Abstract. A complex polynomial P is a
strong uniqueness polynomial for entire functions if one cannot find two
distinct nonconstant entire functions f> and g and a
nonezero 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 (XY)
when c=1 (cf. W. Cherry and J. T.Y. Wang, Uniqueness polynomials
for entire functions, Inter. J. Math 13 (3) (2002),
323332). 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 (XY) when c=1.
