Hilbertʼs Tenth Problem for rational function fields over p-adic fields

We study the behaviour of the iterates of the Chebyshev polynomials of the first kind in p-adic fields. In particular, we determine in the field of complex p-adic numbers for p > 2, the periodic points of the p-th Chebyshev polynomial of the first kind. These periodic points are attractive points. We describe their basin of attraction. The classification of finite field extensions of the field of p-adic numbers ℚ p , enables one to locate precisely, for any integer ν ≥ 1, the ν-periodic points of T p : they are simple and the nonzero ones lie in the unit circle of the unramified extension of ℚ p , (p > 2) of degree ν. This generalizes a result, stated by M. Zuber in his PhD thesis, giving the fixed points of T p in the field ℚ p , (p > 2). As often happens, we consider separately the case p = 2. Also, if the integer n ≥ 2 is not divisible by p, then any fixed point w of T n is indifferent in the field of p-adic complex numbers and we give for p ≥ 3, the p-adic Siegel disc around w.

Tensors associated with a pair of quadratic differential forms

We present a well-parallelizable algorithm which, taking a straight-line program for the evaluation of a vectorial field of rational functions of Q(X 1,...,X n ) as input, decides whether they allow a rational potential function and, in case of affirmative answer, computes it as output. We introduce a mixed model of representation of polynomials to allow the application of integration techniques and show how to perform some basic operations with it. The algorithm is presented as a family of arithmetic networks of polynomial size and poly logarithmic depth in the degree of the occurring polynomials.KeywordsRational FunctionPartial FractionGreat Common DivisorProbabilistic AlgorithmParallel TimeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The p-adic numbers were first considered by Hensel in the 19th century. He observed that the primes play an analogous role in the integers as linear polynomials do in \({\mathbb C}[X]\). The Laurent expansion of a rational function led him to consider the p-adic expansion of a rational number. In this chapter, for a fixed prime p, we will construct block codes over the rings \({\mathbb Z}/p^h{\mathbb Z}\) simultaneously, by constructing codes over the p-adic numbers and then considering the coordinates modulo ph. These codes will be linear over the ring but when mapped to codes over \({\mathbb Z}/p{\mathbb Z}\) will result in codes which are not equivalent to linear codes. We start with a brief introduction to p-adic numbers, which will cover enough background for our purposes. The classical cyclic codes, that we constructed in Chapter 5, lift to cyclic codes over the p-adic numbers. In the case of the cyclic Hamming code, this lift extends to a code over \({\mathbb Z}/4{\mathbb Z}\) which, when mapped to a binary code, gives a non-linear code with a set of parameters for which no linear code exists.

We prove lower bounds for the transcendence degree of fields generated by values of the -adic exponential function. In particular, we estimate the transcendence degree of the field , where are algebraic (over the field of rational numbers) -adic numbers that form a basis of a finite extension of .

According to the theory of electrical networks, we can explicitly get the state equation for a time-invariant RLC network. A computer analysis program is compiled for finding the state equation of RLC network over the field of the rational function in multi-parameters. The program has no limits to the orders of state equation and the complex degree of RLC network. It is operated easily and obtains a good interface.

p-adic dynamical systems of (2,2)-rational functions with unique fixed point

Introduction. In the real case, it is well known that there exists an intimate relation between poles of complex powers of polynomial functions and the roots of the b-functions (cf. [B]). Many examples suggest that a similar relation between them would exist also in the p-adic case. Recently the case of polynomials over a p-adic field in two variables has been closely examined (cf. F. Loeser [L], W. Veys [V]). In this paper we shall deal with the case of relative invariants of prehomogeneous vector spaces over a p-adic field. Let us explain our problem more precisely. Let K be a p-adic field, i.e., a finite algebraic extension of Qp. We denote by ?)K the maximal compact subring of K, by rrT)K the ideal of nonunits of )K, and put q = #(DKI/'XC)K). We denote by | IKthe absolute value on K normalized as ITWK = q-. Let fQ(KX) be the set of quasi-characters of KX. Then the identity component fl(KX)O of fl(KX) consists of w,(t) = Itls for all s in C and fl(Kx)o CX under ws q-S. Let (G, p, V) be a prehomogeneous vector space (abbrev. P.V.) defined over K, where we assume that

This chapter develops tools for a more penetrating study of algebraic number theory than was possible in Chapter V and concludes by formulating two of the main three theorems of Chapter V in the modern setting of “adeles” and “ideles” commonly used in the subject.Sections 1–5 introduce discrete valuations, absolute values, and completions for fields, always paying attention to implications for number fields and for certain kinds of function fields. Section 1 contains a prototype for all these notions in the construction of the fieldQp of p-adic numbers formed out of the rationals. Discrete valuations in Section 2 are a generalization of the order-of-vanishing function about a point in the theory of one complex variable. Absolute values in Section 3 are real-valued multiplicative functions that give a metric on a field, and the pair consisting of a field and an absolute value is called a valued field. Inequivalent absolute values have a certain independence property that is captured by the Weak Approximation Theorem. Completions in Section 4 are functions mapping valued fields into their metric-space completions. Section 5 concerns Hensel’s Lemma, which in its simplest form allows one to lift roots of polynomials over finite prime fields Fp to roots of corresponding polynomials over p-adic fields Qp.Section 6 contains the main theorem for investigating the fundamental question of how prime ideals split in extensions. Let K be a finite separable extension of a field F, let R be a Dedekind domain with field of fractions F, and let T be the integral closure of R in K. The question concerns the factorization of an ideal pT in T when p is a nonzero prime ideal in R. If Fp denotes the completion of F with respect to p, the theorem explains how the tensor product K ⊗F Fp splits uniquely as a direct sum of completions of valued fields. The theorem in effect reduces the question of the splitting of pT in T to the splitting of Fp in a complete field in which only one of the prime factors of pT plays a role.Section 7 is a brief aside mentioning additional conclusions one can draw when the extension K/F is a Galois extension.Section 8 applies the main theorem of Section 6 to an analysis of the different of K/F and ultimately to the absolute discriminant of a number field. With the new sharp tools developed in the present chapter, including a Strong Approximation Theorem that is proved in Section 8, a complete proof is given for the Dedekind Discriminant Theorem; only a partial proof had been accessible in Chapter V.Sections 9–10 specialize to the case of number fields and to function fields that are finite separable extensions of Fq (X), where Fq is a finite field. The adele ring and the idele group are introduced for each of these kinds of fields, and it is shown how the original field embeds discretely in the adeles and how the multiplicative group embeds discretely in the ideles. The main theorems are compactness theorems about the quotient of the adeles by the embedded field and about the quotient of the normalized ideles by the embedded multiplicative group. Proofs are given only for number fields. In the first case the compactness encodes the Strong Approximation Theorem of Section 8 and the Artin product formula of Section 9. In the second case the compactness encodes both the finiteness of the class number and the Dirichlet Unit Theorem.KeywordsPrime IdealValuation RingBasic AlgebraDedekind DomainFractional IdealThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This chapter introduces the abstract algebraic concepts that correspond to the function fields of curves and the local rings of points on the curves. The field of rational functions, which corresponds to a straight line, is examined in detail. The chapter concludes with the existence theorem which shows that a function field contains enough structure to serve as an algebraic equivalent for a curve.

This chapter investigates the objects and mappings of algebraic geometry from a geometric point of view, making use especially of the algebraic tools of Chapter VII and of Sections 7–10 of Chapter VIII. In Sections 1–12, k denotes a fixed algebraically closed field. Sections 1–6 establish the definitions and elementary properties of varieties, maps between varieties, and dimension, all over k. Sections 1–3 concern varieties and dimension. Affine algebraic sets, affine varieties, and the Zariski topology on affine space are introduced in Section 1, and projective algebraic sets and projective varieties are introduced in Section 3. Section 2 defines the geometric dimension of an affine algebraic set, relating the notion to Krull dimension and transcendence degree. The actual context of Section 2 is a Noetherian topological space, the Zariski topology on affine space being an example. In such a space every closed subset is the finite union of irreducible closed subsets, and the union can be written in a certainway thatmakes the decomposition unique. Every nonempty closed set has a meaningful geometric dimension. In affine space the irreducible closed sets are the varieties, and each variety acquires a geometric dimension. The discussion in Section 2 applies in the context of projective space as well, and thus each projective variety acquires a geometric dimension. Moreover, any nonempty open subset of a Noetherian space is Noetherian. A nonempty open subset of an affine variety is called quasi-affine, and a nonempty open subset of a projective variety is called quasiprojective. Each quasi-affine variety or quasiprojective variety has a dimension equal to that of its closure, which is a variety. Sections 4–6 take up maps between varieties. Section 4 introduces spaces of scalar-valued functions on quasiprojective varieties–rational functions, functions regular at a point, and functions regular on an open set. The section goes on to relate these notions for the different kinds of varieties. Section 5 introduces morphisms, which are a restricted kind of function between varieties. The tools of Sections 4–5 together show that for many purposes all the different kinds of varieties can be treated as quasiprojective varieties. Section 6 introduces rational maps between varieties; these are not everywhere-defined functions, but each can be restricted to an open dense subset on which it is a morphism. Rational maps with dense image correspond to field mappings of the fields of rational functions, with the order of the mappings reversed. Section 7 concerns singularities at points of varieties, still over the field k. Zariski's Theorem was stated in Chapter VII for affine varieties and partly proved at that time. In the current context it has a meaning for any point of any quasiprojective variety. The section proves the full theorem, which characterizes singular points in a way that shows they remain singular under isomorphisms of varieties. Section 8 concerns classification questions over k for irreducible curves, i.e., quasiprojective varieties of dimension 1. From Section 6 it is known that two irreducible curves are equivalent under rational maps if and only if their fields of rational functions are isomorphic. The main theorem of Section 8 is that each such equivalence class of irreducible curves contains an everywhere nonsingular projective curve, and this curve is unique up to isomorphism of varieties. The points of this curve are parametrized by those discrete valuations of the underlying function field that are defined over k. Sections 9–12 relate the general theory of Sections 1–6 to the topic of solutions of simultaneous solutions of polynomial equations, as treated at length in Chapter VIII. Section 9 treats monomial ideals in k[X1, . . . , Xn], identifying their zero loci concretely and computing their dimension. The section goes on to introduce the affineHilbert function of this ideal, which measures the proportion of polynomials of degree≤ s not in the ideal. In the way that this function is defined, it is a polynomial for large s called the affine Hilbert polynomial of the ideal. Its degree equals the dimension of the zero locus of the ideal. Section 10 extends this theory from monomial ideals to all ideals, again concretely computing the dimension of the zero loci, obtaining an affine Hilbert polynomial, and showing that its degree equals the dimension of the zero locus of the ideal. Section 11 adapts the theory to homogeneous ideals and projective algebraic sets by making use of the cone in affine space over the set in projective space. Section 12 applies the theory of Section 11 to address the question how the dimension of a projective algebraic set is cut down when the set is intersected with a projective hypersurface. A consequence of the theory is the result that a homogeneous system of polynomial equations over an algebraically closed field with more unknowns than equations has a nonzero solution. Section 13 is a brief introduction to the theory of schemes, which extends the theory of varieties by replacing the underlying algebraically closed field by an arbitrary commutative ring with identity.

On zeta functions of modular representations of a discrete group

In this paper we consider dynamical systems generated by $(3,2)$-rational functions on the field of $p$-adic complex numbers. Each such function has three fixed points. We show that Siegel disks of the dynamical system may either coincide or be disjoint for different fixed points. Also, we find the basin of each attractor of the dynamical system. We show that, for some values of the parameters, there are trajectories which go arbitrary far from the fixed points.

We investigate the ramification of embeddings of local lattices or quadratic forms over the ring of -adic numbers. It is proved that every primitive embedding decomposes uniquely into an orthogonal sum of minimal indecomposable embeddings, and all such embeddings are constructed for -elementary lattices. Ramification theory enables us to find the number of orbits of representations for forms and, in particular, for numbers by other quadratic forms over , and to calculate the local multipliers in the weight formula for representations of a form by a genus of quadratic forms.

This book grew out of seminar held at the University of Paris 7 during the academic year 1985-86. The aim of the seminar was to give an exposition of the theory of the Metaplectic Representation (or Weil Representation) over a p-adic field. The book begins with the algebraic theory of symplectic and unitary spaces and a general presentation of metaplectic representations. It continues with exposes on the recent work of Kudla (Howe Conjecture and induction) and of Howe (proof of the conjecture in the unramified case, representations of low rank). These lecture notes contain several original results. The book assumes some background in geometry and arithmetic (symplectic forms, quadratic forms, reductive groups, etc.), and with the theory of reductive groups over a p-adic field. It is written for researchers in p-adic reductive groups, including number theorists with an interest in the role played by the Weil Representation and -series in the theory of automorphic forms.