# Toronto Math Forum

## APM346-2016F => APM346--Lectures => Chapter 8 => Topic started by: Shentao YANG on November 12, 2016, 10:05:03 PM

Title: Must harmonic polynomial be homogeneous?
Post by: Shentao YANG on November 12, 2016, 10:05:03 PM
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 $deg=2$.
Title: Re: Must harmonic polynomial be homogeneous?
Post by: Victor Ivrii on November 13, 2016, 10:08:33 AM
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.