We present an implemented algorithmic method for counting and isolating all p-adic roots of univariate polynomials f over the rational numbers. The roots of f are uniquely described by p-adic isolating balls, that can be refined to any desired precision; their p-adic distances are also computed precisely. The method is polynomial space in all input data including the prime p. We also investigat...

In this article, we present an effective encoding of dendrograms by embedding them into the Bruhat-Tits trees associated to p-adic number fields. As an application , we show how strings over a finite alphabet can be encoded in cyclotomic extensions of Qp and discuss p-adic DNA encoding. The application leads to fast p-adic agglomerative hierarchic algorithms similar to the ones recently used e....

We study the p-adic behavior of Jacobi sums for Q(ζp) and link this study to the p-Sylow subgroup of the class group of Q(ζp) + and to some properties of the jacobian of the Fermat curve Xp + Y p = 1 over Fl where l is a prime number distinct from p. Let p be a prime number, p ≥ 5. Iwasawa has shown that the p-adic properties of Jacobi sums for Q(ζp) are linked to Vandiver’s Conjecture (see [5]...

For ψ a nontrivial additive character on the finite field Fq, observe that the map t 7→ P x∈Fq ψ(f(x) + tx) is the Fourier transform of the map t 7→ ψ(f(t)). As is well-known, this has a cohomological interpretation, producing a continuous l-adic Galois representation. This paper studies the L-function attached to the k-th symmetric power of this representation using both l-adic and p-adic meth...

An effective p-adic encoding of dendrograms is presented through an explicit embedding into the Bruhat-Tits tree for a p-adic number field. This field depends on the number of children of a vertex and is a finite extension of the field of p-adic numbers. It is shown that fixing p-adic representatives of the residue field allows a natural way of encoding strings by identifying a given alphabet w...

We define exact functors from categories of Harish-Chandra modules for certain real classical groups to finite-dimensional modules over an associated graded affine Hecke algebra with parameters. We then study some of the basic properties of these functors. In particular, we show that they map irreducible spherical representations to irreducible spherical representations and, moreover, that they...

Rigged Hilbert space of the free coherent states is investigated. We prove that this rigged Hilbert space is isomorphous to the space of generalized functions over p–adic disk. We discuss the relation of the described isomorphism of rigged Hilbert spaces and noncommutative geometry and show, that the considered example realises the isomorphism of the noncommutative line and p–adic disk. In the ...

In the present work the notion of the computable (primitive recursive, polynomially time computable) p–adic number is introduced and studied. Basic properties of these numbers and the set of indices representing them are established and it is proved that the above defined fields are p–adically closed. Using the notion of a notation system introduced by Y. Moschovakis an abstract characterizatio...