Investigating a problem posed by W. Hengartner (2000), we study the maximal valence (number of preimages of a prescribed point in the complex plane) of logharmonic polynomials, i.e., complex functions that take the form
f
(
z
)
=
p
(
z
)
q
(
z
)
¯
f(z) = p(z) \overline {q(z)}
of a product of an analytic polynomial
p
(
z
)
p(z)
of degree
n
n
and the complex conjugate of another analytic polynomial
q
(
z
)
q(z)
of degree
m
m
. In the case
m
=
1
m=1
, we adapt an indirect technique utilizing anti-holomorphic dynamics to show that the valence is at most
3
n
−
1
3n-1
. This confirms a conjecture of Bshouty and Hengartner (2000). Using a purely algebraic method based on Sylvester resultants, we also prove a general upper bound for the valence showing that for each
n
,
m
≥
1
n,m \geq 1
the valence is at most
n
2
+
m
2
n^2+m^2
. This improves, for every choice of
n
,
m
≥
1
n,m \geq 1
, the previously established upper bound
(
n
+
m
)
2
(n+m)^2
based on Bezout’s theorem. We also consider the more general setting of polyanalytic polynomials where we show that this latter result can be extended under a nondegeneracy assumption.