Talk:Mahler measure

Calculating the Mahler measure of a polynomial via its roots is usually easier than performing the integration.

To calculate the Mahler measure of a polynomial in Maple You can do this:

>Mahler := proc( f ) > local r, n, i, m; >  n := degree( f ); > r := solve( f=0, x ); > m := abs(coeff( f, x ) ); > for i from 1 to n do >   m := m * max(1,abs(r[i])); > od; > end proc:

> f := -x^9-x^8-x^7-x^6-x^5 + x^4 + x^3+x^2-x-1;

9   8    7    6    5    4    3    2         f := -x  - x  - x  - x  - x  + x  + x  + x  - x - 1

> evalf( Mahler( f ) );

2.294787065

???
What is l&alpha; supposed to be? Polynomials are functions, not sequences. Reading it as L&alpha;(X), where X is presumably the unit circle, does not work either, as then
 * $$\lim_{\alpha\to\infty}\Vert p\Vert_\alpha=\Vert p\Vert_\infty=\max_{z\in X}|p(z)|,$$

which is hopelessly different from $$|a|\prod_{|\alpha_i|\ge1}|\alpha_i|$$.

And what does If p is an irreducible polynomial with $$p(0) \ne 0$$ and $$M(p) = 1$$, then p is a cyclotomic polynomial mean? Irreducible over which field? Certainly not the complex numbers. In any case, any polynomial can be normalized to M(p)=1 just by multiplying it with a suitable constant. -- EJ 03:45, 18 January 2006 (UTC)


 * Right on both counts. According to Borwein it's the L0 "norm" (p.3), and irreducible monic integer polynomials (p.15).  I have edited accordingly.  Richard Pinch (talk) 19:11, 29 July 2008 (UTC)