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International Algorithmic Number Theory Symposium

ANTS 1994: Algorithmic Number Theory pp 93–107Cite as

A new modular interpolation algorithm for factoring multivariate polynomials

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 877))

Abstract

In this paper we present a technique that uses a new interpolation scheme to reconstruct a multivariate polynomial factorization from a number of univariate factorizations. Whereas other interpolation algorithms for polynomial factorization depend on various extensions of the Hilbert irreducibility theorem, our approach is the first to depend only upon the classical formulation. The key to our technique is the interpolation scheme for multivalued black boxes originally developed by Ar et. al. [1]. We feel that this combination of the classical Hilbert irreducibility theorem and multivalued black boxes provides a particularly simple and intuitive approach to polynomial factorization.

Research supported by ONR Young Investigator Award N00014-93-1-0590 and United States—Israel Binational Science Foundation Grant 92-00226.

Research supported in part by the Advanced Research Projects Agency of the Department of Defense under ONR Contract N00014-92-J-1989, by ONR Contract N00014-92-J-1839, United States—Israel Binational Science Foundation Grant 92-00234 and in part by the U.S. Army Research Office through the Mathematical Science Institute of Cornell University.

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References

  1. Sigal Ar, Richard J. Lipton, Ronitt Rubinfeld, and Madhu Sudan. Reconstructing algebraic functions from mixed data. In 33th Symposium on Foundations of Computer Science, pages 503–512. ACM, 1992.

    Google Scholar 

  2. Michael Ben Or and Prasoon Tiwari. A deterministic algorithm for sparse multivariate polynomial interpolation. In 20th Symposium on Theory of Computing, pages 301–309. ACM, 1988.

    Google Scholar 

  3. Elwyn Ralph Berlekamp. Factoring polynomials over finite fields. Bell System technical Journal, 46:1853, 1967.

    Google Scholar 

  4. Elwyn Ralph Berlekamp. Factoring polynomials over large finite fields. Mathematics of Computation, 24(III):713–735, July 1970.

    Google Scholar 

  5. David G. Cantor and Hans Zassenhaus. A new algorithm for factoring polynomials over finite fields. Mathematics of Computation, 36(154):587–592, April 1981.

    Google Scholar 

  6. Michael Clausen, A. Dress, Johannes Grabmeier, and Marek Karpinski. On zerotesting and interpolation of fc-sparse multivariate polynomials over finite fields. Research Report 8522-CS, Universität Bonn, May 1988.

    Google Scholar 

  7. S. D. Cohen. The distribution of Galois groups and Hilbert's irreducibility theorem. Proceedings of the London Mathematical Society (3), 43:227–250, 1981.

    Google Scholar 

  8. Richard A. Demillo and Richard J. Lipton. A probabilistic remark on algebraic program testing. Information Processing Letters, 7(4):193–195, June 1978.

    Google Scholar 

  9. K. Dörge. Über die Seltenheit der reduziblen Polynome und der Normalgleichungen. Mathematische Annalen, 95:247–256, 1926.

    Google Scholar 

  10. K. Dörge. Zum Hilbertschen Irreduzibilitätssatz. Mathematische Annalen, 95:84–97, 1926.

    Google Scholar 

  11. K. Dörge. Einfacher Beweis des Hilbertschen Irreduzibilitätssatzes. Mathematische Annalen, 96:176–182, 1927.

    Google Scholar 

  12. Torsten Ekedahl. An effective version of the Hilbert irreducibility theorem. In Catherine Goldstein, editor, Séminaire de Théorie des Nombres, Paris 1988–1989, volume 91 of Progress in Mathematics, pages 241–249, Boston, 1990.

    Google Scholar 

  13. W. Franz. Untersuchugen aum Hilbertschen Irreduzibilitätssatz. Mathematische Zeitschrift, 33:275–293, 1931.

    Google Scholar 

  14. Michael D. Fried. On Hilbert's irreducibility theorem. Journal of Number Theory, 6:211–231, 1974.

    Google Scholar 

  15. Dima Yu. Grigor'ev and Marek Karpinski. The matching problem for bipartite graphs with polynomial bounded perminants is NC. In 28th Symposium on Foundations of Computer Science, pages 166–172. ACM, 1987.

    Google Scholar 

  16. Dima Yu. Grigor'ev, Marek Karpinski, and Michael F. Singer. Past parallel algorithms for sparse multivariate polynomial interpolation over finite fields. SIAM Journal of Computing, 19(6):1059–1063, 1990.

    Google Scholar 

  17. David Hilbert. über die Irreduzibilität ganzer rationaler Funktionen mit ganzzahligen Koeffizienten. Journal für reine und angewante Mathematik, 110:104–129, 1892.

    Google Scholar 

  18. Ming-Deh A. Huang. Factorization of polynomials over finite fields and decomposition of primes in algebraic number fields. Journal of Algorithms, 12(3):482–489, 1991.

    Google Scholar 

  19. Ming-Deh A. Huang. Generalized Riemann hypothesis and factoring polynomials over finite fields. Journal of Algorithms, 12(3):464–481, 1991.

    Google Scholar 

  20. Erich Kaltofen. Computing with polynomials given by straight-line programs II: Sparse factorization. In 26th Symposium on Foundations of Computer Science, pages 451–457. ACM, 1985.

    Google Scholar 

  21. Erich Kaltofen. A polynomial-time reduction from bivariate to univariate integral polynomial factorization. SIAM Journal of Computing, 14(2):469–489, May 1985.

    Google Scholar 

  22. Erich Kaltofen and Barry Marshall Trager. Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators. Journal of Symbolic Computation, 9(3):301–320, March 1990.

    Google Scholar 

  23. Hans-Wilhelm Knobloch. Zum Hilbertschen Irreduzibilitätssatz. Abhandlung Mathematische Seminar Univ. Hamburg, 19:176–190, 1955.

    Google Scholar 

  24. Hans-Wilhelm Knobloch. Die Seltenheit der reduziblen Polynome. Jarhesbericht der Deutsche Mathematische Verneinung, 59(1):12–19, 1956.

    Google Scholar 

  25. Leopold Kronecker. Grundzüge einer arithmetischen Theorie der algebraischen Größen. Journal für reine und angewante Mathematik, 92:1–122, 1882.

    Google Scholar 

  26. Arjen K. Lenstra, Hendrik W. Lenstra, Jr., and Laslo Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen, 261:515–534, 1982.

    Google Scholar 

  27. Victor S. Miller. Factoring polynomials via relation-finding. In Danny Dolev, Zvi Galil, and Michael Rodeh, editors, Theory of Computing and Systems, volume 601 of Lecture Notes in Computer Science, pages 115–121, New York, 1992. Springer-Verlag.

    Google Scholar 

  28. E. Ng, editor. EUROSAM '79, volume 72 of Lecture Notes in Computer Science, Berlin-Heidelberg-New York, 1979. Springer-Verlag.

    Google Scholar 

  29. Lajos Rónyai. Galois groups and factoring polynomials over finite fields. In 30th Symposium on Foundations of Computer Science, pages 99–104. ACM, 1989.

    Google Scholar 

  30. Andrej Schinzel. On Hilbert's irreducibility theorem. Annales Polinici Mathematici, 16:333–340, 1965.

    Google Scholar 

  31. Arnold Schönhage. Factorization of univariate integer polynomials by diophantine approximation and an improved basis reduction algorithm. In Jan Paredaens, editor, Automata, Languages and Programming, volume 172 of Lecture Notes in Computer Science, pages 436–447, Berlin-Heidelberg-New York, 1984. Springer-Verlag.

    Google Scholar 

  32. Jacob T. Schwartz. Probabilistic algorithms for verification of polynomial identities. Journal of the Association for Computing Machinery, 27:701–717, 1980.

    Google Scholar 

  33. V. G. Sprindzuk. Reducibility of polynomials and rational points on algebraic curves. Soviet Mathematics, 21:331–334,1980.

    Google Scholar 

  34. V. G. Sprindzuk. Arithmetic specializations in polynomials. Journal für reine und angewante Mathematik, 340: 26–52, 1983.

    Google Scholar 

  35. Joachim von zur Gathen. Hensel and Newton methods in valuation rings. Mathematics of Computation, 42(166):637–661, April 1984.

    Google Scholar 

  36. Joachim von zur Gathen and Erich Kaltofen. Factoring sparse multivariate polynomials. Journal of Computer and System Sciences, 31:265–287, 1985.

    Google Scholar 

  37. Richard Eliot Zippel. Probabilistic algorithms for sparse polynomials. In Ng [28], pages 216–226.

    Google Scholar 

  38. Richard Eliot Zippel. Newton's iteration and the sparse Hensel algorithm. In Paul Wang, editor, SYMSAC '81: Proceedings of the 1981 ACM Symposium on Symbolic and Algebraic Computation, pages 68–72. Association for Computing Machinery, 1981.

    Google Scholar 

  39. Richard Eliot Zippel. Interpolating polynomials from their values. Journal of Symbolic Computation, 9:375–403, March 1990.

    Google Scholar 

  40. Richard Eliot Zippel. Effective Polynomial Computation. Kluwer Academic Press, Boston, 1993.

    Google Scholar 

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Authors and Affiliations

  1. Dept.of Computer Science, Cornell University, 14853, Ithaca, NY, USA

    Ronitt Rubinfeld & Richard Zippel

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  1. Ronitt Rubinfeld
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  2. Richard Zippel
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Editor information

Leonard M. Adleman Ming-Deh Huang

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© 1994 Springer-Verlag Berlin Heidelberg

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Rubinfeld, R., Zippel, R. (1994). A new modular interpolation algorithm for factoring multivariate polynomials. In: Adleman, L.M., Huang, MD. (eds) Algorithmic Number Theory. ANTS 1994. Lecture Notes in Computer Science, vol 877. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58691-1_47

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  • DOI: https://doi.org/10.1007/3-540-58691-1_47

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-58691-3

  • Online ISBN: 978-3-540-49044-9

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