Must harmonic polynomial be homogeneous?

[Shentao YANG]

Why harmonic polynomial of $deg=n$ must also be homogeneous polynomial of $deg=n$?
Say, $\Delta ({x^2} - {y^2} + z) = 2 - 2 + 0 = 0$, but we do not count $ ({x^2} - {y^2} + z)$ as harmonic polynomial of $deg=2$.

[Victor Ivrii]

In general NO, but we are looking at homogeneous polynomials. Obviously if a polynomial is harmonic, then all it homogeneous components are also are harmonic.