Polynomial time algorithms for finding integer relations among real numbers

  • J. Hastad
  • B. Helfrich
  • J. Lagarias
  • C. P. Schnorr
Contributed Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 210)

Abstract

We present algorithms, which when given a real vector x∈ℝn and a parameter k∈ℕ as input either find an integer relation m∈ℤn, m≠0 with xTm=0 or prove there is no such integer relation with ‖m‖≦2k. One such algorithm halts after at most O(n3(k+n)) arithmetic operations using real numbers. It finds an integer relation that is no more than \(2^{\frac{{n - 2}}{2}}\)times longer than the length of the shortest relation for x. Given a rational input x∈ℚn this algorithm halts in polynomially many bit operations. The basic algorithm of this kind is due to Ferguson and Forcade (1979) and is closely related to the Lovàsz (1982) lattice basis reduction algorithm.

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References

  1. G. Bergman (1980) Notes on Ferguson and Forcade's Generalized Euclidean Algorithm. Unpublished paper. University of California at Berkeley.Google Scholar
  2. A. Brentjes (1981) Multi-dimensional Continued Fraction Algorithms. Math. Centre Tracts No. 145. Universiteit Amsterdam.Google Scholar
  3. H.R.P. Ferguson and R.W. Forcade (1979) Generalization of the euclidean algorithm for real numbers to all dimensions higher than two. Bulletin of the AMS 1,6 pp. 912–914.Google Scholar
  4. H.R.P. Ferguson (1984) A non-inductive GL(n,ℤ) algorithm that constructs integral linear relations. Preprint. Brigham Young UniversityGoogle Scholar
  5. H.R.P. Ferguson (1985) A short proof of the existence of vector Euclidean algorithms to appear in Proceedings of the AMS.Google Scholar
  6. C.G.J. Jacobi (1868) Allgemeine Theorie der Kettenbruchähnlichen Algorithmen. J. reine Angew. Math. 69 (1969), 29–64.Google Scholar
  7. A.K. Lenstra, H.W. Lenstra Jr., and L. Love (1982) Factoring polynomials with rational coefficients. Math. Ann. 21, 515–534.Google Scholar
  8. R. Kannan, A.K. Lenstra, and L. Lovàsz (1984) Polynomial factorization and nonrandomness of bits of algebraic and some transcendental numbers. Proc. 16th Ann. ACM Symp. on Theory of Computing, pp. 191–200.Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • J. Hastad
    • 1
  • B. Helfrich
    • 2
  • J. Lagarias
    • 3
  • C. P. Schnorr
    • 2
  1. 1.Department of Computer ScienceMITCambridge
  2. 2.Fachbereich Mathematik/InformatikUniversität FrankfurtGermany
  3. 3.Bell LaboratoriesAT&TMurray Hill

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