Skip to main content
SpringerLink
Log in

The Uniformity Conjecture

  • Conference paper
  • First Online:
Book cover Computability and Complexity in Analysis (CCA 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2064))

Included in the following conference series:

Abstract

This paper discusses the relationship between the syntactic length of expressions built up from the integers using field operations, radicals and exponentials and logarithms, and the smallness of non zero complex numbers defined by such expressions. The Uniformity Conjecture claims that if the expressions are written in an expanded form in which all the arguments of the exponential function have absolute value bounded by 1, then a small multiple of the syntactic length gives a bound for the number of decimal places needed to distinguish the defined number from zero. The consequences of this conjecture are compared with some known results about closeness of approximation from Liouville, Baker, Waldschmidt, Thue-Siegel-Roth. A few of many practical computational consequences are stated. Also the problem of searching for a possible counterexample to the Uniformity Conjecture is discussed and some preliminary results are given.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Baker, Transcendental number theory, CUP, 1975

    Google Scholar 

  2. J.M Borwein and P.B. Borwein, On the Complexity of Familiar Functions and Numbers, SIAM Review, Vol 30, No 4, Dec 1988, pp 589–601

    Article  MATH  MathSciNet  Google Scholar 

  3. Blum, Cucker, Shub, Smale, Real Complexity and Computation, Springer

    Google Scholar 

  4. T.Y. Chow, What is a Closed-Form Number?, American Mathematical Monthly, Vol 106, No 5, 1999, pp 440–448

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Davenport, Y. Siret, and E. Tournier, Computer Algebra, Academic Press, 1993.

    Google Scholar 

  6. B.M.M. DeWeger, Solving exponential Diophantine equations using lattice basis reduction algorithms, J. Number Theory 26 (1987) 325–367.

    Article  MathSciNet  Google Scholar 

  7. J. Håstad, B. Just, J.C. Lagarias, and C.P. Schnorr, Polynomial time algorithms for finding integer relations among real numbers, SIAM J. Comput. 18 (1989). pp 859–881.

    Article  MATH  MathSciNet  Google Scholar 

  8. M. Hindry and J. Silverman, Diophantine Geometry, An Introduction, Springer Graduate Texts in Mathematics, 2000

    Google Scholar 

  9. E. Hlawka, J. Schoissengeier, and R. Taschner, Geometric and Analytic Number Theory. Springer Verlag, 1991

    Google Scholar 

  10. Namhyun Hur and James H. Davenport, An exact Real Arithmetic with Equality Determination, draft, (Bath U)

    Google Scholar 

  11. J.R. Johnson, Real Algebraic Number Computation using Interval Arithmetic, ISSAC’ 92, pp 195–205

    Google Scholar 

  12. P. Koiran, A weak version of the Blum, Shub and Smale Model, FOCS’ 93, pp 486-495; also in JCSS 54(1), pp 177–189, 1997

    MATH  MathSciNet  Google Scholar 

  13. S. Langley and D. Richardson, Bounds on Algebraic Expressions, preprint

    Google Scholar 

  14. S. Lang, Introduction to Transcendental Numbers, Addison Wesley, 1966

    Google Scholar 

  15. R. Loos, Computing in Algebraic Extensions, Computing, 4, (Suppl) pp 173–187, 1982

    Google Scholar 

  16. Martin Sombra, Estimaciones para el Teorema de Ceros de Hilbert, Tesis, Departamento de Matematica, Facultad de Ciencias Exactas y Naturales, 1998.

    Google Scholar 

  17. P. Philippon, Sur des hauteurs alternatives, I, Math Ann. 289 (1991), pp 255–283; II Ann. Inst. Fourier 44 (1994), pp 1043-1065; III, J. Math. Pures Appl. 74 (1995), pp 345-365

    Article  MATH  MathSciNet  Google Scholar 

  18. P. Philippon, Quelques remarques sur des questions d’approximation diophantine, Bull Austral. Math. Soc., Vol 59(2), (1999), pp 323–334.

    MATH  MathSciNet  Google Scholar 

  19. P. Philippon, Addendum a quelques remarques sur des questions d’approximation diophantine, Bull Austral. Math. Soc., vol 61(1), (2000), pp 167–169.

    Article  MATH  MathSciNet  Google Scholar 

  20. M. Pohst, Algorithmic Algebraic Number Theory

    Google Scholar 

  21. M. Pohst, On Validated Computing in Algebraic Number Fields, J. Symbolic Computation 24, pp 657–666, 1997

    Article  MATH  MathSciNet  Google Scholar 

  22. A. Macintyre and A. Wilkie, On the decidability of the real exponential field, in Kreiseliana, About and Around Georg Kreisel, A.K. Peters, 1996, pp 441–467.

    Google Scholar 

  23. M. Mignotte, Mathematics for Computer Algebra, Springer Verlag, 1991

    Google Scholar 

  24. D. Richardson, How to Recognise Zero, J. Symbolic Computation (1997), 24, pp 627–645

    Article  MATH  Google Scholar 

  25. D. Richardson, Multiplicative Independence of Algebraic Numbers and Expressions, Mega2000 conference Bath June 2000, to appear in Journal of Pure and Applied Algebra.

    Google Scholar 

  26. V.G. Sprindzhuk, Achievements and Problems in Diophantine Approximation Theory, Russian Mathematical Surveys 35 (4), 1980, pp 1–80

    Google Scholar 

  27. A. W. Strsebonski, Computing in the Field of Complex Algebraic Numbers, J. Symbolic Computation 24, pp 647–656, 1997

    Article  Google Scholar 

  28. Joris van der Hoeven, Automatic Numerical Expansions, in J.C. Bajard, D. Michelucci, J.M. Moreau, and J.M. Muller, editors, Proc. of the conference “Real numbers and computers”, Saint-Etienne, France, Pages 261–274, 1995

    Google Scholar 

  29. Joris van der Hoeven, Automatic Asymptotics, Ph.D. thesis, Ecole Polytechnique, 1997.

    Google Scholar 

  30. Joris van der Hoeven, Zero-testing, witness conjectures and differential diophantine approximation, Preprint

    Google Scholar 

  31. R. Zippel, Effective Polynomial Computation, Kluwer Academic Publishers, 1993

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics, Bath University, Bath, BA2 7AY, UK

    Daniel Richardson

Authors
  1. Daniel Richardson
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Wales Swansea, Singleton Park, Swansea, SA2 8PP, UK

    Jens Blanck

  2. Informatikzentrum, FernUniversität Hagen, 58084, Hagen, Germany

    Vasco Brattka  & Peter Hertling  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Richardson, D. (2001). The Uniformity Conjecture. In: Blanck, J., Brattka, V., Hertling, P. (eds) Computability and Complexity in Analysis. CCA 2000. Lecture Notes in Computer Science, vol 2064. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45335-0_15

Download citation

  • DOI: https://doi.org/10.1007/3-540-45335-0_15

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42197-9

  • Online ISBN: 978-3-540-45335-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation