Polynomial-division-based algorithms for computing linear recurrence relations

Sparse polynomial interpolation, sparse linear system solving or modular rational reconstruction are fundamental problems in Computer Algebra. They come down to computing recurrence relations of a sequence with the Berlekamp–Massey algorithm. Likewise, multivariate interpolation and multidimensional cyclic code decoding require guessing sequence. Several algorithms solve this problem. The so-called Berlekamp–Massey–Sakata algorithm (1988) uses additions shifts by monomial. Scalar-FGLM (2015) relies on algebra operations multi-Hankel matrix, generalization Hankel matrix. Artinian Gorenstein border basis (2017) Gram-Schmidt process. We propose new for Gröbner ideal based solely arithmetic. This allows us both revisit through use divisions completely revise without operations. A key observation design is work mirror truncated generating series allowing arithmetic modulo monomial ideal. It appears have some similarities Padé approximants polynomial. As an addition from paper published at ISSAC conference, we give adaptive variant taking into account shape final gradually as it discovered. main advantage that its complexity terms queries only depends output basis. All these been implemented Maple report our comparisons.