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.
