Abstract
Algorithms for factoring polynomials in one or more variables over various coefficient domains are discussed. Special emphasis is given to finite fields, the integers, or algebraic extensions of the rationals, and to multivariate polynomials with integral coefficients. In particular, various squarefree decomposition algorithms and Hensel lifting techniques are analyzed. An attempt is made to establish a complete historic trace for today’s methods. The exponential worst case complexity nature of these algorithms receives attention.
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References
Abdali, S. K., Caviness, B. F., Pridor, A.: Modular Polynomial Arithmetic in Partial Fraction Decomposition. MACSYMA 1977, 253–261.
Adleman, L. M.: On Distinguishing Prime Numbers from Composite Numbers. Proc. 21 st Symp. Foundations Comp. Sci. IEEE 1980, 387–406.
Adleman, L. M., Odlyzko, A. M.: Irreducibility Testing and Factorization of Polynomials. Proc. 22nd Symp. Foundations Comp. Sci. IEEE 1981, 409–418.
Ben-Or, M.: Probabilistic Algorithms in Finite Fields. Proc. 22nd Symp. Foundations Comp. Sci. IEEE 1981, 394–398.
Berlekamp, E. R.: Factoring Polynomials over Large Finite Fields. Math. Comp. 24, 713–735 (1970).
Calmet, J., Loos, R.: Deterministic Versus Probabilistic Factorization of Integral Polynomials. EUROCAM 1982 (to appear).
Cantor, D. G.: Irreducible Polynomials with Integral Coefficients Have Succinct Certificates. J. of Algorithms 2, 385–392 (1981).
Cantor, D. G., Zassenhaus, H.: On Algorithms for Factoring Polynomials over Finite Fields. Math. Comp. 36, 587–592 (1981).
Claybrook, B. G.: Factorization of Multivariate Polynomials over the Integers. ACM SIGSAM Bulletin 10, 13 (Feb. 1976).
Collins, G. E.: Factoring Univariate Polynomials in Polynomial Average Time. EUROSAM 1979, 317–329.
Davenport, J. H., Trager, B. M.: Factorization over Finitely Generated Fields. SYMSAC 1981, 200–205.
Eisenstein, F. G.: Über die Irreductibilität und einige andere Eigenschaften der Gleichung, von welcher die Theilung der ganzen Lemniscate abhängt. J. f. d. reine u. angew. Math. 39, 160–179 (1850).
Gallagher, P. X.: Probabilistic Galois Theory. AMS Proc. Symp. in Pure Math., Analytic Number Theory 1972, 91–102.
Gelfond, A. O.: Transcendental and Algebraic Numbers. New York: Dover 1960.
Heintz, J., Sieveking, M.: Absolute Primality of Polynomials is Decidable in Random Polynomial Time in the Number of Variables. Springer Lecture Notes Comp. Sci., Vol. 115, pp. 16–28. Berlin-Heidelberg-New York: Springer 1981.
Horowitz, E.: Algorithms for Partial Fraction Decomposition and Rational Function Integration. SYMSAM 1971, 441–457.
Johnson, S. C.: Tricks for Improving Kronecker’s Method. Bell Laboratories Report 1966.
Kaltofen, E.: A Polynomial Reduction from Multivariate to Bivariate Integer Polynomial Factorization. ACM Proc. Symp. on Theory Comp. 1982, 261–266.
Kaltofen, E.: On the Complexity of Factoring Polynomials with Integer Coefficients. Ph.D. Thesis, RPI (1982), in preparation.
Kaltofen, E., Musser, D. R., Saunders, B. D.: A Generalized Class of Polynomials that Are Hard to Factor. SYMSAC 1981.
Knuth, D. E.: The Art of Computer Programming, Vol. 2. Seminumerical Algorithms. Reading, Mass.: Addison-Wesley 1969.
Knuth, D. E.: The Art of Computer Programming, Vol. 2. Seminumerical Algorithms. 2nd ed. Reading, Mass.: Addison-Wesley 1981.
Kronecker, L.: Grundzüge einer arithmetischen Theorie der algebraischen Größen. J. f. d. reine u. angew. Math. 92, 1–122 (1882).
Kung, H. T., Tong, D. M.: Fast Algorithms for Partial Fraction Decomposition. SIAM J. Comp. 582–593 (1977).
Lang, S.: Diophantine Geometry. New York: Interscience 1962.
Lazard, D.: On Polynomial Factorization. EUROCAM 1982, to appear.
Lenstra, A. K.: Lattices and Factorization of Polynomials. ACM SIGSAM Bulletin 15, 15–16 (August 1981).
Lenstra, A. K.: Lattices and Factorization of Polynomials. EUROCAM 1982, to appear.
Moenck, R. T.: On the Efficiency of Algorithms for Polynomial Factoring. Math. Comp. 31, 235–250 (1977).
Moore, P. M. A., Norman, A. C.: Implementing a Polynomial Factorization Problem. SYMSAC 1981,109–116.
Moses, J., Yun, D. Y. Y.: The EZGCD Algorithm. Proc. 1973 ACM National Conf., 159–166.
Musser, D. R.: Algorithms for Polynomial Factorization. Ph.D. Thesis and TR # 134, Univ. of Wisconsin 1971.
Musser, D. R.: Multivariate Polynomial Factorization. J. ACM 22, 291–308 (1976).
Musser, D. R.: On the Efficiency of a Polynomial Irreducibility Test. J. ACM 25, 271–282 (1978).
Narkiewicz, W.: Elementary and Analytic Theory of Algebraic Numbers. Warsaw: Polish Science Publ. 1974.
Noether, E.: Ein algebraisches Kriterium für absolute Irreduzibilität. Math. Ann. 85, 26–33 (1922).
Ostrowski, A. M.: On Multiplication and Factorization of Polynomials I. Aequationes Math. 13, 201–228 (1975).
Plaisted, D. A.: Some Polynomial and Integer Divisibility Problems are NP-Hard. SIAM J. Comp. 458–464 (1978).
Plaisted, D. A.: The Application of Multivariate Polynomials to Inference Rules and Partial Tests for Unsatisfiability. SIAM J. Comp. 9, 698–705 (1980).
Pratt, V. R.: Every Prime Has a Succinct Certificate. SIAM J. Comp. 4, 214–220 (1975).
Rabin, M. O.: Probabilistic Algorithms in Finite Fields. SIAM J. Comp. 9, 273–280 (1980).
Trager, B. M.: Algebraic Factoring and Rational Function Integration. SYMSAC 1976, 219–226.
Trotter, H. F.: Algebraic Numbers and Polynomial Factorization. AMS Short Course Series, Ann Arbor 1980.
van der Waerden, B. L.: Modern Algebra, Vol. 1. Engl, transl. by F. Blum. New York: Frederick Ungar Publ. Co. 1953.
Vaughan, R. C.: Bounds for the Coefficients of Cyclotomic Polynomials. Michigan Math. J. 21, 289–295 (1975).
Viry, G.: Factorisation des Polynomes ä Plusieurs Variables ä Coefficient Entiers. RAIRO Informatique Theorique 12, 305–318 (1978).
Viry, G.: Factorisation des Polynomes ä Plusieurs Variables. RAIRO Informatique Theorique 14, 209–223 (1980).
Wang, P. S.: Factoring Multivariate Polynomials over Algebraic. Number Fields. Math. Comp. 30, 324–336 (1976).
Wang, P. S.: Preserving Sparseness in Multivariate Polynomial Factorization. MACSYMA 1977, 55–61.
Wang, P. S.: An Improved Multivariate Polynomial Factoring Algorithm. Math. Comp. 32, 1215–1231 (1978).
Wang, P. S.: Parallel p-adic Constructions in the Univariate Polynomial Factoring Algorithm. MACSYMA 1979, 310–318.
Wang, P. S.: Analysis of the p-adic Construction of Multivariate Correction Coefficients in Polynomial Factorization: Iteration vs. Reeursion. EUROSAM 1979, 291–300.
Wang, P. S., Rothschild, L. P.: Factoring Multivariate Polynomials over the Integers. Math. Comp. 29, 935–950 (1975).
Wang, P. S., Trager, B. M.: New Algorithms for Polynomial Square-Free Decomposition over the Integers. SIAM J. Comp. 8, 300–305 (1979).
Weinberger, P. J.: Finding the Number of Factors of a Polynomial. 1981, submitted.
Weinberger, P. J., Rothschild, L. P.: Factoring Polynomials over Algebraic Number Fields. ACM Trans. Math. Software 2, 335–350 (1976).
Yun, D. Y. Y.: Hensel Meets Newton — Algebraic Construction in an Analytic Setting. In: Analytic Computational Complexity (Traub, J., ed.). New York: Academic Press 1976.
Yun, D. Y. Y.: On Squarefree Decomposition Algorithms. SYMSAC 1976, 26–35.
Yun, D. Y. Y.: On the Equivalence of Polynomial GCD and Squarefree Factorization Problems. MACSYMA 1977, 65–70.
Zassenhaus, H.: On Hensel Factorization I. J. Number Theory 1, 291–311 (1969).
Zassenhaus, H.: A Remark on the Hensel Factorization Method. Math. Comp. 32, 287–292 (1978).
Zassenhaus, H.: Polynomial Time Factoring of Integral Polynomials. ACM SIGSAM Bulletin 15, 6–7(1981).
Zippel, R. E.: Probabilistic Algorithms for Sparse Polynomials. EUROSAM 1979, 216–226.
Zippel, R. E.: Probabilistic Algorithms for Sparse Polynomials. Ph.D. Thesis, M.I.T., 1979.
Zippel, R. E.: Newton’s Iteration and the Sparse Hensel Algorithm. SYMSAC 1981, 68–72.
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Kaltofen, E. (1983). Factorization of Polynomials. In: Buchberger, B., Collins, G.E., Loos, R., Albrecht, R. (eds) Computer Algebra. Computing Supplementa, vol 4. Springer, Vienna. https://doi.org/10.1007/978-3-7091-7551-4_8
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DOI: https://doi.org/10.1007/978-3-7091-7551-4_8
Publisher Name: Springer, Vienna
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