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Survey on Counting Special Types of Polynomials

  • Joachim von zur Gathen
  • Konstantin Ziegler
Chapter
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8942)

Abstract

Most integers are composite and most univariate polynomials over a finite field are reducible. The Prime Number Theorem and a classical result of Gauß count the remaining ones, approximately and exactly. For polynomials in two or more variables, the situation changes dramatically. Most multivariate polynomials are irreducible. This survey presents counting results for some special classes of multivariate polynomials over a finite field, namely the reducible ones, the \(s\)-powerful ones (divisible by the \(s\)th power of a nonconstant polynomial), the relatively irreducible ones (irreducible but reducible over an extension field), the decomposable ones, and also for reducible space curves. These come as exact formulas and as approximations with relative errors that essentially decrease exponentially in the input size.

Furthermore, a univariate polynomial \(f\) is decomposable if \(f = g\,\circ \,h\) for some nonlinear polynomials \(g\) and \(h\). It is intuitively clear that the decomposable polynomials form a small minority among all polynomials.

The tame case, where the characteristic \(p\) of \({\mathbb {F}}_{q}\) does not divide \(n = \deg f\), is fairly well-understood, and we obtain closely matching upper and lower bounds on the number of decomposable polynomials. In the wild case, where \(p\) does divide \(n\), the bounds are less satisfactory, in particular when \(p\) is the smallest prime divisor of \(n\) and divides \(n\) exactly twice. The crux of the matter is to count the number of collisions, where essentially different \((g, h)\) yield the same \(f\). We present a classification of all collisions at degree \(n = p^{2}\) which yields an exact count of those decomposable polynomials.

Keywords

Counting special polynomials Finite fields Combinatorics on polynomials Generating functions Analytic combinatorics Asymptotic behavior Multivariate polynomials Polynomial decomposition Ritt’s Second Theorem 
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Notes

Acknowledgments

This work was funded by the B-IT Foundation and the Land Nordrhein-Westfalen.

References

  1. Alekseyev, M.: A115457–A115472. In: The On-Line Encyclopedia of Integer Sequences. OEIS Foundation Inc. (2006). http://oeis.org. Last download 4 Dec 2012
  2. Artin, E.: Quadratische Körper im Gebiete der höheren Kongruenzen. II. (Analytischer Teil.). Math. Z. 19(1), 207–246 (1924). http://dx.doi.org/10.1007/BF01181075
  3. Avanzi, R.M., Zannier, U.M.: The equation \(f(X)=f(Y)\) in rational functions \(X=X(t)\), \(Y=Y(t)\). Compositio Math. 139(3), 263–295 (2003). http://dx.doi.org/10.1023/B:COMP.0000018136.23898.65
  4. Barton, D.R., Zippel, R.: Polynomial Decomposition Algorithms. J. Symbolic Comput. 1, 159–168 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  5. Blankertz, R.: A polynomial time algorithm for computing all minimal decompositions of a polynomial. ACM Commun. Comput. Algebra 48(1, Issue 187), 13–23 (2014)CrossRefGoogle Scholar
  6. Blankertz, R., von zur Gathen, J., Ziegler, K.: Compositions and collisions at degree \(p^2\). J. Symbolic Comput. 59, 113–145 (2013). ISSN 0747-7171. http://dx.doi.org/10.1016/j.jsc.2013.06.001. http://arxiv.org/abs/1202.5810. Extended Abstract in: Proceedings of the 2012 International Symposium on Symbolic and Algebraic Computation, ISSAC 2012, Grenoble, France, pp. 91–98 (2012)
  7. Bodin, A.: Number of irreducible polynomials in several variables over finite fields. Am. Math. Monthly 115(7), 653–660 (2008). ISSN 0002-9890zbMATHMathSciNetGoogle Scholar
  8. Bodin, A.: Generating series for irreducible polynomials over finite fields. Finite Fields Their Appl. 16(2), 116–125 (2010). http://dx.doi.org/10.1016/j.ffa.2009.11.002 CrossRefzbMATHMathSciNetGoogle Scholar
  9. Bodin, A., Dèbes, P., Najib, S.: Indecomposable polynomials and their spectrum. Acta Arith. 139(1), 79–100 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  10. Cade, J.J.: A new public-key cipher which allows signatures. In: Proceedings of the 2nd SIAM Conference on Applied Linear Algebra. SIAM, Raleigh (1985)Google Scholar
  11. Car, M.: Théorèmes de densité dans \(\mathbb{F}_{q}[X]\). Acta Arith. 48, 145–165 (1987)zbMATHMathSciNetGoogle Scholar
  12. Carlitz, L.: The arithmetic of polynomials in a Galois field. Am. J. Math. 54, 39–50 (1932)CrossRefGoogle Scholar
  13. Carlitz, L.: The distribution of irreducible polynomials in several indeterminates. Ill. J. Math. 7, 371–375 (1963)zbMATHGoogle Scholar
  14. Carlitz, L.: The distribution of irreducible polynomials in several indeterminates II. Can. J. Math. 17, 261–266 (1965)CrossRefzbMATHGoogle Scholar
  15. Cesaratto, E., von zur Gathen, J., Matera, G.: The number of reducible space curves over a finite field. J. Number Theory 133, 1409–1434 (2013). http://dx.doi.org/10.1016/j.jnt.2012.08.027 CrossRefzbMATHMathSciNetGoogle Scholar
  16. Cohen, S.: The distribution of irreducible polynomials in several indeterminates over a finite field. Proc. Edinburgh Math. Soc. 16, 1–17 (1968)CrossRefzbMATHMathSciNetGoogle Scholar
  17. Cohen, S.: Some arithmetical functions in finite fields. Glasgow Math. Soc. 11, 21–36 (1969)CrossRefGoogle Scholar
  18. Cohen, S.D.: Reducibility of sub-linear polynomials over a finite field. Bull. Korean Math. Soc. 22, 53–56 (1985)zbMATHMathSciNetGoogle Scholar
  19. Cohen, S.D.: Exceptional polynomials and the reducibility of substitution polynomials. Enseign. Math. (2) 36(1–2), 53–65 (1990a). ISSN 0013-8584Google Scholar
  20. Cohen, S.D.: The factorable core of polynomials over finite fields. J. Australian Math. Soc. Ser. A 49(02), 309–318 (1990b). http://dx.doi.org/10.1017/S1446788700030585
  21. Cohen, S.D., Matthews, R.W.: A class of exceptional polynomials. Trans. Am. Math. Soc. 345(2), 897–909 (1994). ISSN 0002-9947. http://www.jstor.org/stable/2155005
  22. Coulter, R.S., Havas, G., Henderson, M.: On decomposition of sub-linearised polynomials. J. Australian Math. Soc. 76(3), 317–328 (2004). ISSN 1446-7887. http://dx.doi.org/10.1017/S1446788700009885
  23. Dickson, L.E.: The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group. Part I & Part II. Ann. Math. 11, 65–120, 161–183 (1897). http://www.jstor.org/stable/1967217, http://www.jstor.org/stable/1967224
  24. Dorey, F., Whaples, G.: Prime and Composite Polynomials. J. Algebra 28, 88–101 (1974). http://dx.doi.org/10.1016/0021-8693(74)90023-4
  25. Eisenbud, D., Harris, J.: The dimension of the Chow variety of curves. Compositio Math. 83(3), 291–310 (1992)zbMATHMathSciNetGoogle Scholar
  26. Engstrom, H.T.: Polynomial substitutions. Am. J. Math. 63, 249–255 (1941). http://www.jstor.org/stable/2371520 CrossRefMathSciNetGoogle Scholar
  27. Faugère, J.-C., Perret, L.: High Order Derivatives and Decomposition of Multivariate Polynomials. In: May, J.P. (ed.) Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation ISSAC ’09, Seoul, Korea, pp. 207–214. ACM Press (2009). http://dx.doi.org/10.1145/1576702.1576732. ISBN 978-1-60558-609-0. Extended Abstract in: Extended Abstracts of the Second Workshop on Mathematical Cryptology, WMC 2008, pp. 15–19 (2008)
  28. Flajolet, P., Gourdon, X., Panario, D.: The complete analysis of a polynomial factorization algorithm over finite fields. J. Algorithms 40(1), 37–81 (2001). Meyer auf der Heide, F., Monien, B. (eds.) Extended Abstract in: Proceedings of the 23rd International Colloquium on Automata, Languages and Programming, ICALP 1996. LNCS, vol. 1099, pp. 232–243. Springer, Heidelberg (1996)Google Scholar
  29. Flajolet, P., Sedgewick, R.: Analytic Combinatorics, 824 pp. Cambridge University Press, Cambridge (2009). ISBN 0521898064Google Scholar
  30. Fried, M.D., MacRae, R.E.: On the invariance of chains of fields. Ill. J. Math. 13, 165–171 (1969)zbMATHMathSciNetGoogle Scholar
  31. Gao, S., Lauder, A.G.B.: Hensel lifting and bivariate polynomial factorisation over finite fields. Math. Comput. 71(240), 1663–1676 (2002). http://dx.doi.org/10.1090/S0025-5718-01-01393-X
  32. von zur Gathen, J.: Functional Decomposition of Polynomials: the Tame Case. J. Symbolic Comput. 9, 281–299 (1990a). http://dx.doi.org/10.1016/S0747-7171(08)80014-4. Extended Abstract in: Proceedings of the 28th Annual IEEE Symposium on Foundations of Computer Science, Los Angeles, CA (1987)
  33. von zur Gathen, J.: Functional Decomposition of Polynomials: the Wild Case. J. Symbolic Comput. 10, 437–452 (1990b). http://dx.doi.org/10.1016/S0747-7171(08)80054-5
  34. von zur Gathen, J.: Factorization and Decomposition of Polynomials. In: Mikhalev, A.V., Pilz, G.F. (eds.) The Concise Handbook of Algebra, pp. 159–161. Kluwer Academic Publishers, Dordrecht (2002). ISBN 0-7923-7072-4Google Scholar
  35. von zur Gathen, J.: Counting reducible and singular bivariate polynomials. Finite Fields Their Appl. 14(4), 944–978 (2008). http://dx.doi.org/10.1016/j.ffa.2008.05.005. Extended Abstract in: Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation, ISSAC 2007, Waterloo, ON, Canada, pp. 369–376 (2007)
  36. von zur Gathen, J.: Counting decomposable multivariate polynomials. Appl. Algebra Eng. Commun. Comput. 22(3), 165–185 (2011). http://dx.doi.org/10.1007/s00200-011-0141-9. Abstract in: Abstracts of the Ninth International Conference on Finite Fields and their Applications, pp. 21–22. Claude Shannon Institute, Dublin, July 2009. http://www.shannoninstitute.ie/fq9/AllFq9Abstracts.pdf
  37. von zur Gathen, J.: Lower bounds for decomposable univariate wild polynomials. J. Symbolic Comput. 50, 409–430 (2013). http://dx.doi.org/10.1016/j.jsc.2011.01.008. Extended Abstract in: Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation, ISSAC 2009, Seoul, Korea (2009)
  38. von zur Gathen, J.: Counting decomposable univariate polynomials. Combin. Probab. Comput. Special Issue 01 24, 294–328 (2014a). http://dx.doi.org/10.1017/S0963548314000388. Extended Abstract in: Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation, ISSAC 2009, Seoul, Korea (2009). Preprint (2008). http://arxiv.org/abs/0901.0054
  39. von zur Gathen, J.: Normal form for Ritt’s Second Theorem. Finite Fields Their Appl. 27, 41–71 (2014b). ISSN 1071-5797. http://dx.doi.org/10.1016/j.ffa.2013.12.004. http://arxiv.org/abs/1308.1135
  40. von zur Gathen, J., Gutierrez, J., Rubio, R.: Multivariate polynomial decomposition. Appl. Algebra Eng. Commun. Comput. 14(1), 11–31 (2003). http://www.springerlink.com/content/0a5c0vvp82xbx5je/. Extended Abstract in: Proceedings of the Second Workshop on Computer Algebra in Scientific Computing, CASC 1999, München, Germany (1999)
  41. von zur Gathen, J., Kozen, D., Landau, S.: Functional decomposition of polynomials. In: Proceedings of the 28th Annual IEEE Symposium on Foundations of Computer Science, Los Angeles, CA, pp. 127–131. IEEE Computer Society Press, Washington, DC (1987). http://dx.doi.org/10.1109/SFCS.1987.29
  42. von zur Gathen, J., Viola, A., Ziegler, K.: Counting reducible, powerful, and relatively irreducible multivariate polynomials over finite fields. SIAM J. Discrete Math. 27(2), 855–891 (2013). http://dx.doi.org/10.1137/110854680. http://arxiv.org/abs/0912.3312. Extended Abstract in: Proceedings of LATIN 2010, Oaxaca, Mexico (2010)
  43. Giesbrecht, M.W.: Some Results on the Functional Decomposition of Polynomials. Master’s thesis, Department of Computer Science, University of Toronto. Technical Report 209/88 (1988). http://arxiv.org/abs/1004.5433
  44. Gogia, S.K., Luthar, I.S.: Norms from certain extensions of \(F_{q}(T)\). Acta Arith. 38(4), 325–340 (1981). ISSN 0065-1036Google Scholar
  45. Goss, D.: Basic Structures of Function Field Arithmetic. Springer, Heidelberg (1996). ISBN 3-540-61087-1Google Scholar
  46. Gutierrez, J., Kozen, D.: Polynomial decomposition. In: Grabmeier, J., Kaltofen, E., Weispfenning, V. (eds.) Computer Algebra Handbook, Sect. 2.2.4, pp. 26–28 (2003). http://www.springer.com/978-3-540-65466-7
  47. Gutierrez, J., Sevilla, D.: On Ritt’s decomposition theorem in the case of finite fields. Finite Fields Their Appl. 12(3), 403–412 (2006). http://dx.doi.org/10.1016/j.ffa.2005.08.004. http://arxiv.org/abs/0803.3976
  48. Hayes, D.R.: The distribution of irreducibles in \(\operatorname{GF}[q, x]\). Trans. Am. Math. Soc. 117, 101–127 (1965). http://dx.doi.org/10.2307/1994199
  49. Henderson, M., Matthews, R.: Composition behaviour of sub-linearised polynomials over a finite field. In: Finite Fields: Theory, Applications, and Algorithms, Waterloo, ON, 1997. Contemporary Mathematics, vol. 225, pp. 67–75. American Mathematical Society, Providence (1999)Google Scholar
  50. Hou, X., Mullen, G.L.: Number of irreducible polynomials and pairs of relatively prime polynomials in several variables over finite fields. Finite Fields Their Appl. 15(3), 304–331 (2009). http://dx.doi.org/10.1016/j.ffa.2008.12.004
  51. Kozen, D., Landau, S.: Polynomial decomposition algorithms. J. Symbolic Comput. 7, 445–456 (1989). http://dx.doi.org/10.1016/S0747-7171(89)80027-6 (1989). An earlier version was published as Technical Report 86-773, Cornell University, Department of Computer Science, Ithaca, New York (1986)
  52. Kozen, D., Landau, S., Zippel, R.: Decomposition of algebraic functions. J. Symbolic Comput. 22, 235–246 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  53. Landau, S., Miller, G.L.: Solvability by radicals is in polynomial time. J. Comput. Syst. Sci. 30, 179–208 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  54. Levi, H.: Composite polynomials with coefficients in an arbitrary field of characteristic zero. Am. J. Math. 64, 389–400 (1942)CrossRefzbMATHGoogle Scholar
  55. Mullen, G.L., Panario, D.: Handbook of Finite Fields. Discrete Mathematics and Its Applications. CRC Press, Boca Raton (2013). ISBN 978-1-4398-7378-6 (Hardback). http://www.crcpress.com/product/isbn/9781439873786
  56. Ore, O.: On a special class of polynomials. Trans. Am. Math. Soc. 35, 559–584 (1933)CrossRefMathSciNetGoogle Scholar
  57. Schinzel, A.: Selected Topics on Polynomials. The University of Michigan Press, Ann Arbor (1982). ISBN 0-472-08026-1zbMATHGoogle Scholar
  58. Schinzel, A.: Polynomials with Special Regard to Reducibility. Cambridge University Press, Cambridge (2000). ISBN 0521662257CrossRefzbMATHGoogle Scholar
  59. Tortrat, P.: Sur la composition des polynômes. Colloq. Math. 55(2), 329–353 (1988)zbMATHMathSciNetGoogle Scholar
  60. Turnwald, G.: On Schur’s conjecture. J. Australian Math. Soc. Ser. A 58, 312–357 (1995). http://anziamj.austms.org.au/JAMSA/V58/Part3/Turnwald.html
  61. Wan, D.: Zeta functions of algebraic cycles over finite fields. Manuscripta Math. 74, 413–444 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  62. Williams, K.S.: Polynomials with irreducible factors of specified degree. Can. Math. Bull. 12, 221–223 (1969). ISSN 0008-4395Google Scholar
  63. Zannier, U.: Ritt’s Second Theorem in arbitrary characteristic. J. Reine Angew. Math. 445, 175–203 (1993)zbMATHMathSciNetGoogle Scholar
  64. Zannier, U.: On composite lacunary polynomials and the proof of a conjecture of Schinzel. Invent. Math. 174, 127–138 (2008). ISSN 0020-9910 (Print) 1432-1297 (Online). http://dx.doi.org/10.1007/s00222-008-0136-8. http://arxiv.org/abs/0705.0911v1
  65. Ziegler, K.: Tame decompositions and collisions. In: Nabeshima, K. (ed.) Proceedings of the 2014 International Symposium on Symbolic and Algebraic Computation ISSAC ’14, Kobe, Japan, pp. 421–428. ACM Press (2014). http://dx.doi.org/10.1145/2608628.2608653. http://arxiv.org/abs/1402.5945
  66. Zieve, M.: Personal communication (2011)Google Scholar
  67. Zieve, M.E., Müller, P.: On Ritt’s Polynomial Decomposition Theorems, Submitted, 38 pp. (2008). http://arxiv.org/abs/0807.3578
  68. Zippel, R.: Rational function decomposition. In: Watt, S.M. (ed.) Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation, ISSAC 1991, Bonn, Germany, pp. 1–6. ACM Press, Bonn (1991). ISBN 0-89791-437-6Google Scholar
  69. Zsigmondy, K.: Über die Anzahl derjenigen ganzzahligen Functionen \(n\)-ten Grades von \(x\), welche in Bezug auf einen gegebenen Primzahlmodul eine vorgeschriebene Anzahl von Wurzeln besitzen. Sitzungsber. Kaiserl. Akad. Wiss. Abt. II 103, 135–144 (1894)zbMATHGoogle Scholar

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  1. 1.B-ITUniversität BonnBonnGermany

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