Ta Thi Hoai An

Strong uniqueness polynomials: The complex case

The theory of strong uniqueness polynomials, satisfying the separation condition (first introduced by Fujimoto [H. Fujimoto (2000). On uniqueness of meromorphic functions sharing finite sets. Amer. J. Math., 122, 1175–1203.]), for complex meromorphic functions is quite complete. We construct examples of strong uniqueness polynomials which do not necessary satisfy the separation condition by constructing regular 1-forms of Wronskian type, a method introduced in Ref. [T.T.H. An, J.T.-Y. Wang and P.-M. Wong. Unique range sets and uniqueness polynomials in positive characteristic. Acta Arith. (to appear).] We also use this method to produce a much easier proof in establishing the necessary and sufficient conditions for a polynomial, satisfying the separation condition, to be a strong uniqueness polynomials for meromorphic functions and rational functions.

A defect relation for non-Archimedean analytic curves in arbitrary projective varieties

If f is a non-Archimedean analytic curve in a projective variety X embedded in P N and if D 1 ,..., Dq are hypersurfaces of P N in general position with X, then we prove the defect relation: qΣj=1 δ(f,D j ) ≤ dim X.

UNIQUENESS POLYNOMIALS FOR COMPLEX MEROMORPHIC FUNCTIONS

A polynomial P(X) in is called a strong uniqueness polynomial for meromorphic functions if whenever there exist two non-constant meromorphic functions f and g and a complex non-zero constant c such that P(f) = cP(g), then we must have f = g. In this paper, we give a necessary and sufficient condition for a polynomial to be a strong uniqueness polynomial for meromorphic functions under the assumption that P(X) is injective on the roots of P′(X) = 0.

Algebraic degeneracy of non-archimedean analytic maps

Abstract We prove non-Archimedean analogs of results of Noguchi and Winkelmann showing algebraic degeneracy of rigid analytic maps to projective varieties omitting an effective divisor with sufficiently many irreducible components relative to the rank of the group they generate in the Neron-Severi group of the variety.

Unique range sets and uniqueness polynomials for algebraic curves

We study unique range sets and uniqueness polynomials for algebraic functions on a smooth projective algebraic curve over an algebraically closed field of characteristic zero.

Genus one factors of curves defined by separated variable polynomials

Abstract We give some sufficient conditions on complex polynomials P and Q to assure that the algebraic plane curve P ( x ) − Q ( y ) = 0 has no irreducible component of genus 0 or 1. Moreover, if deg ( P ) = deg ( Q ) and if both P, Q satisfy Hypothesis I introduced by H. Fujimoto, our sufficient conditions are necessary.

New Applications of the p-Adic Nevanlinna Theory

Let IK be an algebraically closed field of characteristic 0 complete for an ultrametric absolute value. Following results obtained in complex analysis, here we examine problems of uniqueness for meromorphic functions having finitely many poles, sharing points or a pair of sets (C.M. or I.M.) defined either in the whole field IK or in an open disk, or in the complement of an open disk. Following previous works in C, we consider functions fn(x)fm(ax + b), gn(x)gm(ax + b) with |a| = 1 and n ≠ m, sharing a rational function and we show that f/g is a n + m-th root of 1 whenever n + m ≥ 5. Next, given a small function w, if n, m ∈ IN are such that |n − m|∞ ≥ 5, then fn(x)fm(ax + b) − w has infinitely many zeros. Finally, we examine branched values for meromorphic functions fn(x)fm(ax + b).

Supplement and Erratum to “Algebraic degeneracy of non-Archimedean analytic maps” [Indag. Math. (N.S.) 19 (2008) 481–492]

We prove non-Archimedean analogs of results of Noguchi and Winkelmann showing algebraic degeneracy of rigid analytic maps to projective varieties omitting an effective divisor with sufficiently many irreducible components relative to the rank of the group they generate in the Néron-Severi group of the variety.

HEIGHTS OF FUNCTION FIELD POINTS ON CURVES GIVEN BY EQUATIONS WITH SEPARATED VARIABLES

Let P and Q be polynomials in one variable over an algebraically closed field k of characteristic zero. Let f and g be elements of a function field K over k such that P(f) = Q(g). We give conditions on P and Q such that the height of f and g can be effectively bounded, and moreover, we give sufficient conditions on P and Q under which f and g must be constant.