Vol 9, No 4 (2017)

In this paper, we compute the p-adic valuation of exponential sums associated to trinomials \(F\left( X \right) = a{X^{{d_1}}} + b{X^{{d_2}}} + c{X^{{d_3}}}\) over Fp. As a byproduct of our results, we obtain restrictions for permutation polynomials of type \(a{X^{{d_1}}} + b{X^{{d_2}}} + c{X^{{d_3}}}\) over F_p.

In present work we investigate one class of nonlinear integral equations with singularity at zero and boundary value conditions at ±∞. Above mentioned class of equations has direct applications in string theory and in the case of concrete structure of the kernel it describes the dynamics of the open-closed p-adic string for the scalar tachyon field. We prove the existence of nontrivial solution in a certain weight class of functions.With an additional restriction on the kernel the uniqueness of the obtained solution is proved.

In this articlewe introduce a new type of local zeta functions and study some connections with pseudodifferential operators in the framework of non-Archimedean fields. The new local zeta functions are defined by integrating complex powers of norms of polynomials multiplied by infinitely pseudo-differentiable functions. In characteristic zero, the new local zeta functions admit meromorphic continuations to the whole complex plane, but they are not rational functions. The real parts of the possible poles have a description similar to the poles of Archimedean zeta functions. But they can be irrational real numbers while in the classical case are rational numbers. We also study, in arbitrary characteristic, certain connections between local zeta functions and the existence of fundamental solutions for pseudodifferential equations.