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In mathematics, the **Mahler measure** **of a polynomial** with complex coefficients is defined as

where factorizes over the complex numbers as

The Mahler measure can be viewed as a kind of height function. Using Jensen's formula, it can be proved that this measure is also equal to the geometric mean of for on the unit circle (i.e., ):

By extension, the **Mahler measure of an algebraic number** is defined as the Mahler measure of the minimal polynomial of over . In particular, if is a Pisot number or a Salem number, then its Mahler measure is simply .

The Mahler measure is named after the German-born Australian mathematician Kurt Mahler.

- The
**Mahler measure**is multiplicative: - where is the norm of .
^{[1]} **Kronecker's Theorem**: If is an irreducible monic integer polynomial with , then either or is a cyclotomic polynomial.*(Lehmer's conjecture)*There is a constant such that if is an irreducible integer polynomial, then either or .- The Mahler measure of a monic integer polynomial is a Perron number.

The Mahler measure of a multi-variable polynomial is defined similarly by the formula^{[2]}

The multi-variable Mahler measure has been shown, in some cases, to be related to special values
of zeta-functions and -functions. For example, in 1981, Smyth^{[3]} proved the formulas

From the definition, the Mahler measure is viewed as the integrated values of polynomials over the torus (also see Lehmer's conjecture). If vanishes on the torus , then the convergence of the integral defining is not obvious, but it is known that does converge and is equal to a limit of one-variable Mahler measures,^{[4]} which had been conjectured by Boyd.^{[5]}^{[6]}

This is formulated as follows: Let denote the integers and define . If is a polynomial in variables and define the polynomial of one variable by

and define by

where .

**Theorem (Lawton)** — Let be a polynomial in *N* variables with complex coefficients. Then the following limit is valid (even if the condition that is relaxed):

Boyd provided more general statements than the above theorem. He pointed out that the classical Kronecker's theorem, which characterizes monic polynomials with integer coefficients all of whose roots are inside the unit disk, can be regarded as characterizing those polynomials of one variable whose measure is exactly 1, and that this result extends to polynomials in several variables.^{[6]}

Define an *extended cyclotomic polynomial* to be a polynomial of the form

**Theorem (Boyd)** — Let be a polynomial with integer coefficients. Then if and only if is an element of .

This led Boyd to consider the set of values

An action of by automorphisms of a compact metrizable abelian group may be associated via duality to any countable module over the ring .^{[7]} The topological entropy (which is equal to the measure-theoretic entropy) of this action, , is given by a Mahler measure (or is infinite).^{[8]} In the case of a cyclic module for a non-zero polynomial the formula proved by Lind, Schmidt, and Ward gives , the logarithmic Mahler measure of . In the general case, the entropy of the action is expressed as a sum of logarithmic Mahler measures over the generators of the principal associated prime ideals of the module. As pointed out earlier by Lind in the case of a single compact group automorphism, this means that the set of possible values of the entropy of such actions is either all of or a countable set depending on the solution to Lehmer's problem. Lind also showed that the infinite-dimensional torus either has ergodic automorphisms of finite positive entropy or only has automorphisms of infinite entropy depending on the solution to Lehmer's problem.^{[9]}

**^**Although this is not a true norm for values of .**^**Schinzel 2000, p. 224.**^**Smyth 2008.**^**Lawton 1983.- ^
^{a}^{b}Boyd 1981a. - ^
^{a}^{b}Boyd 1981b. **^**Kitchens, Bruce; Schmidt, Klaus (1989). "Automorphisms of compact groups".*Ergodic Theory Dynam. Systems*.**9**(4): 691–735. doi:10.1017/S0143385700005290.**^**Lind, Douglas; Schmidt, Klaus; Ward, Tom (1990). "Mahler measure and entropy for commuting automorphisms of compact groups".*Invent. Math*.**101**: 593–629. doi:10.1007/BF01231517.**^**Lind, Douglas (1977). "The structure of skew products with ergodic group automorphisms".*Israel J. Math*. 28, no. 3: 205–248. doi:10.1007/BF02759810.

- Borwein, Peter (2002).
*Computational Excursions in Analysis and Number Theory*. CMS Books in Mathematics. Vol. 10. Springer. pp. 3, 15. ISBN 978-0-387-95444-8. Zbl 1020.12001.

- Boyd, David (1981a). "Speculations concerning the range of Mahler's measure".
*Canadian Mathematical Bulletin*.**24**(4): 453–469. doi:10.4153/cmb-1981-069-5.

- Boyd, David (1981b). "Kronecker's Theorem and Lehmer's Problem for Polynomials in Several Variables".
*Journal of Number Theory*.**13**: 116–121. doi:10.1016/0022-314x(81)90033-0.

- Boyd, David (2002a). "Mahler's measure and invariants of hyperbolic manifolds". In Bennett, M. A. (ed.).
*Number theory for the Millenium*. A. K. Peters. pp. 127–143.

- Boyd, David (2002b). "Mahler's measure, hyperbolic manifolds and the dilogarithm".
*Canadian Mathematical Society Notes*.**34**(2): 3–4, 26–28.

- Boyd, David; Rodriguez Villegas, F. (2002). "Mahler's measure and the dilogarithm, part 1".
*Canadian Journal of Mathematics*.**54**(3): 468–492. doi:10.4153/cjm-2002-016-9. S2CID 10069657. - Brunault, François (2020).
*Many variations of Mahler measures : a lasting symphony*. Cambridge, United Kingdom New York, NY: Cambridge University Press. ISBN 978-1-108-79445-9. OCLC 1155888228.

- Everest, Graham and Ward, Thomas (1999). "Heights of polynomials and entropy in algebraic dynamics". Springer-Verlag London, Ltd., London. xii+211 pp. ISBN: 1-85233-125-9

- "Mahler measure",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994]. - Jensen, J.L. (1899). "Sur un nouvel et important théorème de la théorie des fonctions".
*Acta Mathematica*.**22**: 359–364. doi:10.1007/BF02417878. JFM 30.0364.02.

- Knuth, Donald E. (1997). "4.6.2 Factorization of Polynomials".
*Seminumerical Algorithms*. The Art of Computer Programming. Vol. 2 (3rd ed.). Addison-Wesley. pp. 439–461, 678–691. ISBN 978-0-201-89684-8.

- Lawton, Wayne M. (1983). "A problem of Boyd concerning geometric means of polynomials".
*Journal of Number Theory*.**16**(3): 356–362. doi:10.1016/0022-314X(83)90063-X. Zbl 0516.12018.

- Mossinghoff, M.J. (1998). "Polynomials with Small Mahler Measure".
*Mathematics of Computation*.**67**(224): 1697–1706. doi:10.1090/S0025-5718-98-01006-0. Zbl 0918.11056.

- Schinzel, Andrzej (2000).
*Polynomials with special regard to reducibility*. Encyclopedia of Mathematics and Its Applications. Vol. 77. Cambridge University Press. ISBN 978-0-521-66225-3. Zbl 0956.12001.

- Smyth, Chris (2008). "The Mahler measure of algebraic numbers: a survey". In McKee, James; Smyth, Chris (eds.).
*Number Theory and Polynomials*. London Mathematical Society Lecture Note Series. Vol. 352. Cambridge University Press. pp. 322–349. ISBN 978-0-521-71467-9. Zbl 1334.11081.

- Mahler Measure on MathWorld
- Jensen's Formula on MathWorld