A multivariate Strassmann theorem

TL;DR

This paper extends Strassmann's theorem from one variable to multivariate power series over non-Archimedean fields, providing bounds on zero sets and discussing effective computation methods.

Contribution

It generalizes the classical Strassmann theorem to multiple variables, characterizing zero set finiteness and bounds via ideal reductions.

Findings

Finiteness of zero sets in multivariate power series is characterized by ideal reductions.

Explicit bounds on the number of zeros are provided in the multivariate case.

Effective methods for approximate power series are discussed.

Abstract

By a theorem of Strassmann, a non-zero convergent power series in one variable over a complete non-Archimedean field has finitely many zeros, with an explicit bound on their number. We generalize this result to convergent power series in several variables, characterizing finiteness of the zero set and bounding its cardinality in terms of the reduction of the saturated ideal defined by the power series. We discuss how to make our result effective, under suitable assumptions, when working with approximate power series.