Norms of Roots of Trinomials

TL;DR

This paper characterizes the geometric and topological structure of the space of univariate trinomials with fixed support, revealing connections to amoeba theory and torus knots, and analyzing roots of specific norms.

Contribution

It provides a geometric and topological description of the coefficient space of trinomials using amoeba theory and relates it to torus knots, extending classical algebraic results.

Findings

Roots of fixed norm are parameterized by a hypotrochoid curve.

The space of trinomials with certain roots is deformation retracted to a torus knot.

The topology of the coefficient space varies depending on the roots' norms.

Abstract

The behavior of norms of roots of univariate trinomials $z^{s+t} + p z^t + q \in \mathbb{C}[z]$ for fixed support $A = \{0,t,s+t\} \subset \mathbb{N}$ with respect to the choice of coefficients $p,q \in \mathbb{C}$ is a classical late 19th and early 20th century problem. Although algebraically characterized by P.\ Bohl in 1908, the geometry and topology of the corresponding parameter space of coefficients had yet to be revealed. Assuming $s$ and $t$ to be coprime we provide such a characterization for the space of trinomials by reinterpreting the problem in terms of amoeba theory. The roots of given norm are parameterized in terms of a hypotrochoid curve along a $\mathbb{C}$-slice of the space of trinomials, with multiple roots of this norm appearing exactly on the singularities. As a main result, we show that the set of all trinomials with support $A$ and certain roots of identical norm, as well as its complement can be deformation retracted to the torus knot $K(s+t,s)$, and thus are connected but not simply connected. An exception is the case where the $t$-th smallest norm coincides with the $(t+1)$-st smallest norm. Here, the complement has a different topology since it has fundamental group $\mathbb{Z}^2$.