Canonical height functions for monomial maps

The height function measures the arithmetic complexity of a point on a variety over ℚ. The canonical height function measures the asymptotic height growth (relative to the degree growth) of a point under a dominant rational map. One property desired for the canonical height function is the Northcott finiteness property, which states that there are only finitely many points for a bounded degree and a bounded height. We show that the canonical height function for dominant rational maps does not have the Northcott finiteness property in general. We develop a new canonical height function for monomial maps. In certain cases, this new canonical height function has the desired nice properties.