A Theory and an Algorithm for Computing Sparse Multivariate Polynomial Remainder Sequence

  • Tateaki SasakiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11077)


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.


Multivariate polynomial remainder sequence Cofactor sequence Sparse multivariate polynomials Pseudo remainder Sparse pseudo remainder Subresultant Hearn’s trial-division algorithm 


  1. 1.
    Brown, W.S.: On Euclid’s algorithm and the computation of polynomial greatest common divisors. JACM 18(4), 478–504 (1971)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Brown, W.S., Traub, J.F.: On Euclid’s algorithm and the theory of subresultants. JACM 18(4), 505–515 (1971)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Brown, W.S.: The subresultant PRS algorithm. ACM TOMS 4, 237–249 (1978)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Collins, G.E.: Polynomial remainder sequences and determinants. Am. Math. Mon. 71, 708–712 (1966)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Collins, G.E.: Subresultants and reduced polynomial remainder sequences. JACM 14, 128–142 (1967)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ducos, L.: Optimizations of the subresultant algorithm. J. Pure Appl. Algebra 145, 149–163 (2000)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Habicht, W.: Zur inhomogenen Eliminationstheorie. Comm. Math. Helvetici 21, 79–98 (1948)MathSciNetCrossRefGoogle Scholar
  8. 8.
    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). Scholar
  9. 9.
    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). Scholar
  10. 10.
    Sasaki, T.: A subresultant-like theory for Buchberger’s procedure. JJIAM (Jap. J. Indust. Appl. Math.) 31, 137–164 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Sasaki, T., Furukawa, A.: Theory of multiple polynomial remainder sequence. Publ. RIMS (Kyoto Univ.) 20, 367–399 (1984)MathSciNetCrossRefGoogle Scholar
  12. 12.
    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)Google Scholar
  13. 13.
    Sasaki, T., Suzuki, M.: Three new algorithms for multivariate polynomial GCD. J. Symb. Comput. 13, 395–411 (1992)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.University of TsukubaTsukuba-shiJapan

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