ~ one for each initial value. It is interesting to consider an implicit difference equation over any ring, because the case of implicit equation over a ring is a significantly different from the case of explicit one. The previous results on the difference equations over rings mostly concern to the ring of integers and to the low order equations. In the present article the high order implicit difference equations over some other classes of rings, particularly, ring of polynomials, are studied. To study the difference equation over the ring of integer the idea of considering p-adic integers ~ the completion of the ring of integers with respect to the non-Archimedean p-adic valuation was useful. To find a solution of such an equation over the ring of polynomials it is naturally to consider the same construction for this ring: the ring of formal power series is a completion of the ring of polynomials with respect to a non-Archimedean valuation. The ring of formal power series and the ring of p-adic integers both are the particular cases of the valuation rings with respect to the non-Archimedean valuations of some fields: field of Laurent series and field of p-adic rational numbers respectively. In this article the implicit linear difference equation over a valuation ring of an arbitrary field with the characteristic zero and non-Archimedean valuation are studied. The sufficient conditions for the uniqueness and existence of a solution are formulated. The explicit formula for the unique solution is given, it has a form of sum of the series, converging with respect to the non-Archimedean valuation. Difference equation corresponds to an infinite system of linear equations. It is proved that in a case the implicit difference equation has a unique solution, it can be found using Cramer rules. Also in the article some results facilitating the finding the polynomial solution of the equation are given.

Valuations of Somos 4 sequences and canonical local heights on elliptic curves

AbstractSomos 4 sequences are sequences of numbers defined by a bilinear recurrence relation of order 4 and include elliptic divisibility sequences as a special case. In this paper, we describe valuations of Somos 4 sequences in terms of canonical local heights on the associated elliptic curves. We consider both Archimedean and non-Archimedean valuations. As applications, we study the asymptotic behaviour of valuations of Somos 4 sequences and obtain another proof of the integrality of certain Somos 4 sequences.

Global analytic geometry

Motivated by the well-known lack of archimedean information in algebraic geometry, we define, formalizing Ostrowski's classification of seminorms on Z , a new type of valuation of a ring that combines the notion of Krull valuation with that of a multiplicative seminorm. This definition partially restores the broken symmetry between archimedean and non-archimedean valuations artificially introduced in arithmetic geometry by the theory of schemes. This also allows us to define a notion of global analytic space that reconciles Berkovich's notion of analytic space of a (Banach) ring with Huber's notion of non-archimedean analytic spaces. After defining natural generalized valuation spectra and computing the spectrum of Z and Z [ X ] , we define analytic spectra and sheaves of analytic functions on them.

Differentiability of functions: approximate, global, and differentiability along curves over non-archimedean fields

This paper is devoted to the study of approximate and global smoothness and smoothness along curves of functions f(x 1,...,x m ) of variables x 1,...,x m in infinite fields with nontrivial non-Archimedean valuations and relations between them. Theorems on classes of smoothness C n or \(C_{\rm{b}}^n\) of functions with partial difference quotients continuous or bounded uniformly continuous on bounded domains up to order n are investigated. We prove that from f ○ u ∈ C n(K, K l) or f ○ u ∈ \(C_{\rm{b}}^n\)(K, K l) for each C ∞ or \(C_{\rm {b}}^\infty\) curve u: K → K m it follows that f ∈ C n(K m, K l) or f ∈ \(C_{\rm{b}}^n\)(K m, K l), where m ≥ 2. Then the classes of smoothness C n,r and \(C_{\rm{b}}^{n,r}\) and more general in the sense of Lipschitz for partial difference quotients are considered and theorems for them are proved. Moreover, the approximate differentiability of functions relative to measures is defined and investigated. Its relations with the Lipschitzian property and almost everywhere differentiability are studied. Non-Archimedean analogs of classical theorems of Kirzsbraun, Rademacher, Stepanoff, and Whitney are formulated and proved, and substantial differences between two cases are found. Finally, theorems about relations between approximate differentiability by all variables and along curves are proved.

Linear Forms in two m-adic Logarithms and Applications to Diophantine Problems

We give sharp, explicit estimates for linear forms in two logarithms, simultaneously for several non-Archimedean valuations. We present applications to explicit lower bounds for the fractional part of powers of rational numbers, and to the Diophantine equation (xn − 1)/(x − 1) = yq.

The Lotka-Volterra equation over a finite ring ℤ/pNℤ

Discrete Lotka-Volterra equation over $p$-adic space was constructed since $p$-adic space is a prototype of spaces with the non-Archimedean valuations and the space given by taking ultra-discrete limit studied in soliton theory should be regarded as a space with the non-Archimedean valuations in the previous report (solv-int/9906011). In this article, using the natural projection from $p$-adic integer to a ring $Z/p^n Z$, a soliton equation is defined over the ring. Numerical computations shows that it behaves regularly.

P-adic dynamic systems

Dynamic systems in non-Archimedean number fields (i.e., fields with non-Archimedean valuations) are studied. Results are obtained for the fields of p-adic numbers and complex p-adic numbers. Simple p-adic dynamic systems have a very rich structure—attractors, Siegel disks, cycles, and a new structure called a “fuzzy cycle”. The prime number p plays the role of a parameter of the p-adic dynamic system. Changing p radically changes the behavior of the system: attractors may become the centers of Siegel disks, and vice versa, and cycles of different lengths may appear or disappear.

Geometric families of constant reductions and the Skolem property

Let F | K F|K be a function field in one variable and V \mathcal V be a family of independent valuations of the constant field K . K. Given v ∈ V , v\in \mathcal V , a valuation prolongation v \mathrm v to F F is called a constant reduction if the residue fields F v | K v F\mathrm v |Kv again form a function field of one variable. Suppose t ∈ F t\in F is a non-constant function, and for each v ∈ V v\in \mathcal V let V t V_{t} be the set of all prolongations of the Gauß valuation v t v_{t} on K ( t ) K(t) to F . F. The union of the sets V t V_{t} over all v ∈ V v\in \mathcal V is denoted by V t . \mathbfit {V}_{t}. The aim of this paper is to study families of constant reductions V \mathbfit {V} of F F prolonging the valuations of V \mathcal V and the criterion for them to be principal, that is to be sets of the type V t . \mathbfit {V}_{t}. The main result we prove is that if either V \mathcal V is finite and each v ∈ V v\in \mathcal V has rational rank one and residue field algebraic over a finite field, or if V \mathcal V is any set of non-archimedean valuations of a global field K K satisfying the strong approximation property, then each geometric family of constant reductions V \mathbfit {V} prolonging V \mathcal V is principal. We also relate this result to the Skolem property for the existence of V \mathcal V -integral points on varieties over K , K, and Rumely’s existence theorem. As an application we give a birational characterization of arithmetic surfaces X / S \mathcal X /S in terms of the generic points of the closed fibre. The characterization we give implies the existence of finite morphisms to P S 1 . \mathbb P ^{1}_{S}.

On the classification of the maximal arithmetic subgroups of simply connected groups

Let be a simply connected simple algebraic group defined over a field of algebraic numbers and let be the set of all non-Archimedean valuations of the field . As is well known, each maximal arithmetic subgroup can be uniquely recovered by means of some collection of parachoric subgroups; to be more precise, there exist parachoric subgroups , , that have maximal types and satisfy the relation , where . Thus, there naturally arises the following question: for what collections of parachoric subgroups of maximal types is the above subgroup a maximal arithmetic subgroup of ? Using Rohlfs's cohomology criterion for the maximality of an arithmetic subgroup, necessary and sufficient conditions for the maximality of the above arithmetic subgroup are obtained. The answer is given in terms of the existence of elements of the field with prescribed divisibility properties.

Dissections of regular polygons into triangles of equal areas

This paper answers the question, “If a regular polygon withn sides is dissected intom triangles of equal areas, mustm be a multiple ofn?” Forn=3 the answer is “no,” since a triangle can be cut into any positive integral number of triangles of equal areas. Forn=4 the answer is again “no,” since a square can be cut into two triangles of equal areas. However, Monsky showed that a square cannot be dissected into an odd number of triangles of equal areas.We show that ifn is at least 5, then the answer is “yes.” Our approach incorporates the techniques of Thomas, Monsky, and Mead, in particular, the use of Sperner's lemma and non-Archimedean valuations, but also makes use of affine transformations to distort a given regular polygon into one to which those techniques apply.

Valuation theory: A constructive view

The theory of valuations on fields is developed in the constructive spirit of Errett Bishop. As a consequence of the general theory we are able to construct all nonarchimedean valuations on algebraic number fields and compute their ramification indices and residue class degrees. The notion of a field with a valuation for which the infimum of the values of any polynomial function can be computed plays an important role. Numerous limiting counterexamples are provided.

On the extension property of normed spaces over fields with non-archimedean valuations.

On the extension property of normed spaces over fields with non-archimedean valuations.