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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.
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