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.
