A Polynomial-Division-Based Algorithm for Computing Linear Recurrence Relations

Abstract : Sparse polynomial interpolation, sparse linear system solving or modular rational reconstruction are fundamental problems in Computer Algebra. They come down to computing linear recurrence relations of a sequence with the Berlekamp–Massey algorithm. Likewise, sparse multivariate polynomial interpolation and multidi-mensional cyclic code decoding require guessing linear recurrence relations of a multivariate sequence. Several algorithms solve this problem. The so-called Berlekamp– Massey–Sakata algorithm (1988) uses polynomial additions and shifts by a monomial. The Scalar-FGLM algorithm (2015) relies on linear algebra operations on a multi-Hankel matrix, a multivariate generalization of a Hankel matrix. The Artinian Gorenstein border basis algorithm (2017) uses a Gram-Schmidt process. We propose a new algorithm for computing the Gröbner basis of the ideal of relations of a sequence based solely on multivariate polynomial arithmetic. This algorithm allows us to both revisit the Berlekamp–Massey–Sakata algorithm through the use of polynomial divisions and to completely revise the Scalar-FGLM algorithm without linear algebra operations. A key observation in the design of this algorithm is to work on the mirror of the truncated generating series allowing us to use polynomial arithmetic modulo a monomial ideal. It appears to have some similarities with Padé approximants of this mirror polynomial. Finally, we give a partial solution to the transformation of this algorithm into an adaptive one.
Document type :
Conference papers
Complete list of metadatas

Cited literature [25 references]  Display  Hide  Download

https://hal.inria.fr/hal-01784369
Contributor : Jérémy Berthomieu <>
Submitted on : Friday, June 1, 2018 - 4:11:53 PM
Last modification on : Friday, July 5, 2019 - 3:26:03 PM
Long-term archiving on : Wednesday, September 26, 2018 - 1:30:03 PM

File

main.pdf
Publisher files allowed on an open archive

Identifiers

Citation

Jérémy Berthomieu, Jean-Charles Faugère. A Polynomial-Division-Based Algorithm for Computing Linear Recurrence Relations. ISSAC 2018 - 43rd International Symposium on Symbolic and Algebraic Computation, Jul 2018, New York, United States. ⟨10.1145/3208976.3209017⟩. ⟨hal-01784369v2⟩

Share

Metrics

Record views

634

Files downloads

440