Integrals with Respect to the Euler Characteristic over Spaces of Functions and the Alexander Polynomial

We discuss some results that describe an expression for the Alexander polynomial (and, thus, for the zeta-function of the classical monodromy transformation) of a plane curve singularity in terms of the ring of functions on a curve. They describe the coefficients of the Alexander polynomial as Euler characteristics of some explicitly constructed complements to arrangements of projective hyperplanes in projective spaces. We also discuss the notion of integral with respect to the Euler characteristics over the projectivization of the space of functions (in the spirit of the motivic integration) and its connection with the formulas for the coefficients of the Alexander polynomial.