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Ax's theorem states that $|Z(f)|\\equiv 0\\pmod {q^{\\lceil n/d\\rceil-1}}$, that is, $\\nu_p(|Z(f)|)\\ge m(\\lceil n/d\\rceil-1)$, where $p=\\text{char}\\,\\Bbb F_q$, $q=p^m$, and $\\nu_p$ is the $p$-adic valuation. In this paper, we determine a condition on the coefficients of $f$ that is necessary and sufficient for $f$ to meet Ax's bound, that is, $\\nu_p(|Z(f)|)=m(\\lceil n/d\\rceil-1)$. 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