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| Mahler measure | |
|---|---|
| Name | Mahler measure |
| Field | Number theory |
| Introduced | 1960s |
| Named after | Kurt Mahler |
Mahler measure is a numerical invariant assigned to nonzero multivariate polynomials that plays a central role in algebraic number theory, transcendence, and arithmetic geometry. It links properties of algebraic integers, heights on varieties, and special values of analytic objects through explicit formulae and deep conjectures. The concept has applications in the study of algebraic dynamical systems, regulators of motives, and computational number theory.
The Mahler measure of a single-variable polynomial f(x)=a_0 ∏_{i=1}^n (x−α_i) is defined by an integral over the unit circle and equals |a_0| ∏_{i=1}^n max(1,|α_i|), connecting to the product formula used by Kurt Mahler and classical results of Carl Ludwig Siegel and André Weil. For multivariate polynomials, the Mahler measure is given by a toral integral over copies of the unit circle, a definition used by researchers such as D. H. Lehmer and Walter Rudin in related contexts. The measure is multiplicative under polynomial multiplication and invariant under toral translations and monomial changes of variables analogous to invariances studied by Emil Artin and Alexander Grothendieck. It provides a canonical height for algebraic numbers comparable to heights introduced by Helmut Hasse and John Tate.
For f(x)=x−1 the Mahler measure is 1, mirroring trivial cases analyzed by G. H. Hardy and John Edensor Littlewood in classical analysis. Cyclotomic polynomials have Mahler measure 1, a fact related to work of Leopold Kronecker on algebraic integers all of whose conjugates lie on the unit circle. Lehmer's degree-10 polynomial yields the smallest known nontrivial measure >1, an example studied by D. H. Lehmer and later investigated numerically by Enrique V. Bombieri and Walter M. Schmidt. For two-variable families such as x+y+1, explicit computations use Jensen-type formulas as in research by A. Jensen and examples computed by David Boyd and Fernando Rodriguez Villegas. Many low-degree examples are catalogued and computed using techniques influenced by computational projects at institutions like École Polytechnique and Institute for Advanced Study.
The Mahler measure provides a bridge to Weil heights on projective space, paralleling constructions by André Weil and height inequalities employed by Paul Vojta and Alan Baker. It appears in explicit formulae bounding heights of algebraic numbers in terms of coefficients, reminiscent of estimates by Kurt Mahler and results used in effective versions of the Thue–Siegel–Roth theorem and the Mordell conjecture (Faltings's theorem) proved by Gerd Faltings. Relations between Mahler measure and canonical heights on elliptic curves connect to work of Joseph H. Silverman and applications explored by Niels Hendrik Abel in historical context.
Multivariate Mahler measure uses integration over the real torus and exhibits phenomena absent in one variable, a subject advanced by Don Zagier and David Boyd in conjectures linking measures to special values. For two-variable Laurent polynomials, connections to regulators of algebraic K-theory were developed in the framework of Spencer Bloch and Goncharov's ideas, and further explored by Richard Hain and F. Brown in the study of mixed motives. The behavior under specialization and slice operations relates to deformation techniques used by Pierre Deligne and Kazuya Kato.
Numerous conjectures propose that Mahler measures of specific polynomials equal rational multiples of special values of L-functions of modular forms and elliptic curves, themes pursued by Don Zagier, F. Rodriguez Villegas, and David Boyd. Examples link measures to L'(E,0) for elliptic curves E studied in the work of Andrew Wiles and Ben Green's collaborators on modularity, and to values of Dirichlet L-series examined by Poul Heegaard and classical analysts. These conjectural relations tie Mahler measure to regulators appearing in Beilinson's conjectures formulated by Alexander Beilinson and to motivic cohomology concepts developed by Spencer Bloch.
Lehmer's problem asks whether a noncyclotomic integer polynomial can have Mahler measure arbitrarily close to 1; this question originates with D. H. Lehmer and has motivated work by Smyth, Dobrowolski, Enrique Bombieri, and Walter M. Schmidt. Dobrowolski's lower bound is a landmark result building on techniques linked to Baker's theory and transcendence methods of Alan Baker, while conditional improvements leverage conjectures such as the Generalized Riemann Hypothesis studied by Bernhard Riemann and later analysts. The problem connects to distribution questions for algebraic units investigated by Jean-Pierre Serre and equidistribution results of A. B. Šidák-type in arithmetic dynamics.
Practical computation of Mahler measures employs numerical integration on tori, lattice reduction, and root-finding algorithms used by researchers at Max Planck Institute and members of computational projects such as those led at University of Cambridge and Princeton University. High-precision evaluation of integrals uses techniques from numerical analysis developed by John von Neumann and Donald Knuth while symbolic methods exploit factorization algorithms inspired by Carl Friedrich Gauss and optimized in software from institutions like University of Oxford and Massachusetts Institute of Technology. Continued experimental computation informs conjectures and cross-checks links with L-values in computational explorations by Don Zagier and David Boyd.