Abstract
This paper presents an algorithm for computing the polynomial remainder sequence (PRS) and corresponding cofactor sequences of sparse multivariate polynomials over a number field \({\mathbb K}\). Most conventional algorithms for computing PRSs are based on the pseudo remainder (Prem), and the celebrated subresultant theory for the PRS has been constructed on the Prem. The Prem is uneconomical for computing PRSs of sparse polynomials. Hence, in this paper, the concept of sparse pseudo remainder (spsPrem) is defined. No subresultant-like theory has been developed so far for the PRS based on spsPrem. Therefore, we develop a matrix theory for spsPrem-based PRSs. The computational formula for PRS, regardless of whether it is based on Prem or spsPrem, causes a considerable intermediate expression growth. Hence, we next propose a technique to suppress the expression growth largely. The technique utilizes the power-series arithmetic but no Hensel lifting. Simple experiments show that our technique suppresses the intermediate expression growth fairly well, if the sub-variable ordering is set suitably.
Work supported by Japan Society for Promotion of Science KAKENHI Grant number 15K00005.
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Brown, W.S.: On Euclid’s algorithm and the computation of polynomial greatest common divisors. JACM 18(4), 478–504 (1971)
Brown, W.S., Traub, J.F.: On Euclid’s algorithm and the theory of subresultants. JACM 18(4), 505–515 (1971)
Brown, W.S.: The subresultant PRS algorithm. ACM TOMS 4, 237–249 (1978)
Collins, G.E.: Polynomial remainder sequences and determinants. Am. Math. Mon. 71, 708–712 (1966)
Collins, G.E.: Subresultants and reduced polynomial remainder sequences. JACM 14, 128–142 (1967)
Ducos, L.: Optimizations of the subresultant algorithm. J. Pure Appl. Algebra 145, 149–163 (2000)
Habicht, W.: Zur inhomogenen Eliminationstheorie. Comm. Math. Helvetici 21, 79–98 (1948)
Hearn, A.C.: Non-modular computation of polynomial GCDS using trial division. In: Ng, E.W. (ed.) Symbolic and Algebraic Computation. LNCS, vol. 72, pp. 227–239. Springer, Heidelberg (1979). https://doi.org/10.1007/3-540-09519-5_74
Loos, R.: Generalized polynomial remainder sequence. In: Buchberger, B., Collins, G.E., Loos, R. (eds.) Computer Algebra. Computing Supplementum, vol. 4, pp. 115–137. Springer, Vienna (1982). https://doi.org/10.1007/978-3-7091-3406-1_9
Sasaki, T.: A subresultant-like theory for Buchberger’s procedure. JJIAM (Jap. J. Indust. Appl. Math.) 31, 137–164 (2014)
Sasaki, T., Furukawa, A.: Theory of multiple polynomial remainder sequence. Publ. RIMS (Kyoto Univ.) 20, 367–399 (1984)
Sasaki, T., Inaba, D.: Simple relation between the lowest-order element of ideal \(\langle G, H \rangle \) and the last element of polynomial remainder sequence. In: Proceedings of SYNASC 2017 (Symbolic and Numeric Algorithms for Scientific Computing), IEEE Computer Society (2017, in printing)
Sasaki, T., Suzuki, M.: Three new algorithms for multivariate polynomial GCD. J. Symb. Comput. 13, 395–411 (1992)
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Sasaki, T. (2018). A Theory and an Algorithm for Computing Sparse Multivariate Polynomial Remainder Sequence. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2018. Lecture Notes in Computer Science(), vol 11077. Springer, Cham. https://doi.org/10.1007/978-3-319-99639-4_24
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