Algebraic stability and degree growth of monomial maps and polynomial maps

Given a rational monomial map, we consider the question of finding a toric variety on which it is algebraically stable. We give conditions for when such variety does or does not exist. We also obtain several precise estimates of the degree sequences of monomial maps on ${\mathrm{\P }}^n$. Finally, we characterize polynomial maps which are algebraically stable on $({\mathrm{\P }}^1{)}^n$.