Timeline for Harmonic polynomials on complex 2-space
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Dec 16, 2012 at 22:25 | comment | added | Alexander Isaev | Robert Bryant: Thank you for your latest example. Indeed, you are right, and a simple perturbation of $f_0$ is a counterexample to what I asked. I have to think more about what conditions to impose. The three conditions that I stated come from the geometry of totally real embeddings of $S^3$ in ${\Bbb C}^3$. The fact that these conditions do not guarantee dependence on just $|z|$, $|w|$ means that such embeddings can have a more interesting structure than I previously thought. Once again, thank you very much for your help. | |
| Dec 16, 2012 at 14:53 | history | edited | Robert Bryant | CC BY-SA 3.0 |
added the construction of a new example
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| Dec 16, 2012 at 12:52 | comment | added | Alexander Isaev | Robert Bryant: Once again, thank you for your example and remarks. My objective is to find a reasonable sufficient condition for a complex-valued harmonic polynomial $f$ on ${\Bbb C}^2$ to be a function of $|z|$, $|w|$. I need that to study totally real embeddings of the 3-sphere into ${\Bbb C}^3$. I am now looking at the following three conditions: (i) $f(0)=0$, (ii) $f$ does not vanish on $S^3$, (iii) $f$ contains no purely anti-holomorphic terms. I hope that under these three assumptions $f$ is a function of $|z|$, $|w|$ (modulo a unitary change of $z,w$). | |
| Dec 15, 2012 at 14:38 | comment | added | Robert Bryant | @Alexander Isaev: By the way, the linear symmetry group of the polynomial $f$ that I wrote down is a copy of $\mathrm{O}(3)$, acting as the orthogonal transformations in the variables $x_1,x_2,x_3$. This symmetry group does not contain an abelian $2$-parameter symmetry group, so there is no linear change of variables that would bring it to being a polynomial in $|z|$ and $|w|$. | |
| Dec 15, 2012 at 13:59 | comment | added | Robert Bryant | @Alexander Isaev: I think your comment to Michael just above probably should have been a response to his comment to your question and not to my answer. You might consider moving it. As for 'dependent only on $|z|$ and $|w|$ (after a linear change of variables)', I wasn't sure what you were trying for, but I was suspecting that you wanted to know whether examples of the kind you seek must necessarily have a $2$-parameter symmetry group (such as being invariant under multiplication by unit complex numbers in two different lines, which is what happens for polynomials in $|z|$ and $|w|$). | |
| Dec 15, 2012 at 9:48 | comment | added | Alexander Isaev | Dear Robert, maybe you are right, and one cannot make a linear change to bring your example into the form dependent only on $|z|$, $|w|$. Maybe I should think more carefully about what exactly I want. For example, the polynomials I am dealing with do not have purely anti-holomorphic terms (when written in some complex coordinates), whereas if your example is rewritten through of complex variables, it will always contain anti-holomorphic terms (or so it seems). Thank you for your example anyway. | |
| Dec 15, 2012 at 2:24 | history | answered | Robert Bryant | CC BY-SA 3.0 |