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Compositions and collisions at degree p^2

Raoul Blankertz & Konstantin Ziegler (b-it cosec)

Thursday 24 May 2012, 15.00, b-it 1.25 (cosec meeting room)

A univariate polynomial f over a field is decomposable if f = g o h = g(h) for nonlinear polynomials g and h. In order to count the decomposables, one wants to know the number of equal-degree collisions of the form f = g o h = g* o h* with (g, h) != (g*, h*) and deg g = deg g*. Such collisions only occur in the wild case, where the field characteristic p divides deg f. 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 algorithm that determines whether a given polynomial of degree p^2 has a collision or not.

This is joint work with Joachim von zur Gathen.

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