p-Adic ideals of p-rank d and the p-adic Nullstellensatz

In this paper we introduce and we study the notion of p-adic ideal of p-rank d and the one of p-adic radical of p-rank d of an ideal, by analogy with the notion of real ideal and the one of real radical of an ideal in the real case. Using those two notions and following more closely the real approach of Bochnak et al. (Géométrie algébrique réelle, Springer, Berlin 1987) we give a new proof of the p-adic Nullstellensatz.     

Central units in p-adic group rings

If p is a natural prime and G any p-group then an exact sequence is constructed to describe the group of units in the center of the p-adic group ring _p [G].     

On nonlinear equations of p-adic strings for scalar tachyon fields

We consider boundary value problems for open and closed p-adic strings for scalar tachyon fields. Estimates for solutions to these problems and possible ways of constructing these solutions are obtained by reducing the problems to linear parabolic equations with nonlinear boundary conditions. We give an application of Gauss-type quadrature formulas to the numerical solution of the boundary value problems, and discuss the possibility of using these methods in multidimensional problems (d = 2). Original Russian Text ? V.S. Vladimirov, 2009, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2009, Vol. 265, pp. 254–272.     

Representation theory of p-adic groups and canonical bases

In this paper, we interpret the Gindikin–Karpelevich formula and the Casselman–Shalika formula as sums over Kashiwara–Lusztig?s canonical bases, generalizing the results of Bump and Nakasuji (2010) [7] to arbitrary split reductive groups. We also rewrite formulas for spherical vectors and zonal spherical functions in terms of canonical bases. In a subsequent paper Kim and Lee (preprint) [14], we will generalize these formulas to p-adic affine Kac–Moody groups.     

On minimal decomposition of p-adic polynomial dynamical systems

A polynomial of degree ?2 with coefficients in the ring of p-adic numbers Zp is studied as a dynamical system on Zp. It is proved that the dynamical behavior of such a system is totally described by its minimal subsystems. For an arbitrary quadratic polynomial on Z₂, we exhibit all its minimal subsystems.     

Free-by-finite pro-p groups and integral p-adic representations

Let F be a free pro-p group of finite rank n and C_p^r{C_{p^r}} a cyclic group of order p ^r . In this work we classify p-adic representations C_p^r? GL_n(\mathbbZ_p){ C_{p^r}\longrightarrow GL_n(\mathbb{Z}_{p})} that can be obtained as a composite of an embedding C_p^r? Aut(F){C_{p^r}\longrightarrow {\rm Aut}(F)} with the natural epimorphism Aut(F)? GL_n(\mathbbZ_p){{\rm Aut}(F)\longrightarrow GL_n(\mathbb{Z}_{p})} .     

The Group Ring of SL2(p) over p-adic Integers for p Odd

Let p > 2 be a prime, R =  _p[ζ_p^f − 1], K =  _p[ζ_p^f − 1], and G = SL₂(p^f). The group ring RG is calculated nearly up to Morita equivalence: The projections of RG into the simple components of KG are given explicitly and the endomorphism rings and homomorphism bimodules between the projective indecomposable RG-lattices are described.     

Integer Polynomials with Roots mod p for all Primes p

Let f(X) be an integer polynomial which is a product of two irreducible factors. Assume that f(X) has a root mod p for all primes p. If the splitting field of f(X) over the rationals is a cyclic extension of the stem fields, then the Galois group of f(X) over the rationals is soluble and of bounded Fitting length. Moreover, the fixed groups of the stem extensions are in, some sense, unique.     

On the chaotic behavior of a generalized logistic p-adic dynamical system

In the paper we describe basin of attraction p-adic dynamical system G(x)=(ax)²(x+1). Moreover, we also describe the Siegel discs of the system, since the structure of the orbits of the system is related to the geometry of the p-adic Siegel discs.     

3-Adic Cantor function on local fields and its p-adic derivative

The problem of “rate of change” for fractal functions is a very important one in the study of local fields. In 1992, Su Weiyi has given a definition of derivative by virtue of pseudo-differential operators [Su W. Pseudo-differential operators and derivatives on locally compact Vilenkin groups. Sci China [series A] 1992;35(7A):826–36. Su W. Gibbs–Butzer derivatives and the applications. Numer Funct Anal Optimiz 1995;16(5&6):805–24. [2] and [3]]. In Qiu Hua and Su Weiyi [Weierstrass-like functions on local fields and their p-adic derivatives. Chaos, Solitons & Fractals 2006;28(4):958–65. [8]], we have introduced a kind of Weierstrass-like functions in p-series local fields and discussed their p-adic derivatives. In this paper, the 3-adic Cantor function on 3-series field is constructed, and its 3-adic derivative is evaluated, it has at most order. Moreover, we introduce the definition of the Hausdorff dimension [Falconer KJ. Fractal geometry: mathematical foundations and applications. New York: Wiley; 1990. [1]] of the image of a complex function defined on local fields. Then we conclude that the Hausdorff dimensions of the 3-adic Cantor function and its derivatives and integrals on 3-series field are all equal to 1.There are various applications of Cantor sets in mechanics and physics. For instance, E-infinity theory [El Naschie MS. A guide to the mathematics of E-infinity Cantorian spacetime theory. Chaos, Solitons & Fractals 2005;25(5):955–64. El Naschie MS. Dimensions and Cantor spectra. Chaos, Solitons & Fractals 1994;4(11):2121–32. El Naschie MS. Einstein’s dream and fractal geometry. Chaos, Solitons & Fractals 2005;24(1):1–5. El Naschie MS. The concepts of E infinity: an elementary introduction to the Cantorian-fractal theory of quantum physics. Chaos, Solitons & Fractals 2004;22(2):495–511. [9], [10], [11] and [12]] is based on random Cantor set which takes the golden mean dimension as shown by El Naschie.     

Hilbert's Tenth Problem for function fields of varieties over number fields and p-adic fields

Let k be a subfield of a p-adic field of odd residue characteristic, and let be the function field of a variety of dimension n1 over k. Then Hilbert's Tenth Problem for is undecidable. In particular, Hilbert's Tenth Problem for function fields of varieties over number fields of dimension 1 is undecidable.     

Codes over p-adic Numbers and Finite Rings Invariant under the Full Affine Group

Codes over p-adic numbers and over integers modulo p^d of block length p^m invariant under the full affine group AGL_m(F_p) are described.     

Generating Polynomials and Symmetric Tensor Decompositions

This paper studies symmetric tensor decompositions. For symmetric tensors, there exist linear relations of recursive patterns among their entries. Such a relation can be represented by a polynomial, which is called a generating polynomial. The homogenization of a generating polynomial belongs to the apolar ideal of the tensor. A symmetric tensor decomposition can be determined by a set of generating polynomials, which can be represented by a matrix. We call it a generating matrix. Generally, a symmetric tensor decomposition can be determined by a generating matrix satisfying certain conditions. We characterize the sets of such generating matrices and investigate their properties (e.g., the existence, dimensions, nondefectiveness). Using these properties, we propose methods for computing symmetric tensor decompositions. Extensive examples are shown to demonstrate the efficiency of proposed methods.     

Bounds of weighted multilinear Hardy-Cesàro operators in p-adic functional spaces

We introduce the p-adic weighted multilinear Hardy-Cesàro operator. We also obtain the necessary and sufficient conditions on weight functions to ensure the boundedness of that operator on the product of Lebesgue spaces, Morrey spaces, and central bounded mean oscillation spaces. In each case, we obtain the corresponding operator norms. We also characterize the good weights for the boundedness of the commutator of weighted multilinear Hardy-Cesàro operator on the product of central Morrey spaces with symbols in central bounded mean oscillation spaces.     

Indecomposable and noncrossed product division algebras over function fields of smooth p-adic curves

We construct indecomposable and noncrossed product division algebras over function fields of connected smooth curves X over Zp. This is done by defining an index preserving morphism which splits , where is the completion of K(X) at the special fiber, and using it to lift indecomposable and noncrossed product division algebras over .     

A New Method for Real Root Isolation of Univariate Polynomials

A new algorithm for real root isolation of univariate polynomials is proposed, which is mainly based on exact interval arithmetic and bisection method. Although exact interval arithmetic is usually supposed to be inefficient, our algorithm is surprisingly fast because the termination condition of our algorithm is different from those of existing algorithms which are mostly based on Descartes’ rule of signs or Vincent’s theorem and we decrease the times of Taylor shifts in some cases. We test our algorithm on a large number of examples from the literature and report the performance.      

Operator-semistable, operator semi-selfdecomposable probability measures and related nested classes on p-adic vector spaces

Let V be a finite dimensional p-adic vector space and let τ be an operator in GL(V). A probability measure μ on V is called τ-decomposable or m ? [(L)\tilde]₀(t)\mu\in {\tilde L}_0(\tau) if μ = τ(μ)* ρ for some probability measure ρ on V. Moreover, when τ is contracting, if ρ is infinitely divisible, so is μ, and if ρ is embeddable, so is μ. These two subclasses of [(L)\tilde]₀(t){\tilde L}_0(\tau) are denoted by L ₀(τ) and L ₀ ^#(τ) respectively. When μ is infinitely divisible τ-decomposable for a contracting τ and has no idempotent factors, then it is τ-semi-selfdecomposable or operator semi-selfdecomposable. In this paper, sequences of decreasing subclasses of the above mentioned three classes, [(L)\tilde]_m(t) é L_m(t) é L^#_m(t), 1 ￡ m ￡ ￥{\tilde L}_m(\tau)\supset L_m(\tau) \supset L^\#_m(\tau), 1\le m\le \infty , are introduced and several properties and characterizations are studied. The results obtained here are p-adic vector space versions of those given for probability measures on Euclidean spaces.     

The Group Ring ofSL2(p) over thep-adic Integers

This paper describes the ring-theoretic structure of the group rings ofSL₂(p²) over thep-adic integers.     

Finding a Cluster of Zeros of Univariate Polynomials

A method to compute an accurate approximation for a zero cluster of a complex univariate polynomial is presented. The theoretical background on which this method is based deals with homotopy, Newton's method, and Rouché's theorem. First the homotopy method provides a point close to the zero cluster. Next the analysis of the behaviour of the Newton method in the neighbourhood of a zero cluster gives the number of zeros in this cluster. In this case, it is sufficient to know three points of the Newton sequence in order to generate an open disk susceptible to contain all the zeros of the cluster. Finally, an inclusion test based on a punctual version of the Rouché theorem validates the previous step. A specific implementation of this algorithm is given. Numerical experiments illustrate how this method works and some figures are displayed.     

Banach空间中闭算子的多项式的谱分解性

Suppose X is a Banach space. The main results of this paper is as follows: Theorem. A closed operator T has the SDP if and only if for any complex polynomial p(λ) that is not constant, p(T) has the SDP.