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

Ricci-flat Metrics with U(1) Action and the Dirichlet Boundary-value Problem in Riemannian Quantum Gravity and Isoperimetric Inequalities

The Dirichlet boundary-value problem and isoperimetric inequalities for positive definite regular solutions of the vacuum Einstein equations are studied in arbitrary dimensions for the class of metrics with boundaries admitting a U(1) action. We show that in the case of non-trivial bundles Taub-Bolt infillings are double-valued whereas Taub-Nut and Eguchi-Hanson infillings are unique. In the case of trivial bundles, there are two Schwarzschild infillings in arbitrary dimensions. The condition of whether a particular type of filling in is possible can be expressed as a limitation on squashing through a functional dependence on dimension in each case. The case of the Eguchi-Hanson metric is solved in arbitrary dimension. The Taub-Nut and the Taub-Bolt are solved in four dimensions and methods for arbitrary dimension are delineated. For the case of Schwarzschild, analytic formulae for the two infilling black hole masses in arbitrary dimension have been obtained. This should facilitate the study of black hole dynamics/thermodynamics in higher dimensions. We found that all infilling solutions are convex. Thus convexity of the boundary does not guarantee uniqueness of the infilling. Isoperimetric inequalities involving the volume of the boundary and the volume of the infilling solutions are then investigated. In particular, the analogues of Minkowski's celebrated inequality in flat space are found and discussed providing insight into the geometric nature of these Ricci-flat spaces.Comment: 40 pages, 3 figure

On the reducibility type of trinomials

Say a trinomial x^n+A x^m+B \in \Q[x] has reducibility type $(n_1,n_2,...,n_k)$ if there exists a factorization of the trinomial into irreducible polynomials in \Q[x] of degrees $n_1$, $n_2$,...,$n_k$, ordered so that $n_1 \leq n_2 \leq ... \leq n_k$. Specifying the reducibility type of a monic polynomial of fixed degree is equivalent to specifying rational points on an algebraic curve. When the genus of this curve is 0 or 1, there is reasonable hope that all its rational points may be described; and techniques are available that may also find all points when the genus is 2. Thus all corresponding reducibility types may be described. These low genus instances are the ones studied in this paper.Comment: to appear in Acta Arithmetic