Author Topic: Must harmonic polynomial be homogeneous?  (Read 726 times)

Shentao YANG

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Must harmonic polynomial be homogeneous?
« 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 $deg=2$.

Victor Ivrii

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Re: Must harmonic polynomial be homogeneous?
« Reply #1 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.