Polynomial decomposition algorithms
Journal of Symbolic Computation
Finding irreducible polynomials over finite fields
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
Functional decomposition ofpolynomials: the tame case
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Functional decomposition of polynomials: The wild case
Journal of Symbolic Computation
Rational function decomposition
ISSAC '91 Proceedings of the 1991 international symposium on Symbolic and algebraic computation
Nearly Optimal Algorithms For Canonical Matrix Forms
SIAM Journal on Computing
Factoring in skew-polynomial rings over finite fields
Journal of Symbolic Computation
On x6 + x + a in Characteristic Three
Designs, Codes and Cryptography
Polynomial Decomposition Algorithms
Polynomial Decomposition Algorithms
The number of decomposable univariate polynomials. extended abstract
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Lower bounds for decomposable univariate wild polynomials
Journal of Symbolic Computation
Compositions and collisions at degree p2
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Compositions and collisions at degree p 2
Journal of Symbolic Computation
Self-dual skew codes and factorization of skew polynomials
Journal of Symbolic Computation
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The functional decomposition of polynomials has been a topic of great interest and importance in pure and computer algebra and their applications. The structure of compositions of (suitably normalized) polynomials f = g o h in Fq[x] is well understood in many cases, but quite poorly when the degrees of both components are divisible by the characteristic p. This work investigates the decomposition of polynomials whose degree is a power of p. An (equal-degree) i-collision is a set of i distinct pairs (g, h) of polynomials, all with the same composition and deg g the same for all (g, h). Abhyankar (1997) introduced the projective polynomials xn+ ax + b, where n is of the form (rm -- 1)/(r -- 1) and r is a power of p. Our first tool is a bijective correspondence between i-collisions of certain additive trinomials, projective polynomials with i roots, and linear spaces with i Frobenius-invariant lines. Bluher (2004b) has determined the possible number of roots of projective polynomials for m = 2, and how many polynomials there are with a prescribed number of roots. We generalize her first result to arbitrary m, and provide an alternative proof of her second result via elementary linear algebra. If one of our additive trinomials is given, we can efficiently compute the number of its decompositions, and similarly the number of roots of a projective polynomial. The runtime of these algorithms depends polynomially on the sparse input size, and thus on the input degree only logarithmically. For non-additive polynomials, we present certain decompositions and conjecture that these comprise all of the prescribed shape.