A Short Proof of the Bernstein Inequality for Formal Power Series

TL;DR

This paper presents a concise proof of the Bernstein inequality, which states that for finitely generated modules over the ring of differential operators on formal power series, the module's dimension is at least the number of variables.

Contribution

The paper offers a simplified and shorter proof of the Bernstein inequality for modules over differential operator rings on formal power series.

Findings

Proves the Bernstein inequality in a concise manner

Establishes the lower bound of module dimension as the number of variables

Simplifies previous proofs of the inequality

Abstract

Let $k$ be a field of characteristic zero, let $R$ be the ring of formal power series in $n$ variables over $k$ and let $D(R,k)$ be the ring of $k-$linear differential operators in $R$. If $M$ is a finitely generated $D(R,k)-$module then $d(M)\geq n$ where $d(M)$ is the dimension of $M$. This inequality is called the Bernstein inequality. We provide a short proof.