Norms of Roots of Trinomials

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 $\mathbb{Z}^2$.Comment: Minor revision, final version, 28 pages, 8 figure

A Polyhedral Homotopy Algorithm For Real Zeros

We design a homotopy continuation algorithm, that is based on numerically tracking Viro's patchworking method, for finding real zeros of sparse polynomial systems. The algorithm is targeted for polynomial systems with coefficients satisfying certain concavity conditions. It operates entirely over the real numbers and tracks the optimal number of solution paths. In more technical terms; we design an algorithm that correctly counts and finds the real zeros of polynomial systems that are located in the unbounded components of the complement of the underlying A-discriminant amoeba.Comment: some cosmetic changes are done and a couple of typos are fixed to improve readability, mathematical contents remain unchange

Intersections of Amoebas

Amoebas are projections of complex algebraic varieties in the algebraic torus under a Log-absolute value map, which have connections to various mathematical subjects. While amoebas of hypersurfaces have been intensively studied in recent years, the non-hypersurface case is barely understood so far. We investigate intersections of amoebas of $n$ hypersurfaces in $(\mathbb{C}^*)^n$, which are canonical supersets of amoebas given by non-hypersurface varieties. Our main results are amoeba analogs of Bernstein's Theorem and B\'ezout's Theorem providing an upper bound for the number of connected components of such intersections. Moreover, we show that the \emph{order map} for hypersurface amoebas can be generalized in a natural way to intersections of amoebas. In particular, analogous to the case of amoebas of hypersurfaces, the restriction of this generalized order map to a single connected component is still $1$-to-$1$.Comment: Revision; Appendix added; 26 pages, 5 figure