Skip to main content

Abstract

Consider ℤ[x 1,...,x n], the multivariate polynomial ring over integers involvingn variables. For a fixedn, we show that the ideal membership problem as well as the associated representation problem for ℤ[x 1,...,x n] are primitive recursive. The precise complexity bounds are easily expressible by functions in the Wainer hierarchy.

Thus, we solve a fundamental algorithmic question in the theory of multivariate polynomials over the integers. As a direct consequence, we also obtain a solution to certain foundational problem intrinsic to Kronecker's programme for constructive mathematics and provide an effective version of Hilbert's basis theorem. Our original interest in this area was aroused by Edwards' historical account of theKronecker's problem in the context of Kronecker's version of constructive mathematics.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Ayoub, C. W.: On Constructing Bases for Ideals in Polynomial Rings over the Integers. J. Number Theory17, 204–225 (1983)

    Google Scholar 

  2. Bayer, D., Stillman, M.: On the Complexity of Computing Syzygies. J. Symbolic Comput.6, 135–147 (1988)

    Google Scholar 

  3. Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Ph.D. thesis, University of Innsbruck, Austria, 1965

    Google Scholar 

  4. Dubé, T. W.: Quantitative Analysis Problems in Computer Algebra: Gröbner Bases and the Nullstellensatz. Ph.D. thesis, Courant Institute of Mathematical Sciences, New York University, New York, 1989

    Google Scholar 

  5. Dubé, T., Mishra, B., Yap, C. K.: Admissible Orderings and Bounds for Gröbner Bases Normal Form Algorithm. Technical Report No. 88, Courant Institute of Mathematical Sciences, New York University, New York, 1986

    Google Scholar 

  6. Edwards, H. M.: Dedekind's Invention of Ideals. Bull Lond. Math. Soc.15, 8–17 (1983)

    Google Scholar 

  7. Edwards, H. M.: Kronecker's Views on the Foundations of Mathematics. In: Proceedings of a Conference held at Vassar College in June 1988, Rowe, D., McCleary, J. (eds.). Boston, Massachusetts: Academic Press 1990

    Google Scholar 

  8. Gallo, G.: Complexity Issues in Computational Algebra. Ph.D. thesis, Courant Institute of Mathematical Sciences, New York University, New York, 1992

    Google Scholar 

  9. Gallo, G., Mishra, B.: A Solution to Kronecker's Problem. Technical Report No. 600, Courant Institute of Mathematical Sciences, New York University, New York, 1992

    Google Scholar 

  10. Kandri-Rody, A., Kapur, D.: Algorithms for Computing the Gröbner Bases of Polynomial Ideals over Various Euclidean Rings. In: Proceedings of EUROSAM '84, Lecture Notes in Computer Science, Vol. 174. Fitch, J. (ed.) pp. 195–206, Berlin, Heidelberg, New York: Springer 1984

    Google Scholar 

  11. Keaton, J., Solovay, R.: Rapidly Growing Ramsey Functions. Unpublished manuscript, 1980

  12. Kronecker, L., Hensel, K.: Vorlesungen über Zahlentheorie. Leipzig, 1901

  13. Kronecker, L.: Leopld Kronecker's Werke. Hensel, K. (ed.) Vol. 3 Leipzig, Druck und Verlag von B. G. Teubner, 1895

    Google Scholar 

  14. Lankford, D.: Generalized Gröbner Bases: Theory and Applications. In: Rewriting Techniques and Applications, Lecture Notes in Computer Science, Vol 355. Dershowitz, N. (ed.) pp. 203–221. Berlin, Heidelberg, New York: Springer 1989

    Google Scholar 

  15. Lazard, D.: Gröbner Bases, Gaussian Elimination and Resolution of Systems of Algebraic Equations. In: Proceedings for EUROCAL '83, Lecture Notes in Computer Science, Vol. 162, pp. 146–156. Berlin, Heidelberg, New York: Springer 1983

    Google Scholar 

  16. Mishra, B.: Algorithmic Algebra. Berlin, Heidelberg, New York: Springer 1993

    Google Scholar 

  17. Mishra, B., Yap, C.: Notes on Gröbner Bases. Information Sciences48, 219–252 (1989)

    Google Scholar 

  18. Mayr, E. W., Meyer, A. R.: The Complexity of the Word Problems for Commutative Semigroups and Polynomial Ideals. Adv. Math.64, 305–329 (1982)

    Google Scholar 

  19. Möller, H. M.: On the Construction of Gröbner Bases Using Syzygies. J. Symb. Comput.6, 345–360 (1988)

    Google Scholar 

  20. Moreno Socias, G.: Length of Polynomial Ascending Chains and Primitive Recursiveness. Notes Informelles De Calcul Formel 19, Équipe de Calcul Formel, Centre de Mathématiques, École Polytechnique, F-91128 Palaiseau cedex, France, 1992

    Google Scholar 

  21. Paris, J., Harrington, L.: A Mathematical Incompleteness in Peano Arithmetic. In: Handbook of Mathematical Logic. Barwise, J. (ed.) pp. 1113–1142 (1977)

  22. Richman, F.: Constructive Aspects of Noetherian Rings. Proceedings Am. Math. Soc.,44, 436–441 (1974)

    Google Scholar 

  23. Scidenberg, A.: What Is Noetherian? Rend. Sem. Mat. Fis. Milano44, 55–61 (1974)

    Google Scholar 

  24. Scidenberg, A.: An Elimination Theory for Differential Algebra. University of California Pub. in Math. New series, University of California, Berkeley and Los Angeles3, 31–66 (1956)

    Google Scholar 

  25. Sims, C.: The Role of Algorithms in the Teaching of Algebra. In: Topics in Algebra. Newman, M. F. (ed.) pp. 95–107. Lecture Notes in Mathematics, Vol. 697. Canberra: Proc 1978. Berlin, Heidelberg, New York: Springer 1978

    Google Scholar 

  26. Simmons, H.: The Solution of a Decision Problems for Several Classes of Rings. Pacific J. Math.34, 547–557 (1970)

    Google Scholar 

  27. Szekeres, G.: A Canonical Basis for the Ideals of a Polynomial Domain. Am. Mathematical Monthly59(6), 379–386 (1952)

    Google Scholar 

  28. Szekeres, G.: Metabelian Groups with Two Generators. In: Proceedings of the International Conference Theory of Groups, Canberra '65, pp. 323–346, Gordon and Breach 1967

    Google Scholar 

  29. Trotter, P. G.: Ideals in ℤ [x,y]. Acta Math. Acad. Sci. Hungar32, 63–73 (1978)

    Google Scholar 

  30. Wainer, S. S.: A Classification of the Ordinal Recursive Functions. Arch. Math. Logik13, 136–153 (1970)

    Google Scholar 

  31. Weber, H.: Leopold Kronecker. Jahresber. D.M.-V., Vol. 2, 1892

  32. Weispfenning, V.: Some Bounds for the Construction of Gröbner Bases, Preprint, Mathematisches Institut der Universität, Heidelberg, Germany, 1987

    Google Scholar 

  33. Yap., C. K.: A New Lower Bound Construction for Commutative Thue Systems with Applications. J. Symp. Comput.12, 1–27 (1991)

    Google Scholar 

  34. Zacharias, G.: Generalized Gröbner Bases in Commutative Polynomial Rings. Master's thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1978

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Supported by an Italian grant Italian MURST 40% Calcolo Algebraico e Simbolico 1993 and an NSF grant: #CCR-9002819.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gallo, G., Mishra, B. A solution to Kronecker's problem. AAECC 5, 343–370 (1994). https://doi.org/10.1007/BF01188747

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01188747

Keywords