Timeline for Harmonic polynomials on complex 2-space
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Dec 14, 2012 at 18:32 | comment | added | Alexander Isaev | Alexandre Eremenko, a non-constant entire function (for example, $z+iw$) cannot have a non-empty compact zero set, so if a holomorphic polynomial vanishes at the origin, it also vanishes somewhere on the unit sphere. The question that I asked above is an analogue of this statement for harmonic polynomials. In a sense, harmonic polynomials that are not functions of $|z|$, $|w|$ (after a suitable linear change of the real coordinates) should behave like holomorphic functions: their zero sets should be non-compact. I hope this explains my motivation for the question. | |
| Dec 14, 2012 at 16:49 | comment | added | Alexander | Dima, the main condition is the condition of non-vanishing on the sphere. That rules out a lot of homogeneous polynomials. | |
| Dec 14, 2012 at 16:44 | comment | added | Alexander | Yes, Michael is right, $z+iw$ does vanish on the sphere. Certainly, my question must be understood up to a linear change of the real coordinates. Otherwise, one can take a function of $|z|,|w|$ on-vanishing on the sphere and then simply swap, say, the real part of $z$ with the imaginary part of $w$. The resulting function will not be a function of $|z|,|w|$ any more. So, my question needs to be understood modulo such things. | |
| Dec 14, 2012 at 15:35 | comment | added | Michael Renardy | $z=1/\sqrt{2}$, $w=i/\sqrt{2}$. | |
| Dec 14, 2012 at 14:18 | history | answered | Alexandre Eremenko | CC BY-SA 3.0 |