# Comparing topologies on linearly recursive sequences

## TL;DR

This paper compares two natural topologies on the space of linearly recursive sequences of complex numbers, demonstrating that they are fundamentally different and not equivalent.

## Contribution

The paper establishes that the adic topology and the Krull topology on linearly recursive sequences are not equivalent, clarifying their distinct properties.

## Key findings

- The adic and Krull topologies are not equivalent on the space of linearly recursive sequences.
- The adic topology is induced by sequences with zero first term.
- The Krull topology is derived from the embedding into complex power series.

## Abstract

The space of linearly recursive sequences of complex numbers admits two distinguished topologies. Namely, the adic topology induced by the ideal of those sequences whose first term is $0$ and the topology induced from the Krull topology on the space of complex power series via a suitable embedding. We show that these topologies are not equivalent.