where factorizes over the complex numbers as
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 multiplicative:
- where is the norm of .
- (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.
Higher-dimensional Mahler measure
The Mahler measure of a multi-variable polynomial is defined similarly by the formula
It inherits the above three properties of the Mahler measure for a one-variable polynomial.
where is the Dirichlet L-function, and
where is the Riemann zeta function. Here is called the logarithmic Mahler measure.
Some results by Lawton and Boyd
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, which had been conjectured by Boyd.
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
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.
Define an extended cyclotomic polynomial to be a polynomial of the form
where is the m-th cyclotomic polynomial, the are integers, and the are chosen minimally so that is a polynomial in the . Let be the set of polynomials that are products of monomials and extended cyclotomic polynomials.
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
and the union . He made the far-reaching conjecture that the set of is a closed subset of . An immediate consequence of this conjecture would be the truth of Lehmer's conjecture, albeit without an explicit lower bound. As Smyth's result suggests that , Boyd further conjectures that
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