On the arithmetic and the geometry of skew-reciprocal polynomials

Liechti, Livio

doi:10.1090/proc/14668

On the arithmetic and the geometry of skew-reciprocal polynomials
HTML articles powered by AMS MathViewer

by Livio Liechti PDF

Proc. Amer. Math. Soc. 147 (2019), 5131-5139 Request permission

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 understanding that the difference between orientation-preserving and orientation-reversing mapping classes is at least as complicated as answering these questions.

References

R. Ackermann, Achievable spectral radii of symplectic Perron-Frobenius matrices, New York J. Math. 17 (2011), 683–697. MR 2851068
Robert Breusch, On the distribution of the roots of a polynomial with integral coefficients, Proc. Amer. Math. Soc. 2 (1951), 939–941. MR 45246, DOI 10.1090/S0002-9939-1951-0045246-9
Benson Farb and Dan Margalit, A primer on mapping class groups, Princeton Mathematical Series, vol. 49, Princeton University Press, Princeton, NJ, 2012. MR 2850125
Eriko Hironaka, Small dilatation mapping classes coming from the simplest hyperbolic braid, Algebr. Geom. Topol. 10 (2010), no. 4, 2041–2060. MR 2728483, DOI 10.2140/agt.2010.10.2041
L. Kronecker, Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, J. Reine Angew. Math. 53 (1857), 173–175 (German). MR 1578994, DOI 10.1515/crll.1857.53.173
Erwan Lanneau and Jean-Luc Thiffeault, On the minimum dilatation of pseudo-Anosov homeromorphisms on surfaces of small genus, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 1, 105–144 (English, with English and French summaries). MR 2828128, DOI 10.5802/aif.2599
D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. (2) 34 (1933), no. 3, 461–479. MR 1503118, DOI 10.2307/1968172
L. Liechti and B. Strenner, Minimal pseudo-Anosov stretch factors on nonoriented surfaces, preprint, https://arxiv.org/abs/1806.00033, 2018 (to appear in Algebr. Geom. Topol.).
R. C. Penner, Bounds on least dilatations, Proc. Amer. Math. Soc. 113 (1991), no. 2, 443–450. MR 1068128, DOI 10.1090/S0002-9939-1991-1068128-8
A. Schinzel and H. Zassenhaus, A refinement of two theorems of Kronecker, Michigan Math. J. 12 (1965), 81–85. MR 175882, DOI 10.1307/mmj/1028999247
Chris Smyth, The Mahler measure of algebraic numbers: a survey, Number theory and polynomials, London Math. Soc. Lecture Note Ser., vol. 352, Cambridge Univ. Press, Cambridge, 2008, pp. 322–349. MR 2428530, DOI 10.1017/CBO9780511721274.021
William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, DOI 10.1090/S0273-0979-1988-15685-6