A polyanalytic polynomial is a complex-valued function of the form
 |
where the 
are polynomials and 
is the complex conjugate
of 
(Sète and Zur 2024). Thus 
is polynomial in 
and 
.
If 
and 
when 
and 
are independent variables in 
,
then 
is its bidegree. This differs from the total polynomial
degree.
For 
, Lundberg et al. (2026) proved that
a finite zero set has at most
 |
distinct roots. The bound is asymptotically sharp as 
for each fixed 
.
The finiteness hypothesis excludes polynomials whose
zeros contain a curve. Lundberg et al. (2026) credit GPT-6.1 with suggesting
the proof method based on a topological bound on genus.
