Abstract
Let A, B ∈ K[ X, Y ] be two bivariate polynomials over an effective field K, and let G be the reduced Gröbner basis of the ideal I := 〈 A, B 〉 generated by A and B with respect to the usual degree lexicographic order. Assuming A and B sufficiently generic, we design a quasi-optimal algorithm for the reduction of P ∈ K[ X, Y ] modulo G, where "quasi-optimal" is meant in terms of the size of the input A, B, P. Immediate applications are an ideal membership test and a multiplication algorithm for the quotient algebra A := K[ X, Y ]/〈 A, B 〉, both in quasi-linear time. Moreover, we show that G itself can be computed in quasi-linear time with respect to the output size.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.