# On the arithmetic and the geometry of skew-reciprocal polynomials

## TL;DR

This paper reformulates classical questions about Mahler measures and polynomial houses in terms of skew-reciprocal polynomials, linking these problems to the complexity of mapping class orientation types.

## Contribution

It introduces a new perspective by connecting Lehmer's and Schinzel-Zassenhaus questions to skew-reciprocal polynomials and mapping class orientation, offering a novel approach.

## Key findings

- Reformulation of Lehmer's question in terms of skew-reciprocal polynomials
- Connection between Mahler measures and polynomial houses
- Link between polynomial properties and mapping class orientation complexity

## Abstract

We reformulate Lehmer's question from 1933 and a question due to Schinzel and Zassenhaus from 1965 in terms of a comparison of the Mahler measures and the houses, respectively, of monic integer reciprocal and skew-reciprocal polynomials of the same degree. This entails that understanding the difference between orientation-preserving and orientation-reversing mapping classes is at least as complicated as answering these questions.