The Kroó homogeneous polynomial approximation conjecture states that every continuous real function on the boundary of an origin-symmetric convex
body in 
is uniformly approximable
by sums of homogeneous polynomials of degrees
and 
as 
. The body has nonempty interior,
but no smoothness or curvature hypothesis is imposed (Kroó 2023).
Colbrook (2026) proved the conjecture. Even functions can be approximated by homogeneous polynomials
of degree 
, and odd functions by homogeneous polynomials of degree 
. The combination of consecutive degrees permits approximation
of a general function, since homogeneous polynomials
of a fixed degree have a fixed parity under 
. Colbrook (2026) reports ChatGPT assistance, through
Codex, with the accompanying Lean formalization.
