For any proper polynomial map$f: C$^{$k$}→$C$^{$k$}define the function α as $$\alpha (z): = \mathop {\lim \sup }\limits_{n \to \infty } \frac{{\log ^ + \log ^ + \left| {f^n (z)} \right|}}{n},where log^ + : = \max \{ \log , 0\} .$$ Let$f=(P$_{1},...,$P$_{$k$}) be a proper polynomial map. We define a notion of$s$-regularity using the extension of$f$to P^{k}. When$f$is (maximally) regular we show that the function α is lower semicontinuous and takes only finitely many values: 0 and$d$_{1},...,$d$_{$k$}, where$d$_{$i$}:=deg$P$_{$i$}. We then describe dynamically the sets {α≤$d$_{$i$}}. We give a concrete description of regular maps. If$d$_{$i$}>1, this allows us to construct the equilibrium measure μ associated with$f$as a generalized intersection of positive currents. We then give an estimate of the Hausdorff dimension of μ. We extend the approach to the larger class of (π,$s$)-regular maps. This gives an understanding of the largest values of α. The results can be applied to construct dynamically interesting measures for automorphisms.