= Bernstein inequality for Weyl algebra modules
{c}
{title2=$d(M)\geq n$}
For a nonzero finitely generated <module> over the complex <Weyl algebra> $A_n$, its <Bernstein growth dimension of a Weyl algebra module> is at least $n$. The <faithful finite-step action of a Weyl algebra> gives $\dim F_jA_n\leq(\dim M_j)(\dim M_{2j})$. Comparing degrees $2n$ and $2d(M)$ proves the inequality. The polynomial <module> has dimension $n$, so the bound is sharp. This is an algebraic growth inequality, distinct from probabilistic Bernstein inequalities.
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