Bernstein filtration
= Bernstein filtration
{c}
{title2=$F_jA_n=\operatorname{span}\{x^{\mathbf a}\partial^{\mathbf b}:|\mathbf a|+|\mathbf b|\leq j\}$}
Give both position and differentiation generators of a <Weyl algebra> degree one. The ordered <monomial> <basis> gives $\dim F_jA_n=\binom{j+2n}{2n}$ and $\operatorname{gr}_F A_n$ equal to a polynomial ring in $2n$ variables. Commuting a filtered element with any generator lowers its degree by at least one.