The topological zeta function and Igusa's local zeta function are respectively a geometrical invariant associated to a complex polynomial $f$ and an arithmetical invariant associated to a polynomial $f$ over a $p$-adic field.

When $f$ is a polynomial in two variables we prove a formula for both zeta functions in terms of the so-called log canonical model of $f^{-1} \{ 0 \}$ in $\Bbb A^2$. This result yields moreover a conceptual explanation for a known cancellation property of candidate poles for these zeta functions. Also in the formula for Igusa's local zeta function appears a remarkable non-symmetric ‘$q$-deformation’ of the intersection matrix of the minimal resolution of a Hirzebruch-Jung singularity.

1991 Mathematics Subject Classification: 32S50 11S80 14E30 (14G20)