Counting invariant subspaces and decompositions of additive polynomials. (July 2021) - von zur Gathen, Joachim; Giesbrecht, Mark; Ziegler, Konstantin
Abstract: The functional (de)composition of polynomials is a topic in pure and computer algebra with many applications. The structure of decompositions of (suitably normalized) polynomials f = g ∘ h in F [ x ] over a field F is well understood in many cases, but less well when the degree of f is divisible by the positive characteristic p of F . This work investigates the decompositions of r - additive polynomials, where every exponent and also the field size is a power of r, which itself is a power of p . The decompositions of an r -additive polynomial f are intimately linked to the Frobenius-invariant subspaces of its root space V in the algebraic closure of F . We present an efficient algorithm to compute the rational Jordan form of the Frobenius automorphism on V . A formula of Fripertinger (2011) then counts the number of Frobenius-invariant subspaces of a given dimension and we derive the number of decompositions with prescribed degrees.