Julie Tzu-Yueh Wang

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.

Truncated second main theorem with moving targets

We prove a truncated Second Main Theorem for holomorphic curves intersecting a finite set of moving or fixed hyperplanes. The set of hyperplanes is assumed to be non-degenerate. Previously only general position or subgeneral position was considered.

UNIQUENESS POLYNOMIALS FOR ENTIRE FUNCTIONS

A polynomial P is called a strong uniqueness polynomial for the family of non-constant entire functions on the complex plane if one cannot find two distinct non-constant entire functions f and g and a non-zero constant c such that P(f)=cP(g). We give necessary and sufficient geometric conditions on the divisor of zeros of P that P be a strong uniqueness polynomial for the family of non-constant entire functions on the complex plane.

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.

The ABC theorem for higher-dimensional function fields

We generalize the ABC theorems to the function field of a variety over an algebraically closed field of arbitrary characteristic which is non-singular in codimension one. We also obtain an upper bound for the minimal order sequence of Wronskians over such function fields of positive characteristic.

HENSLEY'S PROBLEM FOR FUNCTION FIELDS

Buchis square problem asks if there exists a positive integer M such that all x1, …, xM ∈ ℤ satisfying the equations for all 3 ≤ r ≤ M must also satisfy for some integer x and for all 1 ≤ r ≤ M. Hensleys problem asks if there exists a positive integer M such that, for any integers ν and a, if (ν + r)2 - a is a square for all 1 ≤ r ≤ M, then a = 0. It is not difficult to see that a positive answer to Hensleys problem implies a positive answer to Buchis square problem. One can ask a more general version of Hensleys problem by replacing the square power by an nth power for any integer n ≥ 2 which is called Hensleys problem for nth powers. In this paper, we will study Hensleys problem for nth powers over function fields of any characteristic.

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.

Generalizations of rigid analytic Picard theorems

Berkovichs Picard theorem states that there are no non-constant analytic maps from the affine line to the complement of two points on a non-singular projective curve. The purpose of this article is to find generalizations of this result in higher dimensional varieties.

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.

On Non-Archimedean Curves Omitting Few Components and their Arithmetic Analogues

Let k be an algebraically closed field complete with respect to a non-Archimedean absolute value of arbitrary characteristic. Let D_1,...,D_n be effective nef divisors intersecting transversally in an n-dimensional nonsingular projective variety X. We study the degeneracy of non-Archimedean analytic maps from k into