Let X be a normal projective variety of dimension d over an algebraically closed field and f an automorphism of X. Suppose that the pullback \(f^*|_{{\textsf{N}}^1(X)_{\textbf{R}}}\) of f on the real Néron–Severi space \({\textsf{N}}^1(X)_{\textbf{R}}\) is unipotent and denote the index of the eigenvalue 1 by \(k+1\) . We establish the following upper bound for the polynomial volume growth \(\textrm{plov}(f)\) of f: \(\begin{aligned} \textrm{plov}(f) \le (k/2 + 1)d. \end{aligned}\) This inequality is optimal in certain cases. Moreover, we prove that \(k\le 2(d-1)\) , extending a result of Dinh–Lin–Oguiso–Zhang for compact Kähler manifolds to arbitrary characteristic. By combining these two inequalities, we obtain the optimal bound \(\begin{aligned} \textrm{plov}(f) \le d^2, \end{aligned}\) that affirmatively answers the questions of Cantat–Paris-Romaskevich and Lin–Oguiso–Zhang.
