The authors have shown recently that the canonical p-henselian valuation is uniformly ∅-definable in the elementary class of fields which have characteristic p or contain a primitive pth root of unity ζp. In order to do this, we proved a classification of the canonical p-henselian valuation via case distinction. One of the cases discussed was previously not even known for algebraic extensions o...

Let p be a fixed odd prime number. Throughout this paper, Zp, Qp, C, and Cp, will, respectively, denote the ring of p-adic integers, the field, of p-adic rational numbers, the complex number field and the completion of algebraic closure of Qp. Let νp be the normalized exponential valuation of Cp such that |p|p p−νp p 1/p see 1–16 . As well-known definition, the Euler numbers and Genocchi number...

Let p be a fixed prime number. Throughout this paper, the symbols Z, Zp, Qp, and Cp denote the ring of rational integers, the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure ofQp, respectively. LetN be the set of natural numbers andZ N∪{0}. Let νp be the normalized exponential valuation ofCp with |p|p p−νp p p−1. Let UD Zp be the space of u...

Let p be a fixed prime number. Throughout this paper, the symbols Z, Zp, Qp, and Cp denote the ring of rational integers, the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp, respectively. Let N be the set of natural numbers. The normalized valuation in Cp is denoted by | · |p with |p|p 1/p. Let UD Zp be the space of uniformly differe...

Let p be a fixed odd prime number. Throughout this paper Zp, Qp, C and Cp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field and the completion of algebraic closure of Qp. Let vp be the normalized exponential valuation of Cp with |p|p = p −vp(p) = 1 p . When one talks of q-extension, q is variously considered as and in...

In this paper we define the p-density of a finite subset D ⊂ Nr , and show that it gives a good lower bound for the p-adic valuation of exponential sums over finite fields of characteristic p. We also give an application: when r = 1, the p-density is the first slope of the generic Newton polygon of the family of Artin-Schreier curves associated to polynomials with their exponents in D. 0. Intro...

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gröbner bases taking into account the valuation of K. While generalizing the classical theory of Gröbner bases, it is not clear how modern algorithms for computing Gröbner bases can be adapted to the tropical case. Among them, one of the most efficient is the celebrated F5 Algorithm of Faugère....

In this article we prove a Grothendieck trace formula for L-functions of not necessarily commutative adic sheaves.

We compute the p-adic regulator of cyclic cubic extensions of Q with discriminant up to 1016 for 3 < p < 100, and observe the distribution of the p-adic valuation of the regulators. We find that for almost all primes the observation matches the model that the entries in the regulator matrix are random elements with respect to the obvious restrictions. Based on this random matrix model a conject...