Cartan’s method is used to prove a several variable, non-Archimedean, Nevanlinna Second Main Theorem for hyperplanes in projective space. The corresponding defect relation is derived, but unlike in the complex case, we show that there can only be finitely many non-zero non-Archimedean defects. We then address the non-Archimedean Nevanlinna inverse problem, by showing that given a set of defects...

Certain estimates involving the derivative f 7→ f ′ of a meromorphic function play key roles in the construction and applications of classical Nevanlinna theory. The purpose of this study is to extend the usual Nevanlinna theory to a theory for the exact difference f 7→ ∆f = f(z+ c)− f(z). An a-point of a meromorphic function f is said to be c-paired at z ∈ C if f(z) = a = f(z+c) for a fixed co...

In this paper we obtain the discrete integrable self-dual network hierarchy associated with a spectral problem. On basis of theory algebraic curves, continuous flow and related to are straightened using Abel-Jacobi coordinates. The meromorphic function Baker-Akhiezer introduced on hyperelliptic curve. Quasi-periodic solutions constructed help asymptotic properties algebra-geometric characters f...

Abstract. Concerning the uniqueness and sharing values of meromorphic functions, many results about meromorphic functions that share more than or equal to two values have been obtained. In this paper, we shall study meromorphic functions that share only one value, and prove the following result: For n ≥ 11 and two meromorphic functions f(z) and g(z) , if ff ′ and gg share the same nonzero and f...

Let K be a p-adic field. We explore Igusa’s p-adic zeta function, which is associated to a K-analytic function on an open and compact subset of Kn. First we deduce a formula for an important coefficient in the Laurent series of this meromorphic function at a candidate pole. Afterwards we use this formula to determine all values less than −1/2 for n = 2 and less than −1 for n = 3 which occur as ...

The Nevanlinna defect relation and other results on the Nevanlinna theory assert that each meromorphic mapping f of C into P(C) has few deficient hyperplanes in P(C) . However, it seems to me that meromorphic mappings with a deficient hyperplane must be very few. In this paper, we show that for an arbitrary given transcendental meromorphic mapping f (which may be linearly degenerate), we can el...

By definition, an autonomous function is a differentially algebraic function Jon iw (or on C), every translate J, of which satisfies every algebraic differential equation that J satisfies. We find several equivalent formulations of the property of being autonomous. Our main result is that if Jis differentially algebraic and meromorphic in the full complex plane, and if g is an autonomous entire...

Let q 1 , q 2 , ..., q N be the coordinates of N particles on the circle, interacting with the integrable potential N j<k ℘(q j − q k), where ℘ is the Weierstrass elliptic function. We show that every symmetric elliptic function in q 1 , q 2 , ..., q N is a meromorphic function in time. We give explicit formulae for these functions in terms of genus N − 1 theta functions.