View source for Mahler measure

Given a [[Polynomial|polynomial]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m1200701.png" /> with complex coefficients, the logarithmic Mahler measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m1200702.png" /> is defined to be the average over the unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m1200703.png" />-torus of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m1200704.png" />, i.e.

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m1200705.png" /></td> </tr></table>

The Mahler measure is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m1200706.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m1200707.png" /> is the [[Geometric mean|geometric mean]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m1200708.png" /> over the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m1200709.png" />-torus. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007011.png" />, Jensen's formula gives the explicit formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007012.png" /></td> </tr></table>

so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007013.png" />.

The Mahler measure is useful in the study of polynomial inequalities because of the multiplicative property <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007014.png" />. The important basic inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007015.png" /></td> </tr></table>

[[#References|[a9]]] relates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007016.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007017.png" />, the sum of the absolute values of the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007019.png" /> denotes the total degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007020.png" />, i.e. the sum of the degrees in each variable separately. A recent inequality for polynomials of one variable is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007023.png" /> is the sum of the degrees of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007025.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007026.png" /> is the best possible constant [[#References|[a2]]].

Specializing to polynomials with integer coefficients, in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007028.png" /> is the logarithm of an [[algebraic integer]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007029.png" />, there are few explicit formulas known, but those that do exist suggest that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007030.png" /> has intimate connections with [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007031.png" />-theory]]. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007033.png" /> is the [[Dirichlet L-function|Dirichlet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007034.png" />-function]] for the odd primitive character of conductor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007035.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007036.png" />, and it has been conjectured that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007037.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007038.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007039.png" />-function of an [[Elliptic curve|elliptic curve]] of conductor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007040.png" />. This formula has not been proved but has been verified to over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007041.png" /> decimal places [[#References|[a3]]], [[#References|[a4]]].

The Mahler measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007042.png" /> occurs naturally as the growth rate in many problems, for example as the entropy of certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007044.png" />-actions [[#References|[a10]]]. The set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007045.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007046.png" /> is known: in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007047.png" />, a theorem of Kronecker shows that these are products of cyclotomic polynomials and monomials. In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007048.png" />, these are the generalized cyclotomic polynomials [[#References|[a1]]]. An important open question, known as [[Lehmer conjecture|Lehmer's problem]], is whether there is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007049.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007050.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007051.png" />. This is known to be the case if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007052.png" /> is a non-reciprocal polynomial, where a polynomial is reciprocal if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007053.png" /> is a monomial. In this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007054.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007055.png" /> is the smallest [[Pisot number|Pisot number]], the real root of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007056.png" /> [[#References|[a6]]], [[#References|[a1]]]. A possible value for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007057.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007058.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007059.png" /> is the smallest known [[Salem number|Salem number]], a number of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007060.png" /> known as Lehmer's number.

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007061.png" />, the best result in this direction is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007062.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007063.png" /> is an explicit absolute constant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007064.png" /> is the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007065.png" /> [[#References|[a5]]]. A result that applies to polynomials in any number of variables is an explicit constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007066.png" /> depending on the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007067.png" /> of non-zero coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007068.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007069.png" /> [[#References|[a7]]], [[#References|[a1]]].

A recent development is the elliptic Mahler measure [[#References|[a8]]], in which the torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007070.png" /> is replaced by an elliptic curve. It seems likely that this will have an interpretation as the entropy of a dynamical system but this remains as of yet (1998) a future development.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D.W. Boyd, "Kronecker's theorem and Lehmer's problem for polynomials in several variables" ''J. Number Th.'' , '''13''' (1981) pp. 116–121</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> D.W. Boyd, "Two sharp inequalities for the norm of a factor of a polynomial" ''Mathematika'' , '''39''' (1992) pp. 341–349 {{MR|1203290}} {{ZBL|0758.30003}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D.W. Boyd, "Mahler's measure and special values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007071.png" />-functions" ''Experim. Math.'' , '''37''' (1998) pp. 37–82</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> C. Deninger, "Deligne periods of mixed motives, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007072.png" />-theory and the entropy of certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120070/m12007073.png" />-actions" ''J. Amer. Math. Soc.'' , '''10''' (1997) pp. 259–281 {{MR|1415320}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> E. Dobrowolski, "On a question of Lehmer and the number of irreducible factors of a polynomial" ''Acta Arith.'' , '''34''' (1979) pp. 391–401 {{MR|0543210}} {{ZBL|0416.12001}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> C.J. Smyth, "On the product of the conjugates outside the unit circle of an algebraic integer" ''Bull. London Math. Soc.'' , '''3''' (1971) pp. 169–175 {{MR|0289451}} {{ZBL|0235.12003}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> E. Dobrowolski, "Mahler's measure of a polynomial in function of the number of its coefficients" ''Canad. Math. Bull.'' , '''34''' (1991) pp. 186–195 {{MR|1113295}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> G. Everest, Ni Fhlathúin Brid, "The elliptic Mahler measure" ''Math. Proc. Cambridge Philos. Soc.'' , '''120''' : 1 (1996) pp. 13–25 {{MR|1373343}} {{ZBL|0865.11068}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> K. Mahler, "On some inequalities for polynomials in several variables" ''J. London Math. Soc.'' , '''37''' : 2 (1962) pp. 341–344 {{MR|0138593}} {{ZBL|0105.06301}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> K. Schmidt, "Dynamical systems of algebraic origin" , Birkhäuser (1995) {{MR|1345152}} {{ZBL|0833.28001}} </TD></TR></table>