Some Results on the Functional Decomposition of Polynomials

Details Some Results on the Functional Decomposition of Polynomials Some Results on the Functional Decomposition of Polynomials

2010-04-30

arxiv.org/abs/1004.5433v1

Computer Science

Symbolic Computation

Masters Thesis, University of Toronto, 1988

Scientific paper

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.

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