Quasi-Linear Guessing of Minimal Lexicographic Gröbner Bases of Ideals of C-Relations of Random Bi-Indexed Sequences
Résumé
Computing recurrence relations for sequences is a central problem in computer algebra, with applications in error-correcting codes, Gröbner basis computation, and sparse interpolation. While uni-indexed C-recursive sequences benefit from quasi-linear algorithms leveraging the half-gcd method, the extension to multi-indexed sequences remains computationally challenging. Existing methods for bi-indexed sequences achieve quadratic complexity at best, limiting their practical use. This paper presents a quasi-linear algorithm for computing lexicographic Gröbner bases of the ideal of C-relations associated to bi-indexed sequences. Our approach extends the half-gcd algorithm in $K^N$[$y$] by integrating a pseudo-Euclidean division. This approach shows how to leverage the bi-Hankel structure of the matrix, significantly improving the efficiency of computing minimal C-relations closing the complexity gap between the uni- and bi-indexed cases. Our algorithm is restricted to bi-indexed sequences whose associated bi-Hankel matrix has generic row rank profile.
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