An upper bound for the height for regular affine automorphisms of {{{\mathbb{A}}^ n}}

Kawaguchi (Math. Ann. 335(2):285–310, 359–374, 2006) proved a height inequality for ${h\bigl(f(P)\bigr)}$ when f is a regular affine automorphism of ${{\mathbb{A}}^2}$ , and he conjectured that a similar estimate is also true for regular affine automorphisms of ${{\mathbb{A}}^n}$ for n ≥ 3. In this paper we prove Kawaguchi’s conjecture. This implies that Kawaguchi’s theory of canonical heights for regular affine automorphisms of projective space is true in all dimensions.