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I am working with a function $F(z;a)$, for $z\in \mathbb{C}$ and $a$ being a set of parameters, from which I need to analyze the level set $\text{Re}(F(z))=0$ (for a fixed set of parameters $a$, which I will drop the notation now). The function $F$ is composed of elliptic functions and algebraic functions. To achieve my goal, I need to be able to say some concrete things about this level set. Here is what I know about the level sets right now:

  1. The real line is always part of this level set.

  2. The following symmetry exists: if $\text{Re}(F(z))=0$ then $\text{Re}(F(z^*))=0$ where $z^*$ is the complex conjugate of $z$.

  3. There are 4 points in $\mathbb{C}\setminus\mathbb{R}$ which are part of this level set (2 are conjugates of the other two). I can prove that the 2 conjugate points are endpoints of a "band" (arc) of the level set connecting the points.

  4. The curve parameterized by $\text{Re}(F(z))=0$ has four saddle points at which $\textrm{d}(\text{Re}(F(z)))/\textrm{d}z = 0$. For some values of $a$, two of these saddle points are on the real axis and I can show that the bands mentioned in the previous item intersect this point.

I need to know more about this level set, in particular I would like to answer the following questions:

  1. Can I prove that there are $n$ connected components of this level set? ($n$ might depend on $a$).
  2. Is there any (nonreal) component of this level set outside of, for instance, the unit ball?
  3. Is there reason to believe that any additional components of the set (not listed above) should be connected to those listed above ($\textit{i.e.}$ the real line or one of the 4 points)? Or, is there reason to believe that there should only be bands of the set branching off of the real line at saddle points?

Right now I am using software to get my hands on the geometry of this set. I have been looking at contour plots of $\text{Re}(F(z_r + i z_i))=0$ for different choices of $a$. The above questions are conjectures made from looking at different plots, but I know that the software misses things (for instance, it rarely shows the real line as being part of the desired set).

Help in the right direction would be appreciated, my knowledge of complex analysis theorems is woefully inadequate to accomplish my goal here. Any other ideas of what can be said (that I may have missed) would also be appreciated.

Edit: The expression was omitted originally because it is different in different contexts. By request, the expression for one of these cases has been added. It is:

$$F(z;a) = -2iz\omega_1 + 2\Gamma(\omega_1\zeta(\alpha;\Lambda) - \zeta(\omega_1;\Lambda)\alpha),$$ where $\Gamma = \pm 1$, $\zeta(\cdot;\Lambda)$ is the Weierstrass-zeta function defined on a rectangular lattice ($\omega_1\in \mathbb{R}, \omega_3\in i\mathbb{R}$) and $$ \alpha = \wp^{-1}(f(z;a); \Lambda)$$ where $f(z;a)$ has 4 branch points (corresponding to the 4 points mentioned in (3) above), has poles at $z=\infty$ and may also have poles at $z=0$. In general, $f(z;a)$ has points $\hat z$ and $\hat a$ such that $f(\hat z; \hat a) = 0$.

Jeremy Upsal
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  • Where do the parameters exist? – Marra May 03 '18 at 21:26
  • The parameters $a$ are in an open subset of $\mathbb{R}^2$. – Jeremy Upsal May 04 '18 at 20:29
  • Is there any reason not to tell us what this $;F(z;a);$ function is? Is it analytic? Does it have poles? – Somos May 04 '18 at 22:50
  • No real reason, only because it's a little bit messy and has a different form in different cases. I will edit the question now to include this. There are poles at infinity. – Jeremy Upsal May 05 '18 at 01:59
  • What is $\zeta_w(\alpha)$? The Weierstrass $\zeta$ function I know is denoted $\zeta(z;\Lambda)$ where $z$ is complex and $\Lambda$ is a lattice. Similarly, in $\wp^{-1}$ you have not specified the lattice. – Somos May 08 '18 at 13:13
  • @Somos the notation has been changed to fit the convention you are comfortable with. – Jeremy Upsal May 08 '18 at 16:18
  • The level set of a real analytic function $f:\mathbb{R}^n\to\mathbb{R}$ is stratified in countable union of (analytic) manifolds. This is called Lojasiewicz's structure theorem, and you can find the proof of it in "Krantz, Parks, A primer in real analytic functions" pages 168-170. Maybe you can view $\mathbb{C}$ as $\mathbb{R}^2$ and work use the structure theorem. However it really depends on what you really need. – Diesirae92 May 23 '18 at 10:52

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