 Mahler measure

For a nonzero complex polynomial $f(X)=a_d{\prod }_{j=1}^d(X-{\alpha }_j)$, its Mahler measure isThe roots of a polynomial inside the unit disk contribute a factor of one; those outside contribute their moduli. The leading coefficient is included, so the definition also applies to nonmonic polynomials.

For example, $M(X^2-2)=2$, since both roots have modulus $\sqrt{2}$. By contrast, $M(X^2+X+1)=1$, since its two roots are roots of unity. A nonzero polynomial with integer coefficients has $M(f)\ge 1$.

The root formula proves multiplicativity immediately: the roots of $fg$ combine the roots of the two factors, with their multiplicities, and the leading coefficients multiply. Hence $M(fg)=M(f)M(g)$. The logarithmic version $m(f)=\log M(f)$ turns this product into a sum.

The Mahler measure is a concept from number theory and algebraic geometry that provides a way to measure the "size" or "complexity" of a polynomial or a rational function.