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Counting Decomposable Univariate Polynomials

Published online by Cambridge University Press:  05 September 2014

JOACHIM VON ZUR GATHEN
JOACHIM VON ZUR GATHEN*
Affiliation:
B-IT, Universität Bonn, D-53113 Bonn, Germany (e-mail: [email protected]) http://cosec.bit.uni-bonn.de/
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Abstract

A univariate polynomial f over a field is decomposable if it is the composition f = g ○ h of two polynomials g and h whose degree is at least 2. We determine an approximation to the number of decomposables over a finite field. The tame case, where the field characteristic p does not divide the degree n of f, is reasonably well understood, and we obtain exponentially decreasing relative error bounds. The wild case, where p divides n, is more challenging and our error bounds are weaker.

Keywords

Primary 68W30Secondary 11T0612E10
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Special Issue 1: Honouring the Memory of Philippe Flajolet - Part 3 , January 2015 , pp. 294 - 328
Copyright
Copyright © Cambridge University Press 2014 

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References

[1] Blankertz, R., von zur Gathen, J. and K., Ziegler, (2013) Compositions and collisions at degree p2 . J. Symbol. Comput. 59 113–145. Extended abstract in Proc. 2012 International Symposium on Symbolic and Algebraic Computation: ISSAC2012, pp. 91–98.CrossRefGoogle Scholar
[2] Bodin, A., Dèbes, P. and Najib, S. (2009) Indecomposable polynomials and their spectrum. Acta Arithmetica 139 79–100.CrossRefGoogle Scholar
[3] Dorey, F. and Whaples, G. (1974) Prime and composite polynomials. J. Algebra 28 88–101.CrossRefGoogle Scholar
[4] von zur Gathen, J. (1990) Functional decomposition of polynomials: The tame case. J. Symbol. Comput. 9 281–299.CrossRefGoogle Scholar
[5] von zur Gathen, J. (1990) Functional decomposition of polynomials: The wild case. J. Symbol. Comput. 10 437–452.CrossRefGoogle Scholar
[6] von zur Gathen, J. (2002) Factorization and decomposition of polynomials. In The Concise Handbook of Algebra (Mikhalev, A. V. and Pilz, G. F., eds), Kluwer Academic, pp. 159–161.Google Scholar
[7] von zur Gathen, J. (2008). Counting reducible and singular bivariate polynomials. Finite Fields and Their Applications 14 944–978. Extended abstract in Proc. 2007 International Symposium on Symbolic and Algebraic Computation: ISSAC2007, pp. 369–376.CrossRefGoogle Scholar
[8] von zur Gathen, J. (2009). The number of decomposable univariate polynomials: Extended abstract. In Proc. 2009 International Symposium on Symbolic and Algebraic Computation: ISSAC2009 (May, J. P., ed.), ACM Press, pp. 359–366.CrossRefGoogle Scholar
[9] von zur Gathen, J. (2010). Counting decomposable multivariate polynomials. Applicable Algebra in Engineering, Communication and Computing 22 165–185. Abstract in Abstracts of the Ninth International Conference on Finite Fields and their Applications, Claude Shannon Institute, pp. 21–22. www.shannoninstitute.ie/fq9/AllFq9Abstracts.pdf CrossRefGoogle Scholar
[10] von zur Gathen, J. (2013) Lower bounds for decomposable univariate wild polynomials. J. Symbol. Comput. 50 409–430.CrossRefGoogle Scholar
[11] von zur Gathen, J. (2014) Finite Fields and Their Applications 27 41–71. ISSN 1071-5797.CrossRefGoogle Scholar
[12] von zur Gathen, J., Kozen, D. and Landau, S. (1987) Functional decomposition of polynomials. In Proc. 28th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, pp. 127–131. Final version in J. Symbol. Comput. Google Scholar
[13] Giesbrecht, M. W. (1988) Complexity results on the functional decomposition of polynomials. Technical Report 209/88, Department of Computer Science, University of Toronto. arXiv:1004.5433Google Scholar
[14] Grabmeier, J., Kaltofen, E. and Weispfenning, V. (eds) (2003) Computer Algebra Handbook: Foundations, Applications, Systems, Springer.CrossRefGoogle Scholar
[15] Gutierrez, J. and Kozen, D. (2003) Polynomial decomposition. In [14], Section 2.2.4, pp. 26–28.Google Scholar
[16] Gutierrez, J. and Sevilla, D. (2006) On Ritt's decomposition theorem in the case of finite fields. Finite Fields Appl. 12 403–412.CrossRefGoogle Scholar
[17] Kozen, D. and Landau, S. (1986) Polynomial decomposition algorithms. Technical Report 86-773, Department of Computer Science, Cornell University.Google Scholar
[18] Kozen, D. and Landau, S. (1989) Polynomial decomposition algorithms. J. Symbol. Comput. 7 445–456.CrossRefGoogle Scholar
[19] Kozen, D., Landau, S. and Zippel, R. (1996) Decomposition of algebraic functions. J. Symbol. Comput. 22 235–246.CrossRefGoogle Scholar
[20] Ritt, J. F. (1922) Prime and composite polynomials. Trans. Amer. Math. Soc. 23 51–66.CrossRefGoogle Scholar
[21] Schinzel, A. (1982) Selected Topics on Polynomials, The University of Michigan Press.CrossRefGoogle Scholar
[22] Schinzel, A. (2000) Polynomials With Special Regard to Reducibility, Cambridge University Press.CrossRefGoogle Scholar
[23] Tortrat, P. (1988) Sur la composition des polynômes. Colloq. Math. 55 329–353.CrossRefGoogle Scholar
[24] Zannier, U. (1993). Ritt's Second Theorem in arbitrary characteristic. J. Reine Angewandte Mathematik 445 175–203.Google Scholar
[25] Zannier, U. (2008) On composite lacunary polynomials and the proof of a conjecture of Schinzel. Inventio. Math. 174 127–138.CrossRefGoogle Scholar