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Parallel methods for absolute irreducibility testing

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Abstract

A heuristic algorithm for testing absolute irreducibility of multivariate polynomials over arbitrary fields using Newton polytopes was proposed in Gao and Lauder (Discrete Comput. Geom. 26:89–104, [2001]). A preliminary implementation by Gao and Lauder (2003) established a wide range of families of low degree and sparse polynomials for which the algorithm works efficiently and with a high success rate. In this paper, we develop a BSP variant of the absolute irreducibility testing via polytopes, with the aim of producing a more memory and run-time efficient method that can provide wider ranges of applicability, specifically in terms of the degrees of the input polynomials. In the bivariate case, we describe a balanced load scheme and a corresponding data distribution leading to a parallel algorithm whose efficiency can be established under reasonably realistic conditions. This is later incorporated in a doubly parallel algorithm in the multivariate case that achieves similar scalable performance. Both parallel models are analyzed for efficiency, and the theoretical analysis is compared to the performance of our experiments. In the empirical results we report, we achieve absolute irreducibility testing for bivariate and trivariate polynomials of degrees up to 30,000, and for low degree multivariate polynomials with more than 3,000 variables. To the best of our knowledge, this sets a world record in establishing absolute irreducibility of multivariate polynomials.

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References

  1. Abu Salem F, Gao S, Lauder AGB (2004) Factoring polynomials via polytopes. In: Gutierrez J (ed) International symposium on symbolic and algebraic computation. ACM Press, Santander, Spain, pp 4–11

    Chapter  Google Scholar 

  2. Abu Salem FK (2008) An efficient sparse adaptation of the polytope method over \(\mathbb{F}_{p}\) and a record-high binary bivariate factorisation, 43:311–341

  3. Adleman LM (1994) The function field sieve. In: Adleman LM, Huang M-D (eds) Algorithmic number theory. Lecture notes in computer science, vol 877. Springer, Berlin, pp 108–121

    Google Scholar 

  4. Bisseling RH (2004) Parallel scientific computation: a structured approach using BSP and MPI. Oxford University Press, New York

    MATH  Google Scholar 

  5. Duval D (1991) Absolute factorization of polynomials: a geometric approach. SIAM J Comput 20:1–21

    Article  MATH  MathSciNet  Google Scholar 

  6. Gao S (2001) Absolute irreducibility of polynomials via Newton polytopes. J Algebra 237:501–520

    Article  MATH  MathSciNet  Google Scholar 

  7. Gao S (2003) Factoring multivariate polynomials via partial differential equations. Math Comput 72:801–822

    MATH  Google Scholar 

  8. Gao S, Lauder AGB (2001) Decomposition of polytopes and polynomials. Discrete Comput Geom 26:89–104

    MATH  MathSciNet  Google Scholar 

  9. Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. W.H. Freeman, New York

    MATH  Google Scholar 

  10. von zur Gathen J, Kaltofen E (1985) Factorization of multivariate polynomials over finite fields. Math Comput 45:251–261

    Article  MATH  Google Scholar 

  11. Grünbaum B (1967) Convex polytopes. Wiley, New York

    MATH  Google Scholar 

  12. Hirschfeld JWP (1979) Projective geometries over finite fields. Clarendon Press, Oxford

    MATH  Google Scholar 

  13. Kaltofen E (1985) Polynomial-time reductions from multivariate to bi- and univariate integral polynomial factorization. SIAM J Comput 14:469–489

    Article  MATH  MathSciNet  Google Scholar 

  14. Kumar V, Grama A, Gupta A, Karypis G (1994) Introduction to parallel computing: design and analysis of algorithms. The Benjamin/Cummings Publishing Company, Redwood City

    MATH  Google Scholar 

  15. Lenstra AK (1985) Factoring multivariate polynomials over finite fields. J Comput Syst Sci 30:235–248

    Article  MATH  MathSciNet  Google Scholar 

  16. Lenstra AK (1987) Factoring multivariate polynomials over algebraic fields. SIAM J Comput 16:591–598

    Article  MATH  MathSciNet  Google Scholar 

  17. Lidl R, Niederreiter H (1983) Finite fields, encyclopedia of mathematics and its applications, vol 20. Addison-Wesley, Reading

    Google Scholar 

  18. Hill JMD, McColl WF, Skillicorn DB (1996) Questions and answers about BSP. Report PRG-TR-15-96. Oxford University Computing Laboratory, 1996

  19. Hill JMD, McColl WF, Stefanescu DC, Goudrea MW, Lang K, Rao SB, Suel T, Tsantilas T, Bisseling RH (1998) BSPlib: the BSP programming library. Parallel Comput 24:1947–1980

    Article  Google Scholar 

  20. O’Rourke J (1998) Computational geometry in C, 2nd edn. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  21. Ostrowski AM (1921) Über die Bedeutung der Theorie der konvexen Polyeder für die formale Algebra. Jahresber Dtsch Math Ver 30:98–99

    MATH  Google Scholar 

  22. Schneider R (1993) Convex bodies: the Brunn-Minkowski theory. Encyclopedia of Mathematics and its Applications, vol 44. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  23. Stichtenoth H (1993) Algebraic function fields and codes. Universitext. Springer, Berlin

    MATH  Google Scholar 

  24. Szöyi T (1997) Some applications of algebraic curves in finite geometry and combinatorics. In: Bailey RA (ed) Surveys in combinatorics. London mathematical society lecture notes series, vol 241. Cambridge University Press, Cambridge, pp 197–236

    Google Scholar 

  25. Valiant LG (1990) A bridging model for parallel computation. Commun ACM 33:103–111

    Article  Google Scholar 

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Authors and Affiliations

  1. Computer Science Department, American University of Beirut, P.O. Box 11-0236, Riad El Solh Beirut, 1107 2020, Lebanon

    Fatima K. Abu Salem

  2. Department of Computer Science, St. Francis Xavier University, Antigonish, NS, B2G 2W5, Canada

    Laurence T. Yang

Authors
  1. Fatima K. Abu Salem
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  2. Laurence T. Yang
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Correspondence to Laurence T. Yang.

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Abu Salem, F.K., Yang, L.T. Parallel methods for absolute irreducibility testing. J Supercomput 46, 181–212 (2008). https://doi.org/10.1007/s11227-008-0205-1

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  • DOI: https://doi.org/10.1007/s11227-008-0205-1

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