where ,
denoted
by Lehmer (1933) and by Hironaka (2009), is the largest positive root of this
polynomial. The roots of this polynomial, plotted in the left figure above, are very
special, since 8 of the 10 lie on the unit circle
in the complex plane. The roots of the polynomials
(represented by half their coefficients) giving the two next smallest known Mahler
measures are also illustrated above (Mossinghoff 1998, p. S11).
The best current bound is that of Smyth (1971), who showed that , where is a nonzero nonreciprocal polynomial that is not a product
of cyclotomic polynomials (Everest 1999),
and
is the real root of . Generalizations of Smyth's result have been constructed
by Lloyd-Smith (1985) and Dubickas (1997).
In general, the smallest Mahler measures occur for integer polynomials that are small in absolute
value. The histogram above shows the distribution of measures for random -polynomials of random orders 1 to 10. Mossinghoff (1998)
gives a table of the smallest known Mahler measures for polynomial degrees up to
,
and subsequently demonstrated that is the smallest Mahler measure
greater than 1 for all degrees up to 40 (Mossinghoff, Hironaka 2009).
for a univariate
polynomial 
that is not a product of cyclotomic
polynomials. Lehmer (1933) conjectured that if 
is such an integer polynomial,
then
,
denoted 
by Lehmer (1933) and 
by Hironaka (2009), is the largest positive root of this
polynomial. The roots of this polynomial, plotted in the left figure above, are very
special, since 8 of the 10 lie on the unit circle
in the complex plane. The roots of the polynomials
(represented by half their coefficients) giving the two next smallest known Mahler
measures are also illustrated above (Mossinghoff 1998, p. S11).
, where 
is a nonzero nonreciprocal polynomial that is not a product
of cyclotomic polynomials (Everest 1999),
and 
is the real root of 
. Generalizations of Smyth's result have been constructed
by Lloyd-Smith (1985) and Dubickas (1997).
-polynomials of random orders 1 to 10. Mossinghoff (1998)
gives a table of the smallest known Mahler measures for polynomial degrees up to
,
and subsequently demonstrated that 
is the smallest Mahler measure
greater than 1 for all degrees up to 40 (Mossinghoff, Hironaka 2009).
is a Salem constant.
