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A short proof of the Bernstein inequality for formal power series


Author: Peyman Ghahremani
Journal: Proc. Amer. Math. Soc. 148 (2020), 3233-3238
MSC (2010): Primary 13A99
DOI: https://doi.org/10.1090/proc/14970
Published electronically: March 4, 2020
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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 on $ 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.


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Additional Information

Peyman Ghahremani
Affiliation: Department of Mathematics, University of Minnesota, Minneapoles, Minnesota 55455

DOI: https://doi.org/10.1090/proc/14970
Received by editor(s): July 21, 2019
Received by editor(s) in revised form: December 8, 2019
Published electronically: March 4, 2020
Additional Notes: NSF support through grant DMS-1500264 is gratefully acknowledged
Communicated by: Claudia Polini
Article copyright: © Copyright 2020 American Mathematical Society