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Polynomial factorization 1987–1991

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Literature Cited

  • Abbott, J. A., “Factorization of polynomials over algebraic function fields,” Doctoral Thesis, Univ. Bath, England, 1988.

    Google Scholar 

  • Abbott, J. A., “Recovery of algebraic numbers from their p-adic approximations,” Proc. ACMSIGSAM 1989 Internat. Symp. Symbolic Algebraic Comput., pp. 112–120 (1989).

    Google Scholar 

  • Abbott, J. A., Bradford, R. J., and Davenport, J. H., “The Bath algebraic number package,” Proc. 1986 ACM Symp. Symbolic Algebraic Comp., pp. 250–253 (1986).

    Google Scholar 

  • Abhyankar, S. and Bajaj, C., “Automatic rational parameterization of curves and surfaces III: Algebraic plane curves,” Computer Aided Geometric Design 5, pp. 308–321 (1988).

    Google Scholar 

  • Adleman, L. M., Manders, K., and Miller, G. L., “On taking roots in finite fields,” Proc. 18th IEEE Symp. Foundations Comp. Sci., pp. 175–178 (1977).

    Google Scholar 

  • Adleman, L. M. and Odlyzko, A. M., “Irreducibility testing and factorization of polynomials,” Math. Comp. 41, pp. 699–709 (1983).

    Google Scholar 

  • Ankeny, N. C., “The least quadratic non residue,” Ann. of Math. 55/1, pp. 65–72 (1952).

    Google Scholar 

  • Arwin, A., “Über die Kongruenzen von dem fünften und höheren Graden nach einem Primzahlmodulus,” Arkiv f. matematik, astronom. o. fysik 14, pp. 1–46 (1918). In German.

    Google Scholar 

  • Bach, E. and Shoup, V., “Factoring polynomials using fewer random bits,” J. Symbolic Comput. 9/3, pp. 229–239 (1990).

    Google Scholar 

  • Bajaj, C., Canny, J., Garrity, T., and Warren, J., “Factoring rational polynomials over the complexes,” Proc. ACM-SJGSAM 1989 Internat. Symp. Symbolic Algebraic Comput., pp. 81–90 (1989).

    Google Scholar 

  • Batut, C., Bernardi, D., Cohen, H., and Olivier, M., “User's Guide to PARI-GP,” Manual, February 1991.

    Google Scholar 

  • Ben-Or, M., “Probabilistic algorithms in finite fields,” Proc. 22nd IEEE Symp. Foundations Comp. Sci., pp. 394–398 (1981).

    Google Scholar 

  • Ben-Or, M. and Tiwari, P., “A deterministic algorithm for sparse multivariate polynomial interpolation,” Proc. 20th Annual ACM Symp. Theory Comp., pp. 301–309 (1988).

    Google Scholar 

  • Ben-Or, M. and Tiwari, P., “Simple algorithms for approximating all roots of a polynomial with real roots,” J. Complexity 6, pp. 417–442 (1990).

    Google Scholar 

  • Berkowitz, S. J., “On computing the determinant in small parallel time using a small number of processors,” Inform. Process. Letters 18, pp. 147–150 (1984).

    Google Scholar 

  • Berlekamp, E. R., “Factoring polynomials over finite fields,” Bell Systems Tech. J. 46, pp. 18532–1859 (1967). Republished in revised form in: E. R. Berlekamp, Algebraic Coding Theory, Chapter 6, McGraw-Hill Publ., New York 1968.

    Google Scholar 

  • Berlekamp, E. R., “Factoring polynomials over large finite fields,” Math. Comp. 24, pp. 713–735 (1970).

    Google Scholar 

  • Butler, M. C. R., “On the reducibility of polynomials over a finite field,” Quart. J. Math., Oxford Ser. (2) 5, pp. 102–107 (1954).

    Google Scholar 

  • Camion, P., “Un algorithme de construction des idempotents primitifs d'ideaux d'algebres sur Fq,” Ann. Discrete Math 12, pp. 55–63 (1982).

    Google Scholar 

  • Cantor, D. G. and Zassenhaus, H., “A new algorithm for factoring polynomials over finite fields,” Math. Comp. 36, pp. 587–592 (1981).

    Google Scholar 

  • Chistov, A. L., “The construction of a finite field in polynomial time,” Proc. 7 All-Union Conf. on Math. Logic (Novosibirsk), p. 196 (1984). In Russian.

    Google Scholar 

  • Chistov, A. L., “Efficient factorization of polynomials over a local field,” Soviet Math. Doklady (AMS Translation) 37/2, pp. 430–433 (1987).

    Google Scholar 

  • Chistov, A. L. and Grigoryev, D. Yu., “Polynomial-time factoring of multivariable polynomials over a global field,” LOMI Preprints E-5-82, USSR Acad. Sci., Steklov Math. Inst., Leningrad, 1982.

    Google Scholar 

  • Chistov, A. L. and Grigoryev, D. Yu., “Subexponential-time solving of systems of algebraic equations I,” LOMI Preprints E-9-83, USSR Acad. Sci., Steklov Math. Inst., Leningrad, 1983.

    Google Scholar 

  • Claybrook, B. G., “A new approach to the symbolic factorization of multivariate polynomials,” Artificial Intelligence 7, pp. 203–241 (1976).

    Google Scholar 

  • Collins, G. E., “Quantifier elimination for real closed fields by cylindrical algebraic decomposition,” Proc. 2nd GI Conf. Automata, Theory Formal Lang., Springer Lec. Notes Comp. Sci. 33, pp. 515–532 (1975).

    Google Scholar 

  • Collins, G. E., “Infallible calculation of polynomial zeros to specified precision,” in Mathematical Software III, edited by J. R. Rice; Academic Press, New York, pp. 35–68, 1977.

    Google Scholar 

  • Cook, S. A., “A taxonomy of problems with fast parallel algorithms,” Inf. Control 64, pp. 2–22 (1985).

    Google Scholar 

  • Davenport, J. H., Gianni, P., and Trager, B. M., “Scratchpad's view of algebra II: a categorical view of factorization,” in Proc. 1991 Internat. Symp. Symbolic Algebraic Comput., edited by S. M. Watt; ACM Press, pp. 32–38, 1991.

    Google Scholar 

  • Diaz, A., Kaltofen, E., Schmitz, K., and Valente, T., “DSC A System for Distributed Symbolic Computation,” in Proc. 1991 Internat. Symp. Symbolic Algebraic Comput., edited by S. M. Watt; ACM Press, pp. 323–332, 1991.

    Google Scholar 

  • Dicrescenzo, C. and Duval, D., “Le système D5 de calcul formel avec des nombres algébriques,” Chapter 1 of the Doctoral Thesis by D. Duval, Univ. Grenoble, 1987. In French.

    Google Scholar 

  • Duval, D., “Absolute factorization of polynomials: a geometric approach,” SIAM J. Comput. 20/1, pp. 1–21 (1991).

    Google Scholar 

  • Dvornicich, R. and Traverso, C., “Newton symmetric functions and the arithmetic of algebraically closed fields,” in Proc. AAECC-5, Springer Lect. Notes Comput. Sci. 356; pp. 216–224, 1987.

    Google Scholar 

  • Freeman, T. S., Imirzian, G., Kaltofen, E., and Lakshman Yagati, “Dagwood: A system for manipulating polynomials given by straight-line programs,” ACM Trans. Math. Software 14/3, pp. 218–240 (1988).

    Google Scholar 

  • Fröhlich, A. and Shepherdson, J. C., “Effective procedures in field theory,” Phil. Trans. Roy. Soc., Ser. A 248, pp. 407–432 (1955/56).

    Google Scholar 

  • Gantmacher, F. R., The Theory of Matrices, Vol. 1; Chelsea Publ. Co., New York, N. Y., 1960.

    Google Scholar 

  • von zur Gathen, J., “Hensel and Newton methods in valuation rings,” Math. Comp. 42, pp. 637–661 (1984).

    Google Scholar 

  • von zur Gathen, J., “Irreducibility of multivariate polynomials,” J. Comp. System Sci. 31, pp. 225–264 (1985).

    Google Scholar 

  • von zur Gathen, J., “Factoring polynomials and primitive elements for special primes,” Theoretical Comput. Sci. 52, pp. 77–89 (1987).

    Google Scholar 

  • von zur Gathen, J., “Functional decomposition of polynomials: the tame case,” J. Symbolic Comput. 8/3, pp. 281–299 (1990a).

    Google Scholar 

  • von zur Gathen, J., “Functional decomposition of polynomials: the wild case,” J. Symbolic Comput. 10/5, pp. 437–452 (1990b).

    Google Scholar 

  • von zur Gathen, J. and Shoup, V., “Computing Frobenius maps and factoring polynomials,” Manuscript, Toronto, November 1991.

    Google Scholar 

  • Geddes, K. O., Gonnet, G. H., and Smedley, T. J., “Heuristic methods for operations with algebraic numbers,” in Proc. ISSAC '88, Springer Lec. Notes Comput. Sci. 358, edited by P. Gianni; pp. 475–480, 1988.

    Google Scholar 

  • Grigoryev, D. Yu., Karpinski, M., and Singer, M. F., “Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields,” SIAM J. Comput. 19/6, pp. 1059–1063 (1990).

    Google Scholar 

  • Heintz, J. and Sieveking, M., “Absolute primality of polynomials is decidable in random polynomial-time in the number of variables,” Proc. ICALP '81, Springer Lec. Notes Comp. Sci. 115, pp. 16–28 (1981).

    Google Scholar 

  • Huang, M.-D. A., “Generalized Riemann hypothesis and factoring polynomials over finite fields,” J. Algorithms 12/3, pp. 464–481 (1991a).

    Google Scholar 

  • Huang, M.-D. A., “Factorization of polynomials over finite fields and decomposition of primes in algebraic number fields,” J. Algorithms 12/3, pp. 482–489 (1991b).

    Google Scholar 

  • Kaltofen, E., “Polynomial factorization,” in Computer Algebra., 2nd ed., edited by B. Buchberger et al.; Springer Verlag, Vienna, pp. 95–113, 1982.

    Google Scholar 

  • Kaltofen, E., “Effective Hilbert irreducibility,” Information and Control 66, pp. 123–137 (1985c).

    Google Scholar 

  • Kaltofen, E., “Polynomial-time reductions from multivariate to bi-and univariate integral polynomial factorization,” SJAM J. Comp. 14/2, pp. 469–489 (1985a).

    Google Scholar 

  • Kaltofen, E., “Fast parallel absolute irreducibility testing,” J. Symbolic Comput. 1, pp. 57–67 (1985b). Misprint corrections: J. Symbolic Comput. 9, p. 320 (1989).

    Google Scholar 

  • Kaltofen, E., “Deterministic irreducibility testing of polynomials over large finite fields,” J. Symbolic Comp. 4, pp. 77–82 (1987).

    Google Scholar 

  • Kaltofen, E., “Greatest common divisors of polynomials given by straight-line programs,” J. ACM 35/1, pp. 231–264 (1988).

    Google Scholar 

  • Kaltofen, E., “Factorization of polynomials given by straight-line programs,” in Randomness and Computation, Advances in Computing Research 5, edited by S. Micali; JAI Press, Greenwhich, Connecticut, pp. 375–412, 1989.

    Google Scholar 

  • Kaltofen, E., “Polynomial Factorization 1982–1986,” in Computers in Mathematics, Lecture Notes in Pure and Applied Mathematics 125, edited by D. V. Chudnovsky and R. D. Jenks; Marcel Dekker, Inc., New York, N. Y., pp. 285–309, 1990a.

    Google Scholar 

  • Kaltofen, E., “Computing the irreducible real factors and components of an algebraic curve,” Applic. Algebra Engin. Commun. Comput. 1/2, pp. 135–148 (1990b).

    Google Scholar 

  • Kaltofen, E., “Effective Noether irreducibility forms and applications,” Tech. Rep. 91-2, Dept. Comput. Sci., Rensselaer Polytech. Inst., Troy, N. Y., January 1991. Extended abstract in Proc. 23rd Ann. ACM Symp. Theory Comput., ACM Press, pp. 54–63 (1991).

    Google Scholar 

  • Kaltofen, E. and Lakshman Yagati, “Improved sparse multivariate polynomial interpolation algorithms,” Proc. ISSAC '88, Springer Lect. Notes Comput. Sci. 358, pp. 467–474 (1988).

    Google Scholar 

  • Kaltofen, E., Lakshman Y. N., and Wiley, J. M., “Modular rational sparse multivariate polynomial interpolation,” in Proc. 1990 Internat, Symp. Symbolic Algebraic Comput., edited by S. Watanabe and M. Nagata; ACM Press, pp. 135–139, 1990.

    Google Scholar 

  • Kaltofen, E., Musser, D. R., and Saunders, B. D., “A generalized class of polynomials that are hard to factor,” SIAM J. Comp. 12/3, pp. 473–485 (1983).

    Google Scholar 

  • Kaltofen, E. and Trager, B., “Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators,” J. Symbolic Comput. 9/3, pp. 301–320 (1990).

    Google Scholar 

  • Kaltofen, E. and Yui, N., “Explicit construction of Hilbert class fields of imaginary quadratic fields by integer lattice reduction,” in Number Theory New York Seminar 1989–1990, edited by D. V. Chudnovsky, G. V. Chudnovsky, H. Cohn, and M. B. Nathanson; Springer Verlag, New York, pp. 150–202, 1991.

    Google Scholar 

  • Kannan, R., Lenstra, A. K., and Lovász, L., “Polynomial factorization and nonrandomness of bits of algebraic and some transcendental numbers,” Math. Comp. 50, pp. 235–250 (1988).

    Google Scholar 

  • Kerner, I. O., “Ein Gesamtschrittverfahren zur Berechnung der Nullstellen von Polynomen,” Numer. Math. 8, pp. 290–294 (1966). In German.

    Google Scholar 

  • Knuth, D. E., The Art of Programming, Vol. 2, Semi-Numerical Algorithms, Ed. 2; Addison Wesley, Reading, MA, 1981.

    Google Scholar 

  • Kozen, D. and Landau, S., “Polynomial decomposition algorithms,” J. Symbolic Comp. 7/5, pp. 445–456 (1989).

    Google Scholar 

  • Kronecker, L., “Grundzüge einer arithmetischen Theorie der algebraischen Grossen,” J. reine angew. Math. 92, pp. 1–122 (1882).

    Google Scholar 

  • Landau, S., “Factoring polynomials over algebraic number fields,” SIAM J. Comp. 14, pp. 184–195 (1985). Erratum: SIAM J. Comput. 20/5, p. 998 (1991).

    Google Scholar 

  • Landau, S., “Simplification of nested radicals,” Proc. 30th Annual Symp. Foundations of Comp. Sci., pp. 314–319 (1989).

    Google Scholar 

  • Landau, S. and Miller, G. L., “Solvability by radicals,” J. Comp. System Sci. 30, pp. 179–208 (1985).

    Google Scholar 

  • Lenstra, A. K., “Factoring polynomials over algebraic number fields,” Proc. EUROCAL '83, Springer Lec. Notes Comp. Sci. 162, pp. 245–254 (1983).

    Google Scholar 

  • Lenstra, A. K., “Factoring multivariate polynomials over algebraic number fields,” SIAM J. Comp. 16, pp. 591–598 (1987).

    Google Scholar 

  • Lenstra, A. K., Lenstra, H. W., and Lovász, L., “Factoring polynomials with rational coefficients,” Math. Ann. 261, pp. 515–534 (1982).

    Google Scholar 

  • Lickteig, T. M., “On semialgebraic decision complexity,” Tech. Report TR-90-052, Internat. Computer Sci. Inst., Berkeley, California, September 1990. Habilitationsschrift.

    Google Scholar 

  • Lidl, R. and Niederreiter, H., Finite Fields; Addison-Wesley, Reading, MA, 1983.

    Google Scholar 

  • Linwood, D. A., “Roots of a polynomial via a parallel Newton's method,” Manuscript, Dept. Math., California State University, Fresno, CA, July 1990.

    Google Scholar 

  • Lipton, R. and Stockmeyer, L., “Evaluations of polynomials with superpreconditioning,” Proc. 8th ACM Symp. Theory Comp., pp. 174–180 (1976).

    Google Scholar 

  • Lombardi, H., “Algebre elementaire en temps polynomial,” Thèse Doctorat, Université de Franche-Comté, Besançon, France, June 1989. In French.

    Google Scholar 

  • Loos, R., “Computing in algebraic extensions,” in Computer Algebra, 2nd ed., edited by B. Buchberger et al.; Springer Verlag, Vienna, pp. 173–187, 1982.

    Google Scholar 

  • Lucks, M., “A fast implementation of polynomial factorization,” Proc. 1986 ACM Symp. Symbolic Algebraic Comp., pp. 228–232 (1986).

    Google Scholar 

  • Marden, M., The geometry of the zeros of a polynomial in a complex variable; Math. Surveys 3; AMS, Providence, R.I., 1949.

    Google Scholar 

  • Melenk, H., Möller, H. M., and Neun, W., “On Gröbner bases computation on a supercomputer using REDUCE,” Tech. Report SC 88-2, K. Zuse Zentrum Berlin, January 1988.

    Google Scholar 

  • Menezes, A. J., van Oorschot, P. C., and Vanstone, S. A., “Some computational aspects of root finding in GF(q m),” Proc. JSSAC '88, Springer Lec. Notes Comput. Sci. 358, pp. 259–270 (1988).

    Google Scholar 

  • Moenck, R. T., “On the efficiency of algorithms for polynomial factoring,” Math. Comp. 31, pp. 235–250 (1977).

    Google Scholar 

  • Monagan, M. B., “A heuristic irreducibility test for univariate polynomials,” Manuscript, Waterloo, Ontario, 1986. J. Symbolic Comput., to appear.

    Google Scholar 

  • Moore, P. M. A. and Norman, A. C., “Implementing a polynomial factorization problem,” Proc. 1981 ACM Symp. Symbolic Algebraic Comp., pp. 109–116 (1981).

    Google Scholar 

  • Musser, D. R., “Multivariate polynomial factorization,” J. ACM 22, pp. 291–308 (1975).

    Google Scholar 

  • Neff, C. A., “Specified precision polynomial root isolation is in NC,” Proc. 31st Annual Symp. Foundations Computer Sci., pp. 152–162 (1990).

    Google Scholar 

  • Noether, E., “Ein algebraisches Kriterium für absolute Irreduzibilität,” Math. Ann. 85, pp. 26–33 (1922).

    Google Scholar 

  • Pasquini, L. and Trigiante, D., “A globally convergent method for simultaneously finding polynomial roots,” Math. Comput. 44, pp. 135–149 (1985).

    Google Scholar 

  • Pinkert, J. R., “An exact method for finding roots of a complex polynomial,” ACM Trans. Math. Software 2/4, pp. 351–363 (1976).

    Google Scholar 

  • Plaisted, D. A., “Sparse complex polynomials and polynomial reducibility,” J. Comp. System Sci. 14, pp. 210–221 (1977).

    Google Scholar 

  • Rabin, M. O., “Probabilistic algorithms in finite fields,” SIAM J. Comp. 9, pp. 273–280 (1980).

    Google Scholar 

  • Rónyai, L., “Galois groups and factoring polynomials over finite fields,” Proc. 30th Annual Symp. Foundations of Comp. Sci., pp. 99–104 (1989).

    Google Scholar 

  • Schmidt, W. M., Equations over finite fields. An elementary approach; Springer Lect. Notes Math. 536; Springer Verlag, New York, N. Y., 1976.

    Google Scholar 

  • Schoof, R. J., “Elliptic curves over finite fields and the computation of square roots mod p,” Math. Comp. 44, pp. 483–494 (1985).

    Google Scholar 

  • Schwarz, Š., “On the reducibility of polynomials over a finite field,” Quart. J. Math. Oxford Ser. (2) 7, pp. 110–124 (1956).

    Google Scholar 

  • Schönhage, A., “The fundamental theorem of algebra in terms of computational complexity,” Tech. Report, Univ. Tübingen, 1982.

    Google Scholar 

  • Schönhage, A., “Factorization of univariate integer polynomials by diophantine approximation and an improved basis reduction algorithm,” Proc. ICALP '84, Springer Lec. Notes Comp. Sci. 172, pp. 436–447 (1984).

    Google Scholar 

  • Shoup, V., “New algorithms for finding irreducible polynomials over finite fields,” Math. Comput. 54/189, pp. 435–447 (1990a).

    Google Scholar 

  • Shoup, V., “On the deterministic complexity of factoring polynomials over finite fields,” Inform. Process. Letters 33, pp. 261–267 (1990b).

    Google Scholar 

  • Shoup, V., “Smoothness and factoring polynomials over finite fields,” Inform. Process. Letters 38, pp. 39–42 (1991a).

    Google Scholar 

  • Shoup, V., “A fast deterministic algorithm for factoring polynomials over finite fields of small characteristic,” in Proc. 1991 Internat. Symp. Symbolic Algebraic Comput., edited by S. M. Watt; ACM Press, pp. 14–21, 1991b.

    Google Scholar 

  • Shparlinskiy, I. E., “On some problems of theory of finite fields,” Manuscript, Moscow, August 1990.

    Google Scholar 

  • Smedley, T. J., “Fast methods for computation with algebraic numbers,” Ph. D. Thesis, Dept. Comput. Sci., Univ. Waterloo, 1989.

    Google Scholar 

  • Sprindžuk, V. G., “Arithmetic specializations in polynomials,” J. reine angew. Math. 340, pp. 26–52 (1983).

    Google Scholar 

  • Strassen, V., “Algebraic complexity theory,” in Handbook of Theoretical Computer Science, Algorithms and Complexity (Volume A), edited by J. van Leeuwen; Elsevier Science Publ., Amsterdam, pp. 633–672, 1990.

    Google Scholar 

  • Thiong ly, A., “A deterministic algorithm for factorizing polynomials over extensions GF(p m), p a small prime,” J. Inform. Optim. Sci. 10, pp. 337–344 (1989).

    Google Scholar 

  • Tonelli, A., “Bemerkung über die Auflösung quadratischer Congruenzen,” Nachrichten d. Akademie d. Wissenschaften in Göttingen, pp. 344–346 (1891). In German.

    Google Scholar 

  • Trager, B. M., “Algebraic factoring and rational function integration,” Proc. 1976 ACM Symp. Symbolic Algebraic Comp., pp. 219–228 (1976).

    Google Scholar 

  • Trevison, V. and Wang, P., “Practical factorization of univariate polynomials over finite fields,” in Proc. 1991 Internat. Symp. Symbolic Algebraic Comput., edited by S. M. Watt; ACM Press, pp. 22–31, 1991.

    Google Scholar 

  • Trotter, H. F., “Statistics on factoring polynomials mod p and p-adically,” SIGSAM Bulletin 16/3, pp. 24–29 (1982).

    Google Scholar 

  • Viry, G., “Factorization of multivariate polynomials with coefficients in Fp,” Manuscript, Univ. Niamey, Niamey, Niger (West Africa), November 1990. J. Symbolic Comput., to appear.

    Google Scholar 

  • van der Waerden, B. L., “Eine Bemerkung über die Unzerlegbarkeit von Polynomen,” Math. Ann. 102, pp. 738–739 (1930). In German.

    Google Scholar 

  • van der Waerden, B. L., Moderne Algebra; Springer Verlag, Berlin, 1940. English transl. publ. under the title “Modern algebra” by F. Ungar Publ. Co., New York, 1953

    Google Scholar 

  • Wang, P. S., “An improved multivariate polynomial factorization algorithm,” Math. Comp. 32, pp. 1215–1231 (1978).

    Google Scholar 

  • Wang, P. S., “Parallel univariate polynomial factorization on shared-memory multiprocessors,” in Proc. 1990 Internat. Symp. Symbolic Algebraic Comput., edited by S. Watanabe and M. Nagata; ACM Press, pp. 145–151, 1990.

    Google Scholar 

  • Weinberger, P. J. and Rothschild, L. P., “Factoring polynomials over algebraic number fields,” ACM Trans. Math. Software 2, pp. 335–350 (1976).

    Google Scholar 

  • Wiedemann, D., “Solving sparse linear equations over finite fields,” IEEE Trans. Inf. Theory IT-32, pp. 54–62 (1986).

    Google Scholar 

  • Wilf, H. S., “A global bisection algorithm for computing the zeros of polynomials in the complex plane,” J. ACM 25/3, pp. 415–420 (1978).

    Google Scholar 

  • Yokoyama, K., Noro, M., and Takeshima, T., “On factoring multi-variate polynomials over algebraically closed fields,” in Proc. 1991 Internat. Symp. Symbolic Algebraic Comput., edited by S. M. Watt; ACM Press, p. 297, 1990.

    Google Scholar 

  • Yun, D. Y. Y., “The Hensel lemma in algebraic manipulation,” Ph.D. Thesis, M.I.T., 1974. Reprint: Garland Publ., New York 1980.

    Google Scholar 

  • Zassenhaus, H., “On Hensel factorization I,” J. Number Theory 1, pp. 291–311 (1969).

    Google Scholar 

  • Zassenhaus, H., “On Hensel factorization II,” in Instituto Nazionale di Alta Mat., Symposia Mathematica 15; pp. 499–513, 1975.

    Google Scholar 

  • Zassenhaus, H., “Polynomial time factoring of integral polynomials,” SIGSAM Bulletin 15/2, pp. 6–7 (1981).

    Google Scholar 

  • Zippel, R., “Probabilistic algorithms for sparse polynomials,” Proc. EUROSAM '79, Springer Lec. Notes Comp. Sci. 72, pp. 216–226 (1979).

    Google Scholar 

  • Zippel, R., “Newton's iteration and the sparse Hensel algorithm,” Proc. '81 ACM Symp. Symbolic Algebraic Comp., pp. 68–72 (1981).

    Google Scholar 

  • Zippel, R., “Interpolating polynomials from their values,” J. Symbolic Comput. 9/3, pp. 375–403 (1990).

    Google Scholar 

  • Zippel, R., “Rational function decomposition,” in Proc. 1991 Internat. Symp. Symbolic Algebraic Comput., edited by S. M. Watt; ACM Press, pp. 1–6, 1991.

    Google Scholar 

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Kaltofen, E. (1992). Polynomial factorization 1987–1991. In: Simon, I. (eds) LATIN '92. LATIN 1992. Lecture Notes in Computer Science, vol 583. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0023837

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