A solution to Kronecker's problem

  • Giovanni Gallo
  • Bhubaneswar Mishra
Article

Abstract

Consider ℤ[x1,...,xn], 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[x1,...,xn] 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.

Keywords

Ascending chain condition E-bases Gröbner bases Ideal membership problem Rapidly growing functions Ring of polynomials over the integers S-polynomials syzygies Wainer hierarchy 

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References

  1. 1.
    Ayoub, C. W.: On Constructing Bases for Ideals in Polynomial Rings over the Integers. J. Number Theory17, 204–225 (1983)Google Scholar
  2. 2.
    Bayer, D., Stillman, M.: On the Complexity of Computing Syzygies. J. Symbolic Comput.6, 135–147 (1988)Google Scholar
  3. 3.
    Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Ph.D. thesis, University of Innsbruck, Austria, 1965Google Scholar
  4. 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, 1989Google Scholar
  5. 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, 1986Google Scholar
  6. 6.
    Edwards, H. M.: Dedekind's Invention of Ideals. Bull Lond. Math. Soc.15, 8–17 (1983)Google Scholar
  7. 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 1990Google Scholar
  8. 8.
    Gallo, G.: Complexity Issues in Computational Algebra. Ph.D. thesis, Courant Institute of Mathematical Sciences, New York University, New York, 1992Google Scholar
  9. 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, 1992Google Scholar
  10. 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 1984Google Scholar
  11. 11.
    Keaton, J., Solovay, R.: Rapidly Growing Ramsey Functions. Unpublished manuscript, 1980Google Scholar
  12. 12.
    Kronecker, L., Hensel, K.: Vorlesungen über Zahlentheorie. Leipzig, 1901Google Scholar
  13. 13.
    Kronecker, L.: Leopld Kronecker's Werke. Hensel, K. (ed.) Vol. 3 Leipzig, Druck und Verlag von B. G. Teubner, 1895Google Scholar
  14. 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 1989Google Scholar
  15. 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 1983Google Scholar
  16. 16.
    Mishra, B.: Algorithmic Algebra. Berlin, Heidelberg, New York: Springer 1993Google Scholar
  17. 17.
    Mishra, B., Yap, C.: Notes on Gröbner Bases. Information Sciences48, 219–252 (1989)Google Scholar
  18. 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. 19.
    Möller, H. M.: On the Construction of Gröbner Bases Using Syzygies. J. Symb. Comput.6, 345–360 (1988)Google Scholar
  20. 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, 1992Google Scholar
  21. 21.
    Paris, J., Harrington, L.: A Mathematical Incompleteness in Peano Arithmetic. In: Handbook of Mathematical Logic. Barwise, J. (ed.) pp. 1113–1142 (1977)Google Scholar
  22. 22.
    Richman, F.: Constructive Aspects of Noetherian Rings. Proceedings Am. Math. Soc.,44, 436–441 (1974)Google Scholar
  23. 23.
    Scidenberg, A.: What Is Noetherian? Rend. Sem. Mat. Fis. Milano44, 55–61 (1974)Google Scholar
  24. 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. 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 1978Google Scholar
  26. 26.
    Simmons, H.: The Solution of a Decision Problems for Several Classes of Rings. Pacific J. Math.34, 547–557 (1970)Google Scholar
  27. 27.
    Szekeres, G.: A Canonical Basis for the Ideals of a Polynomial Domain. Am. Mathematical Monthly59(6), 379–386 (1952)Google Scholar
  28. 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 1967Google Scholar
  29. 29.
    Trotter, P. G.: Ideals in ℤ [x,y]. Acta Math. Acad. Sci. Hungar32, 63–73 (1978)Google Scholar
  30. 30.
    Wainer, S. S.: A Classification of the Ordinal Recursive Functions. Arch. Math. Logik13, 136–153 (1970)Google Scholar
  31. 31.
    Weber, H.: Leopold Kronecker. Jahresber. D.M.-V., Vol. 2, 1892Google Scholar
  32. 32.
    Weispfenning, V.: Some Bounds for the Construction of Gröbner Bases, Preprint, Mathematisches Institut der Universität, Heidelberg, Germany, 1987Google Scholar
  33. 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. 34.
    Zacharias, G.: Generalized Gröbner Bases in Commutative Polynomial Rings. Master's thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1978Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Giovanni Gallo
    • 1
  • Bhubaneswar Mishra
    • 2
  1. 1.Dipartimento di MatematicaUniversitá di CataniaItalia
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew York

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