Abstract
The composition of two polynomials g(h) = goh is a polynomial. For a given polynomial f we are interested in finding a functional decomposition f = goh. In this paper an algorithm is described, which computes all minimal decompositions in polynomial time. In contrast to many previous decomposition algorithms this algorithm works without restrictions on the degree of the polynomial and the characteristic of the ground field. The algorithm can be iteratively applied to compute all decompositions. It is based on ideas of Landau & Miller (1985) and Zippel (1991). Additionally, an upper bound on the number of minimal decompositions is given.
- M. D. Atkinson (1975). An Algorithm for Finding the Blocks of a Permutation Group. Mathematics of Computation 29(131), 911--913. ISSN 0378-4754. URL http://www.jstor.org/stable/2005304.Google ScholarCross Ref
- David R. Barton & Richard Zippel (1985). Polynomial Decomposition Algorithms. Journal of Symbolic Computation 1, 159--168. Google ScholarDigital Library
- Raoul Blankertz (2011). Decomposition of Polynomials. Diplomarbeit, Universität Bonn, Bonn. Modified version available at http://arxiv.org/abs/1107.0687.Google Scholar
- Raoul Blankertz, Joachim von zur Gathen & Konstantin Ziegler (2012). Compositions and collisions at degree p2. In Proceedings of the 2012 International Symposium on Symbolic and Algebraic Computation ISSAC2012, Grenoble, France, 91--98. ACM Press, New York, USA. Full version available at http://arxiv.org/abs/1202.5810. Google ScholarDigital Library
- A. Bostan, G. Lecerf, B. Salvy, É. Schost & B. Wiebelt (2004). Complexity issues in bivariate polynomial factorization. In Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation ISSAC2004, Santander, Spain, 42--49. ACM Press. ISBN 1-58113-827-X. URL http://dx.doi.org/10.1145/1005285.1005294. Google ScholarDigital Library
- Greg Butler (1992). An analysis of Atkinson's algorithm. ACM SIGSAM Bulletin 26(2), 1--9. ISSN 0163-5824. URL http://dx.doi.org/10.1145/130933.130935. Google ScholarDigital Library
- Michael D. Fried & R. E. MacRae (1969). On the invariance of chains of Fields. Illinois Journal of Mathematics 13, 165--171.Google ScholarCross Ref
- Joachim von zur Gathen (1990a). Functional Decomposition of Polynomials: the Tame Case. Journal of Symbolic Computation 9, 281--299. URL http://dx.doi.org/10.1016/S0747-7171(08)80014-4. Google ScholarDigital Library
- Joachim von zur Gathen (1990b). Functional Decomposition of Polynomials: the Wild Case. Journal of Symbolic Computation 10, 437--452. URL http://dx.doi.org/10.1016/S0747-7171(08)80054-5. Google ScholarDigital Library
- Joachim von zur Gathen (2009). The number of decomposable multivariate polynomials. In Abstracts of the Ninth International Conference on Finite Fields and their Applications, 21--22. Claude Shannon Institute, Dublin. URL http://www.shannoninstitute.ie/fq9/AllFq9Abstracts.pdf.Google Scholar
- Joachim von zur Gathen, Mark Giesbrecht & Konstantin Ziegler (2010). Composition collisions and projective polynomials. Statement of results. In Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation ISSAC2010, Munich, Germany, Stephen Watt, editor, 123--130. ACM Press. URL http://dx.doi.org/10.1145/1837934.1837962. Preprint available at http://arxiv.org/abs/1005.1087. Google ScholarDigital Library
- Mark William Giesbrecht (1988). Complexity Results on the Functional Decomposition of Polynomials. Technical Report 209/88, University of Toronto, Department of Computer Science, Toronto, Ontario, Canada. Available as http://arxiv.org/abs/1004.5433.Google Scholar
- Dexter Kozen & Susan Landau (1989). Polynomial Decomposition Algorithms. Journal of Symbolic Computation 7, 445--456. An earlier version was published as Technical Report 209/88, University of Toronto, Department of Computer Science, Toronto, Ontario, Canada, 1988. Google ScholarDigital Library
- S. Landau & G. L. Miller (1985). Solvability by Radicals is in Polynomial Time. Journal of Computer and System Sciences 30, 179--208.Google ScholarCross Ref
- Susan Landau (1993). Finding maximal subfields. ACM SIGSAM Bulletin 27(3), 4--8. ISSN 0163-5824. URL http://dx.doi.org/10.1145/170906.170907. Google ScholarDigital Library
- Grégoire Lecerf (2007). Improved dense multivariate polynomial factorization algorithms. Journal of Symbolic Computation 42(4), 477--494. ISSN 0747-7171. Google ScholarDigital Library
- Rudolf Lidl & Harald Niederreiter (1997). Finite Fields. Number 20 in Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, UK, 2nd edition. First published by Addison-Wesley, Reading MA, 1983.Google Scholar
- Mark van Hoeij, Jürgen Klüners & Andrew Novocin (2011). Generating subfields. In Proceedings of the 2011 International Symposium on Symbolic and Algebraic Computation ISSAC2011, San Jose CA, 345--352. ACM Press, New York, USA. ISBN 978-1-4503-0675-1. URL http://dx.doi.org/10.1145/1993886.1993937. Google ScholarDigital Library
- Helmut Wielandt (1964). Finite permutation groups. Academic Press, New York. ISBN 0-127-49656-4. Translated from the German by R. Bercov.Google Scholar
- Richard Zippel (1991). Rational Function Decomposition. In Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation ISSAC '91, Bonn, Germany, Stephen M. Watt, editor, 1--6. ACM Press, Bonn, Germany. ISBN 0-89791-437-6. Google ScholarDigital Library
- Richard Zippel (1996). Functional Decomposition. online. URL http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.51.3154. Last visited 31 May 2013.Google Scholar
Index Terms
- A polynomial time algorithm for computing all minimal decompositions of a polynomial
Recommendations
On Polynomial Decompositions
We present a new polynomial decomposition which generalizes the functional and homogeneous bivariate decomposition of irreducible monic polynomials in one variable over the rationals. With these decompositions it is possible to calculate the roots of an ...
Certain classes of polynomial expansions and multiplication formulas
The authors first present a class of expansions in a series of Bernoulli polyomials and then show how this general result can be applied to yield various (known or new) polynomial expansions. The corresponding expansion problem involving the Euler ...
Computing polynomial resultants: Bezout's determinant vs. Collins' reduced P.R.S. algorithm
Algorithms for computing the resultant of two polynomials in several variables, a key repetitive step of computation in solving systems of polynomial equations by elimination, are studied. Determining the best algorithm for computer implementation ...
Comments