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The faithful dimension of a finite group $\mathrm G$ over $\mathbb C$,denoted by $m_\mathrm{faithful}(\mathrm G)$, is the smallest integer $n$ suchthat $\mathrm G$ can be embedded in $\mathrm{GL}_n(\mathbb C)$. Continuing ourprevious work (arXiv:1712.02019), we address the problem of determining thefaithful dimension of a finite $p$-group of the form $\mathcalG_R:=\exp(\mathfrak g_R)$ associated to $\mathfrak g_R:=\mathfrak g\otimes_\mathbb Z R $ in the Lazard correspondence, where $\mathfrak g$ is anilpotent $\mathbb Z$-Lie algebra and $R$ ranges over finite truncatedvaluation rings.  Our first main result is that if $R$ is a finite field with $p^f$ elementsand $p$ is sufficiently large, then $m_\mathrm{faithful}(\mathcal G_R)=fg(p^f)$where $g(T)$ belongs to a finite list of polynomials $g_1,\ldots,g_k$, withnon-negative integer coefficients. The list of polynomials is uniquelydetermined by the Lie algebra $\mathfrak g$. Furthermore, for $1\leq i\leq k$the set of pairs $(p,f)$ for which $g=g_i$ is a finite union of Cartesianproducts $\mathcal P\times \mathcal F$, where $\mathcal P$ is a Frobenius setof prime numbers and $\mathcal F$ is a subset of $\mathbb N$ that belongs tothe Boolean algebra generated by arithmetic progressions.  Next we formulate a conjectural polynomiality property for$m_\mathrm{faithful}(\mathcal G_R)$ in the more general setting where $R$ is afinite truncated valuation ring, and prove special cases of this conjecture. Inparticular, we show that for a vast class of Lie algebras $\mathfrak g $ thatare defined by partial orders, $m_\mathrm{faithful}(\mathcal G_R)$ is given bya single polynomial-type formula.  Finally, we compute $m_\mathrm{faithful}(\mathcal G_R)$ precisely in the casewhere $\mathfrak g$ is the free metabelian nilpotent Lie algebra of class $c$on $n$ generators and $R$ is a finite truncated valuation ring.

Polynomiality of the faithful dimension of nilpotent groups over finite truncated valuation rings