Rudin's homogeneous basis problem asks whether the Hardy space 
on the unit ball of 
, for fixed 
, has an orthonormal
basis of homogeneous polynomials with
a common uniform bound on the boundary of the ball.
The polynomials must be holomorphic
functions, and the inner product uses normalized surface measure on the boundary.
The bound must be independent of both the polynomial's degree and its position in
the basis.
Gonçalves et al. (2026) and Wang et al. (2026) gave affirmative solutions in every fixed dimension. Wang et al.
(2026) obtained a basis 
satisfying
 |
where 
is big-O notation and the implicit constant is
independent of 
. Wang et al. (2026) report ChatGPT assistance with
mathematical methods and with checking intermediate computations and estimates.
