Mahler measure

In mathematics, the Mahler measure is a measure of the complexity of polynomials. It is named after Kurt Mahler (1903-1988) and was originally used in the search for large prime numbers . Today it is the subject of numerous conjectures in analytic number theory because of the connection to special values ​​of L-functions .

definition

The Mahler measure of a polynomial with real or complex coefficients is ${\displaystyle M(p)}$ ${\displaystyle p(x)\in \mathbb {C} [x]}$

${\displaystyle M(p)=\lim _{\tau \rightarrow 0}\|p\|_{\tau }=\exp \left({\frac {1}{2\pi }}\int _{0}^{2\pi }\ln(|p(e^{i\theta })|)\,d\theta \right).}$

Here is

${\displaystyle \|p\|_{\tau }=\left({\frac {1}{2\pi }}\int _{0}^{2\pi }|p(e^{i\theta })|^{\tau }\,d\theta \right)^{1/\tau }\,}$

the norm of . With the help of Jensen's formula one can show that from ${\displaystyle L_{\tau }}$ ${\displaystyle p}$

${\displaystyle p(z)=a(z-\alpha _{1})(z-\alpha _{2})\cdots (z-\alpha _{n})}$

follows:

${\displaystyle M(p)=|a|\prod _{i=1}^{n}\max\{1,|\alpha _{i}|\}=|a|\prod _{|\alpha _{i}|\geq 1}|\alpha _{i}|.}$

The logarithmic Mahler measure of a polynomial is defined as

${\displaystyle m(P)=\log M(P)}$.

The Mahler measure of an algebraic number is defined as the Mahler measure of the minimal polynomial of over . ${\displaystyle \alpha }$ ${\displaystyle \alpha }$ ${\displaystyle \mathbb {Q} }$

properties

• The Mahler measure is multiplicative, i. H. ${\displaystyle M(p\,q)=M(p)\cdot M(q).}$
• For cyclotomic polynomials and their products applies . ${\displaystyle M(p)=1}$
• Kronecker's Theorem: If is an irreducible monic polynomial with integer coefficients and , then either is or is a cyclotomic polynomial. ${\displaystyle p}$ ${\displaystyle M(p)=1}$ ${\displaystyle p(z)=z,}$ ${\displaystyle p}$
• The Lehmersche assumption states that there is a constant is such that each irreducible polynomial with integer coefficients either zyklotomisch or is met. ${\displaystyle \mu >1}$ ${\displaystyle p}$ ${\displaystyle M(p)>\mu }$
• The Mahler measure of a monic polynomial with integer coefficients is a Perron number .

Special values ​​of L-functions

There are numerous assumed and partly also proven relationships between (logarithmic) Mahler measures of polynomials and special values ​​of L-functions .

The first historical example of this was Smyth's formula

${\displaystyle m(1+x_{1}+x_{2})={\frac {3{\sqrt {3}}}{4\pi }}L(\chi _{-3},2)}$

With

${\displaystyle L(\chi _{-3},2)={\frac {1}{1^{s}}}-{\frac {1}{2^{s}}}+{\frac {1}{4^{s}}}-{\frac {1}{5^{s}}}+-\ldots }$.

A conjecture by Chinburg says that for every negative number one has a Laurent polynomial and a rational number with ${\displaystyle -f}$ ${\displaystyle P_{f}\in \mathbb {Z} (x,y)}$ ${\displaystyle r_{f}\in \mathbb {Q} }$

${\displaystyle m(P_{f})=r_{f}d_{F}}$

for the discriminant

${\displaystyle d_{f}={\frac {f{\sqrt {f}}}{4\pi }}L(\chi _{-f},2)}$

of character has. An approach that goes back to Boyd and Rodriguez-Villegas is to represent logarithmic Mahler measures of a certain class of polynomials (especially A-polynomials of hyperbolic manifolds ) as rational linear combinations of values ​​of the Bloch-Wigner dilogarithm on algebraic arguments, and this in turn with the Volume of a hyperbolic manifold and related to special values ​​of zeta functions via Borel's theorem . ${\displaystyle \chi _{-f}(n)=\left({\frac {-f}{n}}\right)}$

Mahler measure for polynomials of several variables

The Mahler measure of a polynomial is defined analogously by the formula ${\displaystyle M(p)}$ ${\displaystyle p(x_{1},\ldots ,x_{n})\in \mathbb {C} [x_{1},\ldots ,x_{n}]}$

${\displaystyle M(p)=\exp \left({\frac {1}{(2\pi )^{n}}}\int _{0}^{2\pi }\int _{0}^{2\pi }\cdots \int _{0}^{2\pi }\ln {\Bigl (}{\bigl |}p(e^{i\theta _{1}},e^{i\theta _{2}},\ldots ,e^{i\theta _{n}}){\bigr |}{\Bigr )}\,d\theta _{1}\,d\theta _{2}\cdots d\theta _{n}\right).}$

It can be shown to converge. ${\displaystyle M(p)}$

For denote ${\displaystyle {\boldsymbol {r}}=(r_{1},\ldots ,r_{n})\in \mathbb {N} ^{n}}$

${\displaystyle q({\boldsymbol {r}}):={\text{min}}\{{\text{max}}\{|s_{j}|:1\leq j\leq N\}:s=(s_{1},\dots ,s_{N})\in \mathbb {Z} ^{N},s\neq (0,\dots ,0)\ {\text{and}}\ \sum _{j=1}^{N}s_{j}r_{j}=0\}}$

Then

${\displaystyle M{\bigl (}p(x_{1},\ldots ,x_{n}){\bigr )}=\lim _{{\boldsymbol {r}}\in \mathbb {N} ^{n} \atop q({\boldsymbol {r}})\to \infty }M{\bigl (}p(x^{r_{1}},x^{r_{2}},\ldots ,x^{r_{n}}){\bigr )}.}$

literature

• Derrick Henry Lehmer : Factorization of certain cyclotomic functions. Ann. of Math. (2) 34 (1933), no. 3, 461-479.
• David W. Boyd : Speculations concerning the range of Mahler's measure. Canad. Math. Bull. 24: 453-469 (1981).
• Klaus Schmidt : Dynamical systems of algebraic origin. Progress in Mathematics, 128. Birkhäuser Verlag, Basel, 1995. ISBN 3-7643-5174-8

Web links

• Mahler's measure and special values ​​of L-functions

Individual evidence

1. ^ W. Lawton: A problem of Boyd concerning geometric means of polynomials. J. Number Theory 16 (1983) no. 3, 356-362.