On the geometry of real polynomials of the 7-th degree with simple complex critical points

Abstract

We consider real monic polynomials with simple complex critical points. Critical values of such polynomial $p$ can be considered as vertices of the uniquely defined polygon, symmetric with respect to the $x$-axis. The inverse image of this polygon is a tree-like system of ovals that touch each other externally in critical points. This system of ovals will be called the geometrical code of $p$. Let $a_1+i\cdot b_1,\ldots,a_n+i\cdot b_n$, $a_1<a_2<\ldots<a_n$ and $b_i>0$, $i=1,\ldots,n$, be the set of critical points of our polynomial $p$ in the upper halfplane, and $c_1+i\cdot d_1,\ldots,c_n+i\cdot d_n$ be their images. The order of numbers $c_k$ and the list of signs of numbers $d_k$ will be called the algebraic code of $p$. In our experimental work we study geometrical and algebraic codes of the 7-th degree polynomials and the relations between them.