SMALL SALEM NUMBERS,
EXCEPTIONAL UNITS,
AND LEHMER'S CONJECTURE

JOSEPH H. SILVERMAN

Abstract:

Lehmer's conjecture says that there is an tex2html_wrap_inline11 so that if an algebraic integer tex2html_wrap_inline13 is not a root of unity, then its Mahler measure tex2html_wrap_inline15 is greater than tex2html_wrap_inline17 . This suggests that if tex2html_wrap_inline19 is small, then tex2html_wrap_inline13 should behave like a root of unity. For example, there might be many small values of n such that tex2html_wrap_inline25 is a unit; that is, such that tex2html_wrap_inline27 is an exceptional unit.

The smallest Mahler measures currently known occur for Salem numbers, and Boyd has constructed a table of small Salem numbers. We verify experimentally that many powers of the numbers in Boyd's table are exceptional units. We also show that if tex2html_wrap_inline13 is an algebraic integer of degree d, then at most tex2html_wrap_inline33 powers of tex2html_wrap_inline13 can be exceptional units. Finally, we consider the Mahler measure (canonical height) associated to arbitrary rational maps tex2html_wrap_inline37 and raise some questions related to tex2html_wrap_inline39 -Salem numbers and the tex2html_wrap_inline39 -Lehmer conjecture.