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Vietnam Journal of Mathematics 37:2&3 (2009) 225-236 

A Note on Uniqueness Polynomials of Entire Functions

Ta Thi Hoai An and Julie Tzu-Yueh Wang

Abstract.  A complex polynomial P is a strong uniqueness polynomial for entire functions if one cannot find two distinct non-constant entire functions f> and g and a none-zero constant c such that P(f)=cP(g). It follows rather easily from Picard's theorem that P(X) is a strong uniqueness polynomial for entire functions if and only if none of the two variable polynomials P(X)-cP(Y) for all complex numbers c 0 have linear or quadratic factors except for the linear factor (X-Y) when c=1 (cf. W. Cherry and J. T.-Y. Wang, Uniqueness polynomials for entire functions, Inter. J. Math 13 (3) (2002), 323--332). In this note, we show that if P(X) is injective on the zeros of P'(X), then P(X) is a strong uniqueness polynomial for entire functions if and only if deg P 4 and none of the two variable polynomials P(X)-cP(Y) for all complex numbers c 0 have linear factors except for the linear factor (X-Y) when c=1.

2000 Mathematics Subject Classification: Primary 30D35, Secondary 30D05.

Keywords: Uniqueness polynomials, entire functions.

 

 

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