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_α supposed to be? Polynomials are functions, not sequences. Reading it as L_α(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 $|a|\prod _{|\alpha _{i}|\geq 1}|\alpha _{i}|$ .

And what does If p is an irreducible polynomial with $p(0)\neq 0$ 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)[reply]

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)[reply]