Author Topic: Must harmonic polynomial be homogeneous?  (Read 2071 times)

Shentao YANG

  • Full Member
  • ***
  • Posts: 24
  • Karma: 0
    • View Profile
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

  • Administrator
  • Elder Member
  • *****
  • Posts: 2607
  • Karma: 0
    • View Profile
    • Personal website of Victor Ivrii
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.