Volume function and Mahler measure of exact polynomials

We study a class of two-variable polynomials called exact polynomials which contains $A$ -polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the two-dimensional torus and give a closed formula for the Mahler measure in terms of these extremal values. This formula shows that the Mahler measure of an irreducible and exact polynomial divided by $\pi$ is greater than the amplitude of the volume function. We also prove a K-theoretic criterion for a polynomial to be a factor of an $A$ -polynomial and give a topological interpretation of its Mahler measure.

Anisotropic tubular neighborhoods of sets

Let $$E \subset {{\mathbb {R}}}^N$$ E ⊂ R N be a compact set and $$C\subset {{\mathbb {R}}}^N$$ C ⊂ R N be a convex body with $$0\in \mathrm{int}\,C$$ 0 ∈ int C . We prove that the topological boundary of the anisotropic enlargement $$E+rC$$ E + r C is contained in a finite union of Lipschitz surfaces. We also investigate the regularity of the volume function $$V_E(r):=|E+rC|$$ V E ( r ) : = | E + r C | proving a formula for the right and the left derivatives at any $$r>0$$ r > 0 which implies that $$V_E$$ V E is of class $$C^1$$ C 1 up to a countable set completely characterized. Moreover, some properties on the second derivative of $$V_E$$ V E are proved.