P-adic Absolute Value Research Articles

Formula omitted]-adic distance and [formula omitted]-Nearest Neighbor classification

The k-Nearest Neighbor (k-NN) is a well-known supervised learning algorithm. The effect of the distance used in the analysis on the k-NN performance is very important. According to Ostrowski’s theorem, there are only two nontrivial absolute values on the field of rational numbers, Q, which are the usual absolute value and the p-adic absolute value for a prime p. In view of this theorem, the p-adic absolute value motivates us to calculate the p-adic distance between two samples for the k-NN algorithm. In this study, the p-adic distance on Q was coupled with the k-NN algorithm and was applied to 10 well-known public datasets containing categorical, numerical, and mixed (both categorical and numerical) type predictive attributes. Moreover, the p-adic distance performance was compared with Euclidean, Manhattan, Chebyshev, and Cosine distances. It was seen that the average accuracy obtained from the p-adic distance ranks first in 5 out of 10 datasets. Especially, in mixed datasets, the p-adic distance gave better results than other distances. For r=1,2,3, the effect of the r-decimal values of the number for the p-adic calculation was examined on numerical and mixed datasets. In addition, the p parameter of the p-adic distance was tested with prime numbers less than 29, and it was found that the average accuracy obtained for each p was very close to each other, especially in categorical and mixed datasets. Also, it can be concluded that k-NN with the p-adic distance may be more suitable for binary classification than multi-class classification.

On effective approximation to quadratic numbers

Let p be a prime number and | · |p the p-adic absolute value on Q and on the p-adic field Qp normalized such that |p|p = p −1 . Let ξ be a quadratic real number and α a quadratic p-adic number. We prove that there exist positive, effectively computable, real numbers c1 = c1(ξ), τ1 = τ1(ξ), c2 = c2(α), τ2 = τ2(α), such that |yξ − x| · |y|p ≥ c1|y| −2+τ1 , for x, y ∈ Z̸=0, and |bα − a|p ≥ c2|ab| −2+τ2 , for a, b ∈ Z̸=0. Both results improve the effective lower bounds which follow from an easy Liouville-type argument.

Euler's factorial series, Hardy integral, and continued fractions

We study p-adic Euler's series Ep(t)=∑k=0∞k!tk at a point pa, a∈Z≥1, and use Padé approximations to prove a lower bound for the p-adic absolute value of the expression cEp(±pa)−d, where c,d∈Z. It is interesting that the same Padé polynomials which p-adically converge to Ep(t), approach the Hardy integral H(t)=∫0∞e−s1−tsds on the Archimedean side. This connection is used with a trick of analytic continuation when deducing an Archimedean bound for the numerator Padé polynomial needed in the derivation of the lower bound for |cEp(±pa)−d|p. Furthermore, we present an interconnection between E(t) and H(t) via continued fractions.

On the t-adic Littlewood conjecture

The p-adic Littlewood conjecture due to De Mathan and Teulié asserts that for any prime number p and any real number α, the equation inf|m|≥1|m|⋅|m|p⋅|⟨mα⟩|=0 holds. Here |m| is the usual absolute value of the integer m, |m|p is its p-adic absolute value, and |⟨x⟩| denotes the distance from a real number x to the set of integers. This still-open conjecture stands as a variant of the well-known Littlewood conjecture. In the same way as the latter, it admits a natural counterpart over the field of formal Laurent series K((t−1)) of a ground field K. This is the so-called t-adic Littlewood conjecture (t-LC). It is known that t-LC fails when the ground field K is infinite. The present article is concerned with the much more difficult case when this field is finite. More precisely, a fully explicit counterexample is provided to show that t-LC does not hold in the case that K is a finite field with characteristic 3. Generalizations to fields with characteristic other than 3 are also discussed. The proof is computer-assisted. It reduces to showing that an infinite matrix encoding Hankel determinants of the paperfolding sequence over F3, the so-called number wall of this sequence, can be obtained as a 2-dimensional automatic tiling satisfying a finite number of suitable local constraints.

On the zeros and singularities of adic trace functions

Given a prime number p we consider the topological completion of the algebraic closure of the field of p-adic numbers with respect to the usual p-adic absolute value. In this paper, we point out new connections between zeros and singularities of the trace functions associated with arbitrary elements of

The frame of the p-adic numbers

The algebraic nature of a frame allows its definition by generators and relations. Joyal used this to introduce the frame of the real numbers and Banaschewski studied this frame with a particular emphasis on the pointfree extension of the ring of continuous real functions; he provided a pointfree version of the Stone-Weierstrass theorem. In this paper, we explore this situation for the case of the p-adic numbers; we define the frame of the p-adic numbers by generators and relations. The field of the p-adic numbers Qp is the completion of Q with respect to the p-adic absolute value |⋅|p, which satisfies |x+y|p≤max⁡{|x|p,|y|p}. Dieudonné proved that the ring Qp[X] of polynomials with coefficients in Qp is dense in the ring C(F,Qp) of continuous functions defined on a compact subset F of Qp with values in Qp, and Kaplansky extended this result by proving that if F is a nonarchimedean valued field and X is a compact Hausdorff space, then any unitary subalgebra A of C(X,F) which separates points is uniformly dense in C(X,F). We give a p-adic version of this theorem in pointfree topology.

On the norm of Krasner analytic functions with applications to transcendence results

Given a prime number p let Cp be the Tate field i.e. the topological completion of the algebraic closure of the field of p-adic numbers with respect to the p-adic absolute value. We give an estimate of the norm of Krasner analytic functions defined on the complement of a fundamental set X of Cp, which are Cauchy transforms obtained by integrating against strongly Lipschitz distributions on X. The functions from a specific class of the above representations are transcendental over Cp(Z) and, as a consequence, we obtain transcendence results related to the twisted p-adic log gamma function and the trace functions. The twisted p-adic log gamma function and its derivatives are linearly independent over Cp(Z) and, moreover, all their zeros are algebraic.

Bounds on the radius of the p-adic Mandelbrot set

Let f be a degree d polynomial defined over the nonarchimedean field C_p, normalized so f is monic and f(0)=0. We say f is post-critically bounded, or PCB, if all of its critical points have bounded orbit under iteration of f. It is known that if p is greater than or equal to d and f is PCB, then all critical points of f have p-adic absolute value less than or equal to 1. We give a similar result for primes between d/2 and d. We also explore a one-parameter family of cubic polynomials over the 2-adic numbers to illustrate that the p-adic Mandelbrot set can be quite complicated when p is less than d, in contrast with the simple and well-understood p > d case.

Some identities on the q-Genocchi polynomials with weak weight \alpha and Bernstein polynomials

Let p be a fixed odd prime number. Throughout this paper, we always make use of the following notations: Z denotes the ring of rational integers, Zp denotes the ring of p-adic rational integers, Qp denotes the field of p-adic rational numbers, and Cp denotes the completion of algebraic closure of Qp, respectively. Let N be the set of natural numbers and Z+ = N ∪ {0}. The padic absolute value is defined by |x|p = 1 pr , where x = p s t ( r ∈ Q and s, t ∈ Z with (s, t) = (p, s) = (p, t) = 1). In this paper we assume that q ∈ Cp with |q − 1|p < 1 as an indeterminate. The q-number is defined by

Exponential sums along p-adic curves

Let K be a p-adic field, R the valuation ring of K, and P the maximal ideal of R. Let Y⊆ R 2 be a non-singular closed curve, and Y m its image in R/ P m × R/ P m , i.e. the reduction modulo P m of Y. We denote by Ψ an standard additive character on K. In this paper we discuss the estimation of exponential sums of type S m ( z, Ψ, Y, g)≔∑ x∈ Y m Ψ( zg( x)), with z∈ K, and g a polynomial function on Y. We show that if the p-adic absolute value of z is big enough then the complex absolute value of S m ( z, Ψ, Y, g) is O( q m(1− β( f, g)) ), for a positive constant β( f, g) satisfying 0< β( f, g)<1.

Lower bounds for the heights of solutions of linear equations

Let K be a number eld of degree d. We de ne as usual the eld height HK ( ) of a number ∈ K by HK ( ) = ∏ v∈M (K) max{1; | |dv v } : Here the v-adic valuations | |v are normalized such that on Q they coincide with either the standard absolute value or with the p-adic absolute value | |p having |p|p = p−1. Moreover dv denotes the local degree. The absolute height of then, is de ned by H ( ) = HK ( ) : Lehmer’s conjecture says the following. Let be a nonzero algebraic number of degree D. Then; if is not a root of unity we have

Tensor products in p-adic Hodge theory

There is a classical relation between the p-adic absolute value of the eigenvalues of Frobenius on crystalline cohomology and Hodge numbers, for a variety in characteristic p: “the Newton polygon lies on or above the Hodge polygon” [14], [1]. For a variety in characteristic p with a lift to characteristic 0, Fontaine conjectured and Faltings proved a more precise statement: there is an inequality which relates the slope of Frobenius on any Frobenius-invariant subspace of the crystalline cohomology to the Hodge filtration, restricted to that subspace [7], [4]. A vector space over a p-adic field together with a σ-linear endomorphism and a filtration which satisfies this inequality is called a weakly admissible filtered isocrystal (see section 1 for the precise definition). The category of such objects is one possible p-adic analogue of the category of Hodge structures: in particular, it is an abelian category. We give a new proof of Faltings’s theorem that the tensor product of weakly admissible filtered isocrystals over a p-adic field is weakly admissible [5]. By a similar argument, we also prove a characterization of weakly admissible filtered isocrystals with G-structure in terms of geometric invariant theory, which was conjectured by Rapoport and Zink [19]. Before Faltings, Laffaille [12] had proved the tensor product theorem in the case of filtered isocrystals over an unramified extension of Qp. Faltings’s proof works by reducing this problem of σ-linear algebra to a different problem of pure linear algebra, the problem of showing that the tensor product of two vector spaces, each equipped with a finite “semistable” set of filtrations, is semistable. The latter problem is solved by constructing suitable integral lattices (in [5]) or hermitian metrics (in [20]) on vector spaces with a semistable set of filtrations, just as one can prove that the tensor product of semistable bundles on an algebraic curve is semistable using Narasimhan-Seshadri’s hermitian metrics ([6], [16]). In this paper, we can avoid the reduction from filtered isocrystals to filtered vector spaces. The point is that Ramanan and Ramanathan’s algebraic proof [17] that the tensor product of semistable vector bundles is semistable can be modified to apply directly to filtered isocrystals. We have an inequality to prove for a class of linear subspaces S of a tensor product V ⊗W . The inequality is obvious for sufficiently general subspaces S and also if S is a very special subspace, say if S is a decomposable subspace S1 ⊗ S2 ⊂ V ⊗W . But it is not clear how to prove the inequality we want if S is somewhere in the middle. The solution, following Ramanan and Ramanathan, is to use geometric invariant theory to give a sharp dichotomy between “general” subspaces and “special” subspaces of V ⊗W , in such a way that we get useful information in either case.

On polynomial-exponential equations

Let M(K) be the set of absolute values of K, normalized such that they extend the standard or a p-adic absolute value of Q. Let S c M(K) be the subset consisting of the archimedean absolute values, as well as of the non-archimedean absolute values v e M(K) having I~ol~ 4= 1 for some ~ej. Thus S is a finite set. An S-unit is an element ~ e K* having I~1~--1 for every v$S. Then in particular the ~o, and therefore every a tXe with x E Z ' , is an S-unit. Therefore (1.2) is related to the S-unit equation

An external approach to unitary representations

The principal ideas of harmonic analysis on a locally compact group G which is not necessarily compact or commutative, were developed in the 1940’s and early 1950’s. In this theory, the role of the classical fundamental harmonics is played by the irreducible unitary representations of G. The set of all equivalence classes of such representations is denoted by Ĝ and is called the dual object of G or the unitary dual of G. Since the 1940’s, an intensive study of the foundations of harmonic analysis on complex and real reductive groups has been in progress (for a definition of reductive groups, the reader may consult the appendix at the end of the second section). The motivation for this development came from mathematical physics, differential equations, differential geometry, number theory, etc. Through the 1960’s, progress in the direction of the Plancherel formula for real reductive groups was great, due mainly to Harish-Chandra’s monumental work, while, at the same time, the unitary duals of only a few groups had been parametrized. With F. Mautner’s work [Ma], a study of harmonic analysis on reductive groups over other locally compact non-discrete fields was started. We shall first describe such fields. In the sequel, a locally compact non-discrete field will be called a local field. If we have a non-discrete absolute value on the field Q of rational numbers, then it is equivalent either to the standard absolute value (and the completion is the field R of real numbers), or to a p-adic absolute value for some prime number p. For r 2 Q× write r = pa/b where α, a and b are integers and neither a nor b are divisible by p. Then the p-adic absolute value of r is jrjp = p−α. A completion of Q with respect to the p-adic absolute value is denoted by Qp. It is called a field of p-adic numbers. Each finite dimensional extension F of Qp has a natural topology of a vector space over Qp. With this topology, F becomes a local field. The topology of F can be also introduced with an absolute value which is denoted by j jF (in the fifth section we shall fix a natural absolute value). The fields of real and complex numbers, together with the finite extensions of p-adic numbers, exhaust all local fields of characteristic zero up to isomorphisms ([We]). Let F be a finite field. Denote by F((X)) the field of formal power series over F. Elements of this field are series of the form f = P∞ n=k anX , an 2 F, for some integer k. Fix q > 1.

On the Leopoldt conjecture on the p-adic regulators

The Leopoldt conjecture for ( K, p) is investigated under the assumption that the Leopoldt conjecture for ( k, p) is valid, where k is a finite algebraic number field and K k is a cyclic extension of prime degree p. A lower bound for the p-adic absolute value of the p-adic regulator R p ( K) of K is also given when k = Q.