Meromorphic solutions of equations over non-Archimedean fields

An, Ta Thi Hoai; Escassut, Alain

doi:10.1007/s11139-007-9086-9

Cited by 6 publications

(3 citation statements)

References 18 publications

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“…The paper is aimed at studying sufficient conditions assuring that if the composition of meromorphic functions of the form h•f and h•g are equal, then f and g are equal. This kind of problem follows many other problems of uniqueness studied in the past years, particularly on unique range sets with (or without) multiplicities and polynomials of uniqueness for analytic or meromorphic functions in the complex field and in an ultrametric field [1], [3], [13], [4], [6], [7], [8], [10], [21]. Polynomials of uniqueness were introduced and studied in l C by X.H Hua and C.C.…”

Section: Introduction and Basic Resultsmentioning

confidence: 99%

Meromorphic functions of uniqueness

Escassut

2007

Bulletin des Sciences Mathématiques

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Let E be an algebraically closed field of characteristic 0 which is either l C or a complete ultrametric field K. We consider the composition of meromorphic functions h • f where h is meromorphic in all E and f is meromorphic either in E or in an open disk of K. We then look for a condition on h in order that if 2 similar functions f, g satisfy h • f (a m ) = h • g(a m ) where (a m ) is a bounded sequence satisfying certain condition, this implies f = g. Particularly we generalize to meromorphic functions previous results on polynomials of uniqueness. The condition on h involves the zeros (c n ) of h ′ and the values h(c n ) but is weaker than this introduced by H.Fujimoto (injectivity on the set of zeros of h ′ ). The main tool is the Nevanlinna Theory but also involves some specific padic properties and basic affine properties. Results concerning p-adic entire functions only suppose a property involving 2 zeros of h ′ . Polynomials of uniqueness for entire functions are characterized. Every polynomial P of prime degree n ≥ 3 is a polynomial of uniqueness for p-adic entire functions, except if is of the form A(x−a) n +B. A polynomial P such that P ′ has exactly two distinct zeros is a polynomial of uniqueness for meromorphic functions in K if and only if both zeros have a multiplicity order greater than 1. Results on p-adic functions have applications to rational functions in any field of characteristic 0.

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Section: Introduction and Basic Resultsmentioning

confidence: 99%

Meromorphic functions of uniqueness

Escassut

2007

Bulletin des Sciences Mathématiques

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“…In order to improve results of [5] on p-adic meromorphic functions and of [6] on complex meromorphic functions, we have to state Propositions P1 and P2 derived from results of [3] and [4].…”

Section: Remarkmentioning

confidence: 99%

“…Let f, g ∈ M(K) be transcendental and let α ∈ M f (K) ∩ M g (K) be non-identically zero. If f P (f ) and g P (g) share α C.M., then f = g. We can check that P (X) = X 10 (X − 1) 5 (X + 1) 4 and…”

Section: Remarkmentioning

confidence: 99%

Complex and $p$-Adic Meromorphic Functions $f'P'(f)$, $g'P'(g)$ Sharing a Small Function

Escassut¹

2014

ATA

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Let K be a complete algebraically closed p-adic field of characteristic zero. We apply results in algebraic geometry and a new Nevanlinna theorem for p-adic meromorphic functions in order to prove results of uniqueness in value sharing problems, both on K and on C. Let P be a polynomial of uniqueness for meromorphic functions in K or C or in an open disk. Let f, g be two transcendental meromorphic functions in the whole field K or in C or meromorphic functions in an open disk of K that are not quotients of bounded analytic functions. We show that if f P (f) and g P (g) share a small function α counting multiplicity, then f = g, provided that the multiplicity order of zeros of P satisfy certain inequalities. A breakthrough in this paper consists of replacing inequalities n ≥ k + 2 or n ≥ k + 3 used in previous papers by Hypothesis (G). In the p-adic context, another consists of giving a lower bound for a sum of q counting functions of zeros with (q − 1) times the characteristic function of the considered meromorphic function. Notation and definition: Let K be an algebraically closed field of characteristic zero, complete with respect to an ultrametric absolute value |. |. We will denote by E a field that is either K or C. Throughout the paper we denote by a a point in K. Given R ∈]0, +∞[ we define disks d(a, R) = {x ∈ K | |x − a| ≤ R} and disks d(a, R −) = {x ∈ K | |x − a| < R}. A polynomial Q(X) ∈ E[X] is called a polynomial of uniqueness for a family of functions F defined in a subset of E if Q(f) = Q(g) implies f = g. The definition of polynomials of uniqueness was introduced in [19] by P. Li and C. C. Yang and was studied in many papers [11], [13], [20] for complex functions and in [1], [2], [9], [10], [17], [18], for p-adic functions. Throughout the paper we will denote by P (X) a polynomial in E[X] such that P (X) is of the form b l i=1 (X − a i) ki with l ≥ 2 and k 1 ≥ 2. The polynomial P will be said to satisfy Hypothesis (G) if P (a i) + P (a j) = 0 ∀i = j. We will improve the main theorems obtained in [5] and [6] with the help of a new hypothesis denoted by Hypothesis (G) and by thorougly examining the situation with p-adic and complex analytic and meromorphic functions in order to avoid a lot of exclusions. Moreover, we will prove a new theorem completing the 2nd Main Theorem for p-adic meromorphic functions. Thanks to this new theorem we will give more precisions in results on value-sharing problems.

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P-Adic Nevanlinna Theory Outside of a Hole

Escassut

2017

Vietnam J. Math.

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Let K be a complete ultrametric algebraically closed field of characteristic 0, take R > 0 and let D be the set {x ∈ K | |x| ≥ R}. Let M(D) be the field of meromorphic functions in D. We contruct a Nevanlinna Theory for M(D) and show many properties similar to those previously obtained with meromorphic functions in K or in an open disk. Functions have at most one Picard value and at most four branched values. The functions with finitely many poles in D have no Picard value and at most one branched value. URSCM and URSIM are similar to those in complex analysis. Fujimoto's way lets obtain polynomials of uniqueness for M(D) with a degree ≥ 5. Many algebraic curves admit no parametrization by functions of M(D). Motzkin Factors, known for analytic elements, here play an essential role.

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Meromorphic solutions of equations over non-Archimedean fields

Cited by 6 publications

References 18 publications

Meromorphic functions of uniqueness

Meromorphic functions of uniqueness

Complex and $p$-Adic Meromorphic Functions $f'P'(f)$, $g'P'(g)$ Sharing a Small Function

P-Adic Nevanlinna Theory Outside of a Hole

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