An upper bound for polynomial volume growth of automorphisms of zero entropy

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^*|N¹(X)R of f on the real N\'eron--Severi space N¹(X)R is unipotent and denote the index of the eigenvalue $1$ by $k+1$. We prove an upper bound for the polynomial volume growth plov(f) of f as follows: \[ plov(f) ≤ (k/2 + 1)d. \] This inequality is optimal in certain cases. Furthermore, we show that k≤ 2(d-1), extending a result of Dinh--Lin--Oguiso--Zhang for compact K\"ahler manifolds to arbitrary characteristic. Combining these two inequalities together, we obtain an optimal inequality that \[ plov(f) ≤ d^2, \] which affirmatively answers questions of Cantat--Paris-Romaskevich and Lin--Oguiso--Zhang.