Counting invariant subspaces and decompositions of additive polynomials. (July 2021)

• Record Type: Journal Article
• Authors: von zur Gathen, Joachim; Giesbrecht, Mark; Ziegler, Konstantin
• Is Part Of: Journal of symbolic computation. Volume 105(2021)
• Page Start: 214
• Page End: 233
• Publication Date: 2021-07
• Subjects: 68W30 -- 12Y05; Univariate polynomial decomposition -- Additive polynomials -- Finite fields -- Rational Jordan form; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2020.06.008
• Languages: English
• ISSNs: 0747-7171

Abstract

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.