In-depth comparison of the Berlekamp -- Massey -- Sakata and the Scalar-FGLM algorithms: the non adaptive variants

TL;DR

This paper provides a detailed comparison of the Berlekamp--Massey--Sakata and Scalar-FGLM algorithms for computing ideals of multi-dimensional linear recurrent sequences, revealing fundamental behavioral differences.

Contribution

It offers the first thorough analysis showing that these algorithms cannot be made to behave identically through simple modifications.

Findings

The algorithms exhibit fundamentally different behaviors.

One cannot modify one algorithm to replicate the other.

The comparison clarifies the distinct properties of each method.

Abstract

We compare thoroughly the Berlekamp -- Massey -- Sakata algorithm and the Scalar-FGLM algorithm, which compute both the ideal of relations of a multi-dimensional linear recurrent sequence. Suprisingly, their behaviors differ. We detail in which way they do and prove that it is not possible to tweak one of the algorithms in order to mimic exactly the behavior of the other.