Decidability of Univariate Real Algebra with Predicates for Rational and Integer Powers

We prove decidability of univariate real algebra extended with predicates for rational and integer powers, i.e., $(x^n \in \mathbb{Q})$ and $(x^n \in \mathbb{Z})$. Our decision procedure combines computation over real algebraic cells with the rational root theorem and witness construction via algebraic number density arguments.Comment: To appear in CADE-25: 25th International Conference on Automated Deduction, 2015. Proceedings to be published by Springer-Verla

Integer Points in Backward Orbits

A theorem of J. Silverman states that a forward orbit of a rational map $\phi(z)$ on $\mathbb P^1(K)$ contains finitely many $S$-integers in the number field $K$ when $(\phi\circ\phi)(z)$ is not a polynomial. We state an analogous conjecture for the backward orbits using a general $S$-integrality notion based on the Galois conjugates of points. This conjecture is proven for the map $\phi(z)=z^d$, and consequently Chebyshev polynomials, by uniformly bounding the number of Galois orbits for $z^n-\beta$ when $\beta\not =0$ is a non-root of unity. In general, our conjecture is true provided that the number of Galois orbits for $\phi^n(z)-\beta$ is bounded independently of $n$.Comment: 13 page

On the unification of quantum 3-manifold invariants

In 2006 Habiro initiated a construction of generating functions for Witten-Reshetikhin-Turaev (WRT) invariants known as unified WRT invariants. In a series of papers together with Irmgard Buehler and Christian Blanchet we extended his construction to a larger class of 3-manifolds. The unified invariants provide a strong tool to study properties of the whole collection of WRT invariants, e.g. their integrality, and hence, their categorification. In this paper we give a survey on ideas and techniques used in the construction of the unified invariants.Comment: 18 page