Let $f: \mathbb{C} \to \mathbb{C}$ be a polynomial and let $\arg(f(z))$ be the phase of $f(z) = | f(z)| \exp(\mathrm{i} \arg(f(z)))$. The zeroes of $f'(z)$ are saddle points of $\arg(f(z))$, i.e. intersections of isochromatic lines (if we color-encode the phase, cf. http://www.ams.org/notices/201106/rtx110600768p.pdf, page 772ff). In the following we assume $f$ to have only simple zeroes.

** Questions: **

a1) Is it correct that through every (generic) saddle point of $\arg(f)$ (= zero of $f’$) there are two isochromatic lines, one of which is unbounded and one of which connects two zeroes of $f$?

a2) If a1) is correct: Is there a simple (geometric) criterion determining which zeroes are directly connected via an isochromatic line?

b1) ** Is it correct that the set of isochromatic lines which connect two zeroes define a spanning tree (of all zeroes)? **

b2) If b1) is correct: Can the (graph theoretical, i.e. topolocial) structure of the spanning tree be solely defined by the position of the zeroes of $f$ (e.g. minimal weigth spanning tree where weight is square of Euclidean distance of zeroes)?

The figure shows the color-encoded phase plot of an example of a polynomial of degree four (the thin black lines are lines of equal absolute value, they are orthogonal to the isochromatic lines). The four zeroes of $f$ can be seen as the minima of the absolute value. The isochromatic lines through the zeroes of $f’$ are the thick black lines. The three chromatic saddle points (=zeroes of $f’$) are clearly visible. In this example the spanning tree has one central vertex (the zero of $f$ below the middle) from which there are edges to the other three zeroes of $f$.

An electrostatic interpretation could possibly be helpful. If we consider the zeroes of $f$ to be equal charges in 2d, the question amounts to how can one know which charges are connected via a field line of the electrostatic field.

** Motivation:**

Every complex polynomial with $n$ distinct zeroes has $n-1$ critial points (zeroes of $f’$). For the location of the zeroes there is huge literature (for a starting point see e.g. Q.I. Rahman und G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press, 2002).

From numerical studies it seems to be interesting to ask which spanning tree of the zeroes could be helpful in describing the location of the critical points (note that the spanning tree through $n$ vertices has $n-1$ edges).

The question was inspired by Alth\“ofer/Voigt, Spiele, R\“atsel, Zahlen, Springer Spektrum 2014 and Kaffka, Zuordnungen zwischen Nullstellen und kritischen Punkten von Polynomen, Diploma Thesis supervised by Prof. Dr. I. Alth\"ofer, Jena, 2012, page 42, http://www.althofer.de/kaffka-diplom.pdf.

In this context I have proven a slight strengthening of the theorem of Gauss-Lucas for polynomials of degree four (http://arxiv.org/abs/1405.0689).