Polynomial decomposition algorithms
Journal of Symbolic Computation
Polynomial decomposition algorithms
Journal of Symbolic Computation
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
The number of decomposable univariate polynomials. extended abstract
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Composition collisions and projective polynomials: statement of results
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Finite Fields and Their Applications
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
Modern Computer Algebra
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A univariate polynomial f over a field is decomposable if f=g@?h=g(h) for nonlinear polynomials g and h. In order to count the decomposables, one wants to know, under a suitable normalization, the number of equal-degree collisions of the form f=g@?h=g^@?@?h^@? with (g,h)(g^@?,h^@?) and degg=degg^@?. Such collisions only occur in the wild case, where the field characteristic p divides degf. Reasonable bounds on the number of decomposables over a finite field are known, but they are less sharp in the wild case, in particular for degree p^2. We provide a classification of all polynomials of degree p^2 with a collision. It yields the exact number of decomposable polynomials of degree p^2 over a finite field of characteristic p. We also present an efficient algorithm that determines whether a given polynomial of degree p^2 has a collision or not.