Some Results on the Functional Decomposition of Polynomials

1 Some Results on the Functional Decomposition of Polynomials by Mark William Giesbrecht Department of Computer Science A Thesis submitted in conformity with the requirements for the Degree of Master’s of Science in the University of Toronto Department of Computer Science University of Toronto Toronto, Ontario, Canada, M5S 1A4 Copyright c⃝1988 Mark Giesbrecht arXiv:1004.5433v1 [cs.SC] 30 Apr 2010 2 Mark Giesbrecht Abstract If g and h are functions over some field, we can consider their composition f = g(h). The inverse problem is decomposition: given f, determine the ex- istence of such functions g and h. In this thesis we consider functional decom- positions of univariate and multivariate polynomials, and rational functions over a field F of characteristic p. In the polynomial case, “wild” behaviour occurs in both the mathematical and computational theory of the problem if p divides the degree of g. We consider the wild case in some depth, and deal with those polynomials whose decompositions are in some sense the “wildest”: the additive polynomials. We determine the maximum number of decompositions and show some polynomial time algorithms for certain classes of polynomials with wild decompositions. For the rational function case we present a definition of the problem, a normalised version of the problem to which the general problem reduces, and an exponential time solution to the normal problem. Functional Decomposition of Polynomials 3 Acknowledgement. I would like to thank my supervisor Dr. Joachim von zur Gathen for the long and fruitful hours he spent helping me with this thesis, and Dr. Rackoff for being my second reader. I would also like to thank my office mates and many others for their helpful suggestions and proof reading. Finally, I would like to thank NSERC for its scholarship support. 4 Mark Giesbrecht Table of Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 Chapter 1. Polynomial Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 A. Definition of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 B. Decomposition and the Subfields of F(x) . . . . . . . . . . . . . . . . . . . . 17 C. Separated Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 D. Multidimensional Block Decompositions . . . . . . . . . . . . . . . . . . . . . 22 E. Chebyshev Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27 F. Complete Rational Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . 30 G. The Number of Indecomposable Polynomials . . . . . . . . . . . . . . . . 32 H. Multivariate Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Chapter 2. Decomposition Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 A. The Model of Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37 B. Computing Right Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38 C. Univariate Decomposition Using Separated Polynomials . . . . . 39 D. Univariate Decomposition in the Tame Case . . . . . . . . . . . . . . . . .40 E. Decomposition Using Block Decomposition . . . . . . . . . . . . . . . . . . 41 F. A Lower Bound on the Degree of Splitting Fields . . . . . . . . . . . . 43 G. Decompositions Corresponding To Ordered Factorisations . . . 46 H. Computing Complete Univariate Decompositions . . . . . . . . . . . . 48 I. Decomposition Multivariate Polynomial in the Tame Case . . . . 48 J. Multivariate Decomposition using Separated Polynomials . . . . 51 Chapter 3. Additive Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53 A. Definition and Root Structure of Additive Polynomials . . . . . . 53 B. Rationality and the Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 C. Rational Decompositions of Additive Polynomials . . . . . . . . . . . 58 D. The Number of Bidecompositions of a Polynomial . . . . . . . . . . . 59 E. Complete Decompositions of Additive Polynomials . . . . . . . . . . .61 F. The Number of Complete Rational Normal Decompositions . . 61 Chapter 4. The Ring of Additive Polynomials . . . . . . . . . . . . . . . . . . . . . . . 67 A. Basic Ring Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 B. The Euclidean Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 C. The Structure of the Set of Decompositions . . . . . . . . . . . . . . . . . .74 D. Completely Reducible Additive Polynomials . . . . . . . . . . . . . . . . . 83 E. The Uniqueness of Transmutation . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 F. The Number of the Complete Decompositions . . . . . . . . . . . . . . . 89 Functional Decomposition of Polynomials 5 Chapter 5. Decompos

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