Counting invariant subspaces and decompositions of additive polynomials
aa r X i v : . [ c s . S C ] D ec Counting invariant subspacesand decompositions of additive polynomials
Joachim von zur Gathen a , Mark Giesbrecht b, ∗ , Konstantin Ziegler c a B-IT, Universität BonnD-53115 Bonn, Germany b Cheriton School of Computer ScienceUniversity of Waterloo, Waterloo, ON, N2L 3G1 Canada c University of Applied Sciences LandshutD-84036 Landshut, Germany
Abstract
The functional (de)composition of polynomials is a topic in pure and computeralgebra with many applications. The structure of decompositions of (suitablynormalized) polynomials f = g ◦ h in F [ x ] over a field F is well understoodin many cases, but less well when the degree of f is divisible by the positivecharacteristic 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 , whichitself is a power of p .The decompositions of an r -additive polynomial f are intimately linked tothe Frobenius-invariant subspaces of its root space V in the algebraic closureof F . We present an efficient algorithm to compute the rational Jordan formof the Frobenius automorphism on V . A formula of Fripertinger (2011) thencounts the number of Frobenius-invariant subspaces of a given dimension andwe derive the number of decompositions with prescribed degrees. Keywords:
Univariate polynomial decomposition, additive polynomials, finitefields, rational Jordan form
1. Introduction
The composition of two polynomials g, h ∈ F [ x ] over a field F is denoted as f = g ◦ h = g ( h ), and then ( g, h ) is a decomposition of f . If g and h have degree ∗ Corresponding author
Email addresses: [email protected] (Joachim von zur Gathen), [email protected] (Mark Giesbrecht), [email protected] (Konstantin Ziegler)
URL: http://cosec.bit.uni-bonn.de/ (Joachim von zur Gathen), https://cs.uwaterloo.ca/~mwg (Mark Giesbrecht), http://zieglerk.net (KonstantinZiegler) t least 2, then f is decomposable and g and h are left and right components of f , respectively.Since the foundational work of Ritt, Fatou, and Julia in the 1920s on com-positions over C , a substantial body of work has been concerned with structuralproperties (e.g., Fried and MacRae (1969), Dorey and Whaples (1974), Schinzel(1982, 2000), Zannier (1993)), with algorithmic questions (e.g., Barton and Zip-pel (1985), Kozen and Landau (1989)), and more recently with enumeration,exact and approximate (e.g., Giesbrecht (1988), Blankertz et al. (2013), von zurGathen (2014), Ziegler (2015, 2016)). A fundamental dichotomy is between the tame case , where the characteristic p of F does not divide deg g , see von zurGathen (1990a), and the wild case , where p divides deg g , see von zur Gathen(1990b).Zippel (1991) suggests that the block decompositions of Landau and Miller(1985) for determining subfields of algebraic number fields can be applied todecomposing rational functions even in the wild case. Blankertz (2014) provesthis formally and shows that this idea can be used to compute all decompo-sitions of a polynomial with an indecomposable right component. Giesbrecht(1998) provides fast algorithms for the decomposition of additive (or linearized)polynomials, where all exponents are powers of p . Subsequent improvements inthe cost of factorization and basic operations have been made in Caruso andLe Borgne (2017, 2018). All these algorithms use time polynomial in the inputdegree.We consider the following counting problem: given f ∈ F [ x ] and a divisor d of its degree, how many ( g, h ) are there with f = g ◦ h and deg g = d ? Undera suitable normalization, the answer in the tame case is simple: at most one.However, we address this question for additive polynomials, in some sense an“extremely wild” case, and determine both the structure and the number of suchdecompositions. This involves three steps: • a bijective correspondence between decompositions of an additive poly-nomial f and Frobenius-invariant subspaces of its root space V f in analgebraic closure of F (Section 2), • a description of the A -invariant subspaces of an F -vector space for a matrix A ∈ F n × n in rational Jordan form (Section 3), and • an efficient algorithm to compute the rational Jordan form of the Frobeniusautomorphism on V f (Section 4). Its runtime is polynomial in log p (deg f ).A combinatorial result of Fripertinger (2011) counts the relevant Frobenius-invariant subspaces of V f and thus our decompositions (Subsection 3.1). We alsocount the number of maximal chains of Frobenius-invariant subspaces and thusthe complete decompositions. Our algorithm deals with squarefree polynomials,and we give a reduction for the general case (Subsection 2.2).Some of the results in the present paper are described in an Extended Ab-stract (von zur Gathen et al., 2010). A version of the present paper is availableat https://arxiv.org/abs/1005.1087 . Implementations of all algorithms in SageMath are available at https://github.com/zieglerk/polynomial_decomposition .2 . Additive polynomials and vector spaces Additive (or linearized) polynomials have a rich mathematical structure. In-troduced by Ore (1933), they play an important role in the theory of finite andfunction fields and have found many applications in coding theory and cryptog-raphy. See Lidl and Niederreiter (1997, Section 3.4) for an introduction andsurvey. In this section, we establish connections between components of addi-tive polynomials, subspaces of root spaces, and factors of so-called projectivepolynomials.We focus on additive polynomials over finite fields F , though some of theseresults hold more generally for any field of characteristic p >
0. Let r be apower of p and let F [ x ; r ] = X ≤ i ≤ n a i x r i : n ∈ Z ≥ , a , . . . , a n ∈ F be the set of r -additive (or r -linearized ) polynomials over F . For F = F r , we fixan algebraic closure F ⊇ F r . Then these are the polynomials f such that f ( aα + bβ ) = af ( α ) + bf ( β ) for any a, b ∈ F r and α, β ∈ F . The r -additive polynomialsform a non-commutative ring under the usual addition and composition. It is aprincipal left (and right) ideal ring with a left (and right) Euclidean algorithm;see Ore (1933, Chapter 1, Theorem 1). For f, h ∈ F [ x ; r ], we find h is a factor of f ⇐⇒ h is a right component of f (2.1)after comparing division with remainder of f by h (in F [ x ]) and decompositionwith remainder of f by h (in F [ x ; r ]). All components of an r -additive polynomialare p -additive, see Dorey and Whaples (1974, Theorem 4) and Giesbrecht (1988,Theorem 3.3).An additive polynomial is squarefree if its derivative is nonzero, meaning thatits linear coefficient a is nonzero. To understand the decomposition behaviorof additive polynomials, it is sufficient to restrict ourselves to monic squarefreeelements of F [ x ; r ]. The general (monic non-squarefree) case is discussed inSubsection 2.2. For f ∈ F [ x ; r ] with deg f = r n , we call n the exponent of f ,denote it by expnf , and write for n ≥ F [ x ; r ] n = { f ∈ F [ x ; r ] : f is monic squarefree with exponent n } . For f ∈ F [ x ; r ] n , the set V f of all roots of f in an algebraic closure F of F forms an F r -vector space of dimension n . From now on, we assume q to be apower of r , and let F = F q be a finite field with q elements. Then V f is invariantunder the q th power Frobenius automorphism σ q , since for α ∈ F with f ( α ) = 0we have f ( σ q ( α )) = f ( α q ) = f ( α ) q = 0, thus σ q ( V f ) ⊆ V f , and σ q is injective.For n ≥
0, we define L [ σ q ; F r ] n = (cid:8) n -dimensional σ q -invariant F r -linear subspaces of F q (cid:9) , n : F q [ x ; r ] n → L [ σ q ; F r ] n ,f V f = { α ∈ F : f ( α ) = 0 } . (2.2)Conversely, for any n -dimensional F r -vector space V ⊆ F , the lowest degreemonic polynomial f V = Q α ∈ V ( x − α ) ∈ F [ x ] with V as its roots is a squarefree r -additive polynomial of exponent n , see Ore (1933, Theorem 8). If V is invariantunder σ q , then f V ∈ F q [ x ; r ] n . For n ≥
0, we define ϕ n : L [ σ q ; F r ] n → F q [ x ; r ] n ,V f V = Y α ∈ V ( x − α ) . (2.3)Ore (1933, Chapter 1, §§ 3–4) gives a correspondence between monic square-free p -additive polynomials and F p -vector spaces which generalizes as follows. Proposition 2.4.
For r a power of a prime p , q a power of r , and n ≥ , themaps ψ n and ϕ n are inverse bijections.2.1. Right components and invariant subspaces The following refinement of Proposition 2.4 is a cornerstone of this paper. Itprovides a bijection between right components of a monic original f ∈ F q [ x ; r ] n and σ q -invariant subspaces of its root space V f ∈ L [ σ q ; F r ] n . The latter areanalyzed with methods from linear algebra in Section 3. Those insights arethen reflected back to questions about decompositions, providing results thatseem hard to obtain directly.For n ≥ d ≥ f ∈ F q [ x ; r ] n , and V ∈ L [ σ q ; F r ] n , we define H d ( f ) = H q,r,d ( f ) = { right components h ∈ F q [ x ; r ] d of f } ⊆ F q [ x ; r ] d ,L d ( V ) = L q,r,d ( V ) = { d -dimensional σ q -invariant F r -linear subspaces of V }⊆ L [ σ q ; F r ] d , where we omit q and r from the notation when they are clear from the context.We also set H q,r,d ( f ) = L q,r,d ( V ) = ∅ for d < Proposition 2.5.
Let n ≥ d ≥ , r be a power of a prime p , q a power of r , and f ∈ F q [ x ; r ] n . Then the restrictions of ψ d and ϕ d are inverse bijectionsbetween H q,r,d ( f ) and L q,r,d ( V f ) .Proof. For h ∈ H d ( f ), we have h | f by (2.1), and thus V h ⊆ V f . Since h ∈ F q [ x ; r ] d , we have dim V h = d and V h ∈ L d ( V f ). Conversely, for W ∈ L d ( f ),we have W ⊆ V f and f W is a squarefree divisor of f with expnf W = d . From(2.1), we have f W ∈ H d ( f ). Thus, ψ d ( ϕ d ( L d ( f ))) ⊆ L d ( f ) and ϕ d ( ψ d ( H d ( f ))) ⊆ H d ( f ). Since both sets are finite and both maps are injective, we have equalitiesand the claim follows.Thus, under the conditions of Proposition 2.5, we have for h ∈ F q [ x ; r ] d h | f ⇐⇒ V h ⊆ V f ⇐⇒ h ∈ H d ( f ) , (2.6)as an extension of (2.1). 4 .2. General additive polynomials We generalize Proposition 2.5 from squarefree to all monic additive polyno-mials. We can write any monic ¯ f ∈ F [ x ; r ] as g ◦ x r m with unique m ≥ g ∈ F [ x ; r ]. Then¯ f = g ◦ x r m = x r m ◦ f (2.7)with unique monic squarefree f ∈ F [ x ; r ] and the coefficients of f are the r m throots of the coefficients of g , see Giesbrecht (1988, Section 3). Composing anadditive polynomial with x r m from the left leaves the root space invariant andwe have V ¯ f = V x rm ◦ f = V f . We now relate the right components of ¯ f to the right components of f . Proposition 2.8.
Let m, n ≥ , m + n ≥ d ≥ , r be a power of a prime p and q a power of r , ≤ d ≤ m + n , and f ∈ F q [ x ; r ] n . For monic ¯ f = x r m ◦ f ∈ F q [ x ; r ] with exponent m + n , we have a bijection between any two of the following threesets:(i) { monic right components ¯ h ∈ F q [ x ; r ] of ¯ f with exponent d } ,(ii) the union of all H i ( f ) for d − m ≤ i ≤ d , and(iii) the union of all L i ( V f ) for d − m ≤ i ≤ d .Proof. We begin with a bijection between (i) and (ii). Following (2.7), we canwrite every ¯ h in (i) as x r d − i ◦ h with unique i satisfying d − m ≤ i ≤ d andunique monic squarefree h ∈ F q [ x ; r ] i . Then V h ⊆ V ¯ f = V f and h ∈ H i ( f )by (2.6). Conversely, let d − m ≤ i ≤ d and h ∈ H i ( f ). Then f = g ◦ h forsome g ∈ F q [ x ; r ] n − i and ¯ f = x r m − d + i ◦ ˜ g ◦ x r d − i ◦ h , where the coefficientsof ˜ g are the r d − i th roots of the coefficients of g . Thus ¯ h = x r d − i ◦ h is amonic right component of ¯ f with exponent d . Together this yields a one-to-onecorrespondence between (i) and (ii).Proposition 2.5 provides a bijection between (ii) and (iii).We note that for d > n , all three sets are empty. As an aside, we exhibit two further sets of polynomials that are in bijectivecorrespondence with H d ( f ); this will not be used beyond this subsection, butillustrates the wide range of applications. Let f = P ≤ i ≤ n a i x r i ∈ F [ x ; r ] and t be a positive divisor of r −
1. We have f = x · ( π t ( f ) ◦ x t ) for π t ( f ) = P ≤ i ≤ n a i x ( r i − /t . Abhyankar (1997) introduced the projective polynomials π r − ( x r n + a x r + a x ) = x ( r n − / ( r − + a x + a , (2.9)5hich may have, over function fields of positive characteristic, nice Galois groupssuch as projective general or projective special linear groups. Projective polyno-mials appear naturally in coding theory (e.g., Helleseth et al. (2008), Zeng et al.(2008)) and the study of difference sets (e.g., Dillon (2002), Bluher (2003)). Theycan be used to construct strong Davenport pairs explicitly (Bluher, 2004a) anddetermine whether a quartic power series is actually hyperquadratic (Bluher andLasjaunias, 2006). The linear shifts of (2.9) are closely related to group actionson irreducible polynomials over F q (Stichtenoth and Topuzoğlu, 2012). The car-dinality of the value set of a (possibly non-additive) polynomial f ∈ F q [ x ] isdetermined by the maximal s , t such that f = x s · ( ¯ f ◦ x t ) for some ¯ f ∈ F q [ x ](Akbary et al., 2009). Bluher (2004b) shows that (2.9) has exactly 0, 1, 2, or r + 1 roots in F q for q a power of r . Helleseth and Kholosha (2010) count theroots for q and r independent powers of 2.The polynomial ρ t ( f ) = x · ( x t ◦ π t ( f )) = x · ( π t ( f )) t is called ( r, t ) -subadditive (or simply subadditive ). We have ρ t ( f ) ◦ x t = x t ◦ f and in particular ρ ( f ) = f . Subadditive polynomials were introduced by Cohen(1990) to study their role as permutation polynomials. Henderson and Matthews(1999) connect their decomposition behavior to that of additive polynomials andprovide the bijection between (i) and (iii) in the following proposition. Coul-ter et al. (2004) use this connection to apply Odoni’s (1999) counting formulafor p -additive polynomials and Giesbrecht’s (1998) decomposition algorithm foradditive polynomials to subadditive polynomials. Proposition 2.10.
Let n ≥ d ≥ , r be a power of a prime p , q a power of r , t a positive divisor of r − , and f ∈ F q [ x ; r ] n . Then we have bijections betweenany two of the following three sets.(i) H d ( f ) ,(ii) the set of monic factors of π t ( f ) that are of the form π t ( h ) for some h ∈ F [ x ; r ] d , and(iii) the set of monic ( r, t ) -subadditive right components of ρ t ( f ) of degree r d .In particular, the maps π t and ρ t are bijections from (i) to (ii) and to (iii),respectively.Proof. For the bijection between (i) and (ii), it is sufficient to show that thefollowing statements are equivalent for h ∈ F q [ x ; r ] d : • h is a right component of f ; • h = x · ( π t ( h ) ◦ x t ) is a factor of f = x · ( π t ( f ) ◦ x t ); • π t ( h ) is a factor of π t ( f ). 6he first two items are equivalent by (2.1), and so are the last two since π t ( h ) π t ( f ) is coprime to x for squarefree h and f .The bijection between (i) and (iii) is due to Henderson and Matthews (1999,Theorem 4.1).Irreducible factors in (ii) correspond to components in (i) and (iii) thatare indecomposable over F q [ x ; r ] and ρ t ( F q [ x ; r ]), respectively. For d = 1 and t = r −
1, this yields the following criterion by Ore.
Fact 2.11 (Ore 1933, Theorem 3) . For n , r , and F as in Proposition 2.10, f ∈ F q [ x ; r ] n and a ∈ F × , we have x r − ax ∈ H ( f ) ⇐⇒ π r − ( f )( a ) = 0 .
3. The rational Jordan form
The usual Jordan (normal) form of a matrix contains the eigenvalues. It isunique up to permutations of the Jordan blocks. The rational Jordan form of amatrix is a generalization, with eigenvalues in a proper extension of the groundfield being represented by the companion matrix of their minimal polynomial.Forms akin to the rational Jordan form were investigated already by Frobenius(1911) and the underlying decomposition of the vector space is described byGantmacher (1959, Chapter VII). A detailed discussion of rational normal formscan be found in Lüneburg (1987, Chapter 6).Let A be a square matrix with entries in F . We factor the minimal polynomial of A over F completely and obtain minpoly ( A ) = u k · · · u k t t ∈ F [ y ] with t pairwise distinct monic irreducible u i ∈ F [ y ] and k i > ≤ i ≤ t . We call u i an eigenfactor of A and ker( u i ( A )) its eigenspace .For any u = P ≤ i ≤ m a i y i ∈ F [ y ] with a m = 1, we have the companionmatrix C u = − a − a − a m − ∈ F m × m with minpoly ( C u ) = u . The rational Jordan block of order ℓ > u is J ( ℓ ) u = C u I m C u . . .. . . I m C u ∈ F ( ℓm ) × ( ℓm ) , I m is the m × m identity matrix. For linear u = y − a ∈ F [ y ], wehave C u = ( a ) and the rational Jordan blocks are the Jordan blocks of theusual Jordan form . The arrangement of rational Jordan blocks along the maindiagonal gives a rational Jordan form.
Definition 3.1. A rational Jordan matrix over F is a matrix of the shape A = diag ( J ( ℓ ) u , . . . , J ( ℓ s ) u , . . . , J ( ℓ t ) u t , . . . , J ( ℓ tst ) u t ) (3.2)with t ≥
1, pairwise distinct monic irreducible u , . . . , u t ∈ F [ y ], s i ≥
1, and ℓ i ≥ ℓ i ≥ · · · ≥ ℓ is i for 1 ≤ i ≤ t .Giesbrecht (1995, Lemma 8.1) shows that minpoly ( J ( ℓ ) u ) = u ℓ , and thus minpoly ( A ) = u ℓ i · · · u ℓ t t . Every matrix over F is similar to a rational Jordanmatrix over F , see, e.g., Giesbrecht (1995, Theorem 8.3), which we call the rational Jordan form of the matrix. The eigenvalues and their multiplicitiesare preserved by this similarity transformation and the rational Jordan formis unique up to permutation of the rational Jordan blocks. Giesbrecht (1995,Corollary 8.6) shows how to transform an n × n matrix over F into rationalJordan form using O ∼ ( n ω + n log r ) field operations, where ω is the exponentof square matrix multiplication over F . This matches the lower bound Ω ( n ω )for this problem up to polylogarithmic factors. The “textbook” method gives ω ≤ ω < . A ∈ F n × n as in (3.2). For 1 ≤ i ≤ t and 1 ≤ j ≤ ℓ i , let λ ij denote the number ofrational Jordan blocks of order j for the eigenfactor u i . The formulae for λ ij over the algebraic closure, see, e.g., Gantmacher (1959, p. 155), generalize as λ ij · deg u i = rk ( u j − i ( A )) − rk ( u ji ( A )) + rk ( u j +1 i ( A ))= 2 nul ( u ji ( A )) − nul ( u j − i ( A )) − nul ( u j +1 i ( A )) , (3.3)where u i ( A ) = I n and nulB = n − rkB is the nullity of B for any B ∈ F n × n .The vector λ ( u i ) = (deg u i ; λ i , λ i , . . . , λ iℓ i ) of positive integers is the species of u i (in A ). This abstracts away the arrangement of the rational Jordan blocksas well as the actual factors u i . The multiset of all the species of eigenfactorsin A is then called the species λ ( A ) of A . This notion was introduced by Kung(1981) over the algebraic closure and generalized to finite fields by Fripertinger(2011).Table 3.4 gives all similarity classes of rational Jordan forms A in F × andtheir species. The notation 3 × (1; 1) indicates that the species (1; 1) occursthree times in the multiset. We also list, for every species, the lattice L ( A ) of A -invariant subspaces in a 3-dimensional F -vector space, the number L ( A ) of1-dimensional A -invariant subspaces, and the number A ) of maximal A -invariant subspace chains (3.6).In the next subsection, we derive the latter from the species. In Section 4, weshow how to compute the rational Jordan form of the Frobenius automorphismon the root space of an additive polynomial without the (costly) computationof a basis. 8 a a a a a b λ ( A ) { (1; 3) } { (1; 0 , , (1; 1) }L ( A ) V. . . h v i ⊥ h v i ⊥ h v r + r +1 i ⊥ . . . h v r + r +1 i h v i h v i{ } V h e , e i h e , e ih e i h e i{ } L ( A ) = L ( A ) r + r + 1 2 A ) ( r + r +1)( r + 1) 3 A a a a a a a λ ( A ) { (1; 0 , , } { (1; 1 , }L ( A ) V h e , e ih e i{ } V h e , e i h e , (0 , , β ) T i . . . h e , (0 , , β r ) T ih e i h ( α , , T i . . . h ( α r , , T i{ } L ( A ) = L ( A ) 1 r + 1 A ) 1 2 r + 19 a a b c c c λ ( A ) { (1; 2) , (1; 1) } { (3; 1) }L ( A ) V. . . h (1 , α , T , e ih e , e i h (1 , α r , T , e i h e , e i . . . h e i h (1 , α , T i h (1 , α r , T i h e i{ } V { } L ( A ) = L ( A ) r + 2 0 A ) 3( r + 1) 1 A a b b a b c λ ( A ) { (1; 1) , (2; 1) } { × (1; 1) }L ( A ) V { }h e i h e , e i V h e , e i h e , e i h e , e ih e i h e i h e i{ } L ( A ) = L ( A ) 1 3 A ) 2 6 Table 3.4: All similarity classes of rational Jordan forms A ∈ F × , where a, b, c ∈ F arepairwise distinct eigenvalues and the eigenfactors y − b y − b and y − c y − c y − c areirreducible over F . .1. The number of invariant subspaces Let r be a power of the prime p and A ∈ F n × nr be a rational Jordan matrixas in (3.2) with minpoly ( A ) = u k · · · u k t t , where u , . . . , u t ∈ F r [ y ] are pairwisedistinct monic irreducible, and k i > ≤ i ≤ t . A operates on every n -dimensional F r -vector space V and we have the corresponding primary vectorspace decomposition V = V ⊕ V ⊕ · · · ⊕ V t , (3.5)where V i = ker( u k i i ( A )) is the generalized eigenspace of u i for 1 ≤ i ≤ t .We ask two counting questions, motivated by the connection to decomposi-tion.(i) What is the number L d ( A ) of d -dimensional A -invariant subspaces of V for a given d ?(ii) What is the number A ) of maximal chains { } = U ( U ( · · · ( U e = V (3.6)of A -invariant subspaces U j for 0 ≤ j ≤ e , where e is the Krull dimensionof V ?The A -invariant subspaces of V constitute the complete lattice L ( A ) with min-imum { } and maximum V . In this lattice’s Hasse diagrams, question (i) asksfor the number of nodes of a given dimension and question (ii) asks for thenumber of paths from the minimum to the maximum.First, we discuss question (i). Let g ( A ) = P ≤ d ≤ n g d z d ∈ Z ≥ [ z ] be the generating function for the number g d = L d ( A ) of d -dimensional A -invariantsubspaces of V . The A -invariant subspace lattice L ( A ) is self-dual, see Brickmanand Fillmore (1967, Theorem 3), and thus the generating function is symmetricwith g d = g n − d for all 0 ≤ d ≤ n .Let A i denote the restriction of A to V i as in (3.5), and L ( A i ) and g ( A i ) bethe lattice and generating function of the A i -invariant subspaces of V i , respec-tively. Brickman and Fillmore (1967, Theorem 1) show that L ( A ) = Y ≤ i ≤ t L ( A i ) and thus g ( A ) = Y ≤ i ≤ t g ( A i ) . (3.7)Thus it suffices to study A -primary vector spaces , where minpoly ( A ) = u k is the k th power of an irreducible polynomial u of some degree m . If an n -dimensional A -primary vector space has species λ ( A ) = { ( m, λ , λ , . . . , λ k ) } , then there isa rational Jordan form B ∈ F n/m × n/mr with species λ ( B ) = { (1 , λ , λ , . . . , λ k ) } and L ( A ) ∼ = L ( B ) and g ( A ) = g ( B ) ◦ z m . (3.8)It is therefore enough to study A -primary vector spaces, where minpoly ( A ) isthe power of a linear polynomial. In this situation, we now compute g ( A ).11rom the theory of q -series, we use the q -bracket (also q -number )[ n ] q = q n − q − n . Lemma 3.9.
Let A ∈ F n × nr be a rational Jordan form as in (3.2) with minpoly ( A ) = u k for some linear u ∈ F r [ y ] , k > , and species λ ( A ) = { (1; λ , λ , . . . , λ k ) } .Then the number of A -invariant lines in an n -dimensional F r -vector space V is g ( A ) = [ s ] r , (3.10) where s = P ≤ j ≤ k λ j .Proof. For v ∈ V { } , the following are equivalent for the line h v i : • h v i is A -invariant. • h v i is in the eigenspace of the linear eigenfactor u (a factor of A ’s minimalpolynomial).For a linear eigenfactor u , the eigenspace has dimension dim(ker( u ( A ))) = P ≤ j ≤ k λ j = s and thus contains ( r s − / ( r −
1) lines.With g = 1, (3.7), and (3.8), we now compute g for a rational Jordan form A with arbitrary minimal polynomial. Proposition 3.11.
Let A ∈ F n × nr be in rational Jordan form as in (3.2) withspecies λ ( A ) = { (deg u i ; λ i , λ i , . . . , λ iℓ i ) : 1 ≤ i ≤ t } . Then the number of A -invariant lines in F nr is g ( A ) = X ≤ i ≤ t deg u i =1 [ s i ] r , (3.12) where s i = P ≤ j ≤ ℓ i λ ij for ≤ i ≤ t . This answers question (i) for d = 1. For d >
1, the number g d of d -dimensional A -invariant subspaces can be derived from the species with theformulas of Fripertinger (2011). We make them available through the SageMath -package accompanying this paper.For perspective, formula (3.12) allows us to determine exactly the possiblevalues for the number of right components of an additive polynomial that haveexponent 1. By Fact 2.11, this is equivalent to finding the possible number ofroots of certain projective polynomials. Let M q,r,n, = { H ( f ) : f ∈ F q [ x ; r ] n } (3.13)be the set of possible numbers of right components of exponent 1 for monicsquarefree r -additive polynomials of exponent n over F q .12or a positive integer m , let Π m be the set of unordered partitions (multisets) π = { π , . . . , π k } of m with positive integers π i and π + · · · + π k = m . Forany partition π ∈ Π m , we define the r -bracket [ π ] r = [ π ] r + [ π ] r + · · · + [ π k ] r .Then (3.12) yields the following theorem. Theorem 3.14.
Let M n = M q,r,n, be as in (3.13) and define c M = { } , c M i = c M i − ∪ { [ π ] r : π ∈ Π m } for ≤ i ≤ n . Then M n ⊆ c M n . Generally, M n = c M q,r,n, for all but a few triples ( q, r, n ), especially oversmall fields F q where not all possible (similarity classes of) Jordan forms mayoccur. As an example, for q = r = n = 2, we have merely two monic squarefreepolynomials under consideration. That is simply not enough to cover all fourcases in ˆ M . A list of the first seven values follows. c M = { } , c M = c M ∪ { [1] r } = { , } , c M = c M ∪ { r , [2] r } = { , , , r + 1 } , (consistent with Bluher (2004b)) c M = c M ∪ { , [2] r + 1 , [3] r } = { , , , , r + 1 , r + 2 , r + r + 1 } , c M = c M ∪ { , [2] r + 2 , r , [3] r + 1 , [4] r } = { , , , , , r + 1 , r + 2 , r + 3 , r + 2 , r + r + 1 , r + r + 2 ,r + r + r + 1 } , c M = c M ∪ { , [2] r + 3 , r + 1 , [3] r + 2 , [3] r + [2] r , [4] r + 1 , [5] r } = { , , , , , , r + 1 , r + 2 , r + 3 , r + 4 , r + 2 , r + 3 ,r + r + 1 , r + r + 2 , r + r + 3 , r + 2 r + 2 ,r + r + r + 1 , r + r + r + 2 , r + r + r + r + 1 } , c M = c M ∪ { , [2] r + 4 , r + 2 , r , [3] r + 3 , [3] r + [2] r + 1 , r , [4] r + 2 , [4] r + [2] r , [5] r + 1 , [6] r } = { , , , , , , , r + 1 , r + 2 , r + 3 , r + 4 , r + 5 , r + 2 , r + 3 , r + 4 , r + 3 ,r + r + 1 , r + r + 2 , r + r + 3 , r + r + 4 , r + 2 r + 2 , r + 2 r + 3 , r + 2 r + 2 , r + r + r + 1 , r + r + r + 2 , r + r + r + 3 ,r + r + 2 r + 2 , r + r + r + r + 1 , r + r + r + r + 2 ,r + r + r + r + r + 1 } . The size of c M n equals P ≤ k ≤ n p ( k ), where p ( k ) is the number of additive par-titions of k . For n → ∞ , p ( n ) grows exponentially as exp( π p n/ / (4 n √ A ∈ F n × nr be in rational Jordanform on V and let A ) denote the number of all maximal A -invariantchains (3.6). If the lattice is a grid, these are the binomial coefficients.The number of A -invariant chains depends only on the species λ ( A ) andwe write λ ( A )) = A ). For every minimal nonzero A -invariantsubspace U , there is a canonical bijection – given by /U and ⊕ U – between thechains for V that start with U = U and chains for V /U . Thus, we have therecursion formula λ ( A )) = X minimal, nonzero A -invariant U ⊆ V λ ( A | V/U )) , (3.15)where A | V/U is A taken as a linear transformation on the quotient vector space V /U , of dimension n − dim( U ).We now have two tasks. • Find all minimal nonzero A -invariant U ⊂ V . • Derive λ ( A | V/U ) for each such U .Every minimal nonzero A -invariant subspace U ⊆ V is contained in theeigenspace V i = ker( u k i i ( A )) for a unique i ≤ t and we can partition the formula(3.15) in the light of the vector space decomposition (3.5) as λ ( A )) = X eigenfactors u i X minimal, nonzero A -invariant U ⊆ V i λ ( A | V/U )) . (3.16)As for question (i) above, we make two simplifications. First, it is sufficientto study A where minpoly ( A ) = u k · · · u k t t is the product of linear u i by (3.7)and (3.8). Second, we will deal only with primary vector spaces, i.e. a singleeigenfactor u i , and thus only the inner sum in (3.16). Example . We have the following base case. If A = ( J ( ℓ ) u ) consists only of asingle Jordan block, i.e. λ = { (1; 0 , . . . , , λ ℓ = 1) } , we have a unique maximalchain of A -invariant subspaces0 ( h e i ( h e , e i ( · · · ( V and A ) = 1. For completeness, we note that U = h e i is the uniqueminimal nonzero A -invariant subspace, A | V/U = ( J ( ℓ − u ), and λ ( A | V/U ) = { (1; 0 , . . . , , λ ℓ − = 1) } .For λ ( A ) = { (1; λ , . . . , λ k ) } , we already know that the number of minimalnonzero A -invariant subspaces is [ P ≤ i ≤ k λ i ] r from (3.8) and (3.10). We need14o scrutinize them further. For A = diag ( J ( ℓ ) u , J ( ℓ ) u , . . . , J ( ℓ s ) u )with u = y − a , ℓ ≥ · · · ≥ ℓ s , minpoly ( A ) = u ℓ , s = P λ j , and λ j ′ = { ℓ j = j ′ : 1 ≤ j ≤ s } , we re-index the basis of V as e , . . . , e ℓ , e , . . . , e ℓ , . . . , e s , . . . , e sℓ s . (3.18)The d -dimensional eigenspace is ker u ( A ) = h e , e , . . . , e s i and contains [ s ] r lines, that is, 1-dimensional subspaces, and these are the only minimal non-zerosubspaces.Let U be an A -invariant subspace. We define its support supp ( U ) (in thebasis (3.18)) as the set of all base vectors for which e ij · U = 0. For a minimal,that is, 1-dimensional, U , we have j = 1 for all e ij in its support, since theseare the base vectors that span the eigenspace.The support links the subspace U to the Jordan blocks that act non-triviallyon U . Of particular interest are the Jordan blocks of minimal size that actnon-trivially on U . We definedepth( U ) = min { ℓ j : e j ∈ supp ( U ) } . Note that there may be several Jordan blocks of size depth( U ) acting on thesupport of U . Example . For A = a a a , we have h e i of depth 2 and h e + αe i for α ∈ F r of depth 1. And these are all r + 1 nonzero minimal A -invariantsubspaces.To make (3.15) applicable, we now determine the number of minimal nonzero A -invariant subspaces of depth j for 1 ≤ j ≤ k . Let λ = (1; λ , . . . , λ k ) be thespecies of the eigenvalue under consideration. The possible values for the depthof a nonzero minimal A -invariant subspace range from 1 to k , where k = max ℓ j and the following counting formula follows easily by inclusion-exclusion. Proposition 3.20.
Let A be primary on V , with species λ ( A ) = { (1; λ , . . . , λ k ) } .(i) The number of A -invariant subspaces with depth i is λ, i ) = r λ i +1 + ··· + λ k [ λ i ] r . (ii) Let U be an A -invariant subspace with depth i . Then A is well-defined on V /U and has species λ ( A | V/U ) = λ ˆ i = ( (1; λ − , λ , . . . , λ k ) if i = 1 , (1; λ , . . . , λ i − + 1 , λ i − , . . . , λ k ) otherwise. iii) The number of maximal A -invariant chains is given by the recursion { (1; 1) } ) = 1 , λ ( A )) = X ≤ j ≤ k λ, j ) · λ ˆ j ) . Proof. (i) For i = k , we have [ λ k ] r eigenspaces of depth k . For i < k , we have[ λ i + λ i +1 + · · · + λ k ] r eigenspaces of depth at least i and we find by directcomputation λ, i ) = [ λ i + λ i +1 + · · · + λ k ] r − [ λ i +1 + · · · + λ k ] r = r λ i + λ i +1 + ··· + λ k − r − − r λ i +1 + ··· + λ k − r − r λ i +1 + ··· + λ k · [ λ i ] r , as claimed.(ii) We dealt with the base case in Example 3.17.Without loss of generality, we assume that e is in the support of U andthat the corresponding first Jordan block has size equal to the depth of U .We have U = h e + α e + · · · + α s e s i for some α j ∈ F r and α j = 0 only if the corresponding Jordan block islarger than the first one. We turn (3.18) into the following basis for V /U : e + α e + · · · + α s e s + U,e + α e + · · · + α s e s + U, ... e ℓ + α e ℓ + · · · + α s e sℓ + U,e + U, . . . , e ℓ + U, ... e s + U, . . . , e sℓ s + U. In other words, we drop the projection of the first base vector (due to thelinear dependence introduced by U ) and modify the base vectors for thefirst Jordan block. A direct computation shows that A | V/U is in rationalJordan form, its first Jordan block is equal to the first Jordan block of A reduced by size 1, and all other Jordan blocks “remained” unchanged.(iii) This follows from (3.16) using (i) and (iii).In the general case of several eigenfactors we obtain A ) by (3.16)using the formulae in Proposition 3.20 (iii) for each eigenfactor.16 . The Frobenius automorphism on the root space In this section, we present an efficient algorithm to compute the rationalJordan form of the Frobenius automorphism on the root space of a squarefreemonic additive polynomial f . With the results of Subsection 3.1, this yieldsthe number of right components of f of a given degree. The straightforwardapproach suffers from possibly exponential costs for the description of the rootspace V f , see Example 4.12.The centre of the Ore ring F q [ x ; r ] will be a useful tool. For q a power of r ,so that F r ⊆ F q , it equals F r [ x ; q ] = (cid:8) X ≤ i ≤ n a i x q i : n ∈ Z ≥ , a , . . . , a n ∈ F r (cid:9) ⊆ F q [ x ; r ] , see, e.g., Giesbrecht (1998, Section 3). Every element f ∈ F q [ x ; r ] has a unique minimal central left component f ∗ ∈ F r [ x ; q ], the unique monic polynomial in F r [ x ; q ] of minimal degree such that f ∗ = g ◦ f for some nonzero g ∈ F q [ x ; r ].For squarefree f , it is the monic generator of the largest two-sided ideal I ( f )contained in the left ideal generated by f . The ideal I ( f ) is then known as the bound of f , see Jacobson (1943, page 83). Fact 4.1 (Giesbrecht 1998, Lemma 4.2) . Let r be a power of a prime p and q = r d . For f ∈ F q [ x ; r ] of exponent n , we can find its minimal central left component f ∗ ∈ F r [ x ; q ] with O ( n d M ( d ) + n d M ( d ) log d ) ⊆ O ∼ ( n d + n d ) operationsin F r , where M ( d ) is the number of operations to multiply two polynomials over F r with degree at most d each. The “schoolbook” method gives M ( d ) = O ( d ) and Harvey and van derHoeven (2019a) show M ( d ) = O ( d log d log ∗ d ). The recent, as yet unpublished,preprint of Harvey and van der Hoeven (2019b) claims M ( d ) = O ( d log d ), whichmany consider to be the best achievable asymptotic bound.Le Borgne (2012, Theorem II.3.2) gives an algorithm for f ∗ with O ∼ ( n ω d ω + n d log r ) operations in F r , where d and n are as above and ω is an exponentof square matrix multiplication over F r .The centre F r [ x ; q ] is a commutative subring of F q [ x ; r ] and isomorphic to F r [ y ] with the usual addition and multiplication via τ : F r [ x ; q ] → F r [ y ] ,f = X ≤ i ≤ n a i x q i τ ( f ) = X ≤ i ≤ n a i y i , see McDonald (1974, pages 24–25). The isomorphic image F r [ y ] is a unique fac-torization domain and factorizations in F r [ y ] are in one-to-one correspondencewith decompositions in F r [ x ; q ] into central components. The following maintheorem shows the close relationship between the minimal central left compo-nent of an additive polynomial and the minimal polynomial of the Frobeniusautomorphism on its root space. 17 heorem 4.2. Let r be a power of a prime p and q a power of r . Let f ∈ F q [ x ; r ] n be monic squarefree of exponent n with root space V f ⊆ F q and minimalcentral left component f ∗ ∈ F r [ x ; q ] . Then the image τ ( f ∗ ) ∈ F r [ y ] is theminimal polynomial of the q th power Frobenius automorphism σ q on the F r -vector space V f .Proof. For a central g = P ≤ i ≤ m g i x q i ∈ F r [ x ; q ], we have τ ( g ) = P ≤ i ≤ m g i y i ∈ F r [ y ] and ( τ ( g ))( σ q ) = g , and the following are equivalent: • g is a right or left component of f ; • g ( α ) = 0 for all α ∈ V f ; • ( τ ( g )( σ q ))( α ) = 0 for all α ∈ V f .The first two items are equivalent by (2.1) and the squarefreeness of f and since g is central. The last two items are equivalent since τ ( g )( σ q ) = g .Thus, g is a central left component of f if and only if τ ( g ) annihilates σ q on V f .Since f ∗ and the minimal polynomial of σ q are the unique monic polynomials ofminimal degree with these properties, respectively, we have the claimed equality.It is useful to recall a little more about the ring F q [ x ; r ]. Ore (1933) showsthat for any f, g ∈ F q [ x ; r ], there exists a unique monic h ∈ F q [ x ; r ] of maximaldegree, called the greatest common right component ( gcrc ) of f and g , such that f = u ◦ h and g = v ◦ h for some u, v ∈ F q [ x ; r ]. Also, h = gcrc ( f, g ) = gcd( f, g ),and the roots of h are those in the intersection of the roots of f and g , in otherwords V gcrc ( f,g ) = V f ∩ V g . In fact, there is an efficient Euclidean-like algorithmfor computing the gcrc ; see Ore (1933) and Giesbrecht (1998) for an analysis.The usual Euclidean algorithm for gcd( f, g ) is insufficient, since the degrees of f and g may be exponential in their exponents. Fact 4.3 (Giesbrecht 1998, Lemma 2.1) . Let r be a power of a prime p and q = r d . For f, g ∈ F q [ x ; r ] of exponent n , we can find gcrc ( f, g ) ∈ F q [ x ; r ] with O ( n M ( d ) d log d ) ⊆ O ∼ ( n d ) operations in F r , where M ( d ) is as in Fact 4.1.4.1. A fast algorithm for the rational Jordan form of σ q on V f We now determine the rational Jordan form of the Frobenius automorphismon the root space of an additive polynomial. We begin with a factorization ofthe minimal polynomial and then compute every eigenfactor’s species indepen-dently. The following proposition deals with the base case, where the minimalpolynomial is the power of an irreducible polynomial.
Proposition 4.4.
Let r be a power of a prime p , q a power of r , f ∈ F q [ x ; r ] n monic squarefree of exponent n with minimal central left component f ∗ ∈ F r [ x ; q ] ,and σ q the q th power Frobenius automorphism on V f . If τ ( f ∗ ) = u k for an ir-reducible u ∈ F r [ y ] and k > , then τ − ( u j )) = u j ( σ q ) , (4.5)18er( u j ( σ q )) = V gcrc ( f,τ − ( u j )) , (4.6) Y α ∈ ker( u j ( σ q )) ( x − α ) = gcrc ( f, τ − ( u j )) (4.7) for all j with ≤ j ≤ k + 1 , where u ( σ q ) is the identity on V f .Proof. Let 0 ≤ j ≤ k + 1. If we write u j = P i w i y i with all w i ∈ F r , then τ − ( u j ) = P i w i x q i = u j ( σ q ), which is (4.5). The kernels of these two mapson V f form the same subset of V f , so that V τ − ( u j ) ∩ V f = V gcrc ( f,τ − ( u j )) . Thisshows (4.6).Furthermore, the bijection ϕ dim(ker( u j ( σ q ))) from (2.3) maps the left and righthand sides of (4.6) to the left and right hand sides of (4.7), respectively. Corollary 4.8.
In the notation and under the assumption of Proposition 4.4, let u be irreducible of degree m and ν j = expn ( gcrc ( f, τ − ( u j ))) for ≤ j ≤ k + 1 .Then the species of the rational Jordan form of σ q on V f is { ( m ; λ , λ , . . . , λ k ) } ,where λ j = (2 ν j − ν j − − ν j +1 ) /m, (4.9) for ≤ j ≤ k .Proof. For monic squarefree g ∈ F q [ x ; r ], we have expng = dim V g due to thebijection (2.2). For 0 ≤ j ≤ k + 1, gcrc ( f, τ − ( u j )) is monic squarefree and thus ν j = dim( V gcrc ( f,τ − ( u j )) ) = dim(ker( u j ( σ q ))) = nul ( u j ( S ))by (4.6). The claim follows from (3.3).In the case of a minimal polynomial with arbitrary factorization, we treatevery eigenfactor separately with Corollary 4.8, see Giesbrecht (1998, Theo-rem 4.1). The result is Algorithm 4.10. It computes the rational Jordan formof the Frobenius automorphism on the root space of a given f ∈ F q [ x ; r ] n . Theorem 4.11.
Algorithm 4.10 works correctly as specified and takes an ex-pected number of O ∼ ( n d ) field operations in F r .Proof. The notation in the algorithm corresponds to that of the rational Jor-dan form (3.2) and Corollary 4.8. In Step 1, we know from Theorem 4.2 that f ∗ is the minimal polynomial of S . Therefore all rational Jordan blocks corre-spond to factors of f ∗ (determined in Step 2) and we only need to figure outevery eigenfactor’s species. By Giesbrecht (1998, Theorem 4.1), we can treatevery eigenfactor separately (Steps 4–11) and align the resulting rational Jordanblocks along the main diagonal (Step 11, initialized in Step 3).For every eigenfactor u i the first inner loop (Steps 5–7) determines ν j asdefined in Corollary 4.8 for 0 ≤ j ≤ k i + 1. The second inner loop (Steps 9–11)derives the number λ j of rational Jordan blocks of order j for u i (Step 10) viaformula (4.9) and extends S along its main diagonal accordingly (Step 11).Doing this for all eigenfactors and all possible orders returns the specifiedoutput in Step 12. 19 lgorithm 4.10: RationalJordanForm
Input: r -additive monic squarefree f ∈ F q [ x ; r ] n of exponent n , where q = r d and r is a power of a prime p Output: rational Jordan form S ∈ F n × nr as in (3.2) of the q th powerFrobenius automorphism on V f f ∗ ← minimal central left component of f u k u k · · · u k t t ← factorization of τ ( f ∗ ) into pairwise distinct monicirreducible u i ∈ F r [ y ] with k i > ≤ i ≤ t S ← ∅ // initialize “empty matrix” for i ← to t do // determine the species of u i for j ← to k i + 1 do h j ← gcrc ( f, τ − ( u ji )) ν j ← expnh j // equal to nul ( u ji ( S )) m ← deg y u i for j ← to k i do λ j ← (2 ν j − ν j − − ν j +1 ) /m // employ (4.9) S ← diag ( S, J ( j ) u i , . . . , J ( j ) u i | {z } λ j -times ) // append Jordan blocks return S We assume that the isomorphism τ and its inverse are free operations. Ifthe polynomials are stored as vectors of coefficients, these operations merelychange the way this information is interpreted. We also take for granted afree operation to determine the exponent of an additive and the degree of an“ordinary” polynomial in Steps 7 and 8, respectively. Finally, we neglect the(cheap) integer arithmetic in Step 10.Step 1 uses O ∼ ( n d + n d ) field operations in F r , see Fact 4.1. We have expnf ∗ ≤ dn and thus deg y τ ( f ∗ ) ≤ n . The factorization in Step 2 can be donein random polynomial time with O ∼ ( n + n log r ) field operations in F r , see, e.g.von zur Gathen and Gerhard (2013, Corollary 14.30). The worst case occurswhen τ ( f ∗ ) is the n th power of a linear eigenfactor u . The n + 2 powers of u can be obtained with O ∼ ( n ) field operations in F r . The additive polynomial τ − ( u j ) has exponent dj and each gcrc in Step 6 requires O ∼ (max( n, dj ) d ) ⊆ O ∼ ( n d ) field operations in F r , see Step 4.3. The complete inner loop thusrequires O ∼ ( n d ) field operations which dominates the costs of the previoussteps.Only the distinct-degree factorization in Step 2 requires randomization. Butthis granularity is necessary for our approach as the following example shows.20et A = (cid:18) a a a b (cid:19) , B = (cid:18) a a b b (cid:19) ∈ F × r , with distinct nonzero a, b ∈ F r . Then A and B are two rational Jordan formswith distinct species { (1; 3) , (1; 1) } and { × (1; 2) } , respectively, but equal mini-mal polynomial u = ( y − a )( y − b ). The single equal-degree factor has multiplicity1 and yields only the information dim ker u ( A ) = dim ker u ( B ) = 4, that is thesum of orders of blocks corresponding to eigenfactors of degree 1.Caruso and Le Borgne (2017) give an algorithm for the species of the Frobe-nius operator on the n -dimensional module F q [ x ; r ] / ( F q [ x ; r ] · f ), as in von zurGathen et al. (2010), and count complete decompositions, as in Fripertinger(2011). Related counting problems are also considered in Le Borgne (2012).The costs of Algorithm 4.10 are only polynomial in expnf and log q , despitethe fact that the actual roots of f may lie in an extension of exponential degreeover F q as illustrated in the following example and Figure 4.13. Example . Let q = r and f ∈ F q [ y ] be primitive of degree n . Its additive q -associate τ − ( f ) factors into x and the irreducible τ − ( f ) /x of degree q n − F q , see Lidl and Niederreiter (1997, Theorem 3.63). Thus, the splittingfield of the additive τ − ( f ) is an extension of F q of degree q n − f ∈ F q [ x ; r ] V f ⊆ F q f ∗ ∈ F r [ x ; q ] ∼ = F r [ y ] ∋ τ ( f ∗ ) ∈ F n × nr σ q on V f ∗ gcrc ( f, · ) Figure 4.13: Algorithm 4.10 computes the rational Jordan form of the Frobenius automor-phism on the root space V f of f while avoiding the expensive computation (dashed) of andon the root space itself. r of any r -additive polynomial f ∈ F q [ x ; r ] of exponent n . This also yields a fast algorithm to compute thenumber of certain factors and right components of projective and subadditivepolynomials as described in Subsection 2.3. Example . Boucher and Ulmer (2014) build self-dual codes from factoriza-tions of x r n − ax beating previously known minimal distances. Over F [ x ; 2],they exhibit 3, 15, 90, and 543 complete decompositions for x + x , x + x , x + x , and x + x , respectively.In this section, we assume the field size q to be a power of the parameter r .As in Bluher’s (2004b) work, our methods go through for the general situation,where q = p d and r = p e are independent powers of the characteristic. Then F q ∩ F r = F s for s = p gcd( d,e ) and the centre of F q [ x ; r ] is F s [ x ; t ] for t = p lcm( d,e ) .
5. Conclusion and open questions
We investigated the structure and number of all right components of anadditive polynomial. This involved three steps: • a bijective correspondence between decompositions of an additive poly-nomial f and Frobenius-invariant subspaces of its root space V f in analgebraic closure of F (Section 2), • a description of the A -invariant subspaces of an F -vector space for a ra-tional Jordan form A ∈ F n × n (Section 3), and • an efficient algorithm for the rational Jordan form of the Frobenius auto-morphism on V f (Section 4). Its runtime is polynomial in log p (deg f ).A combinatorial result of Fripertinger (2011) counts the relevant Frobenius-invariant subspaces of V f and thus our decompositions (Subsection 3.1). Wealso count the number of maximal chains of Frobenius-invariant subspaces andthus the complete decompositions.In Theorem 3.14, we describe the small set of possible values for the numberof right components of exponent r of a given additive polynomial. The natural“inverse” question asks for the number of additive polynomials that admit agiven number of right components.The root space V f has exponentially (in the exponent of f ) many elements,and the field over which it is defined may have exponential degree. The effi-ciency of our algorithms in Section 4 is mainly achieved by avoiding any directcomputation with V f .
6. Acknowledgments
This work was supported by the German Academic Exchange Service (DAAD)in the context of the German-Canadian PPP program. In addition, Joachim22on zur Gathen and Konstantin Ziegler were supported by the B-IT Foundationand the Land Nordrhein-Westfalen. Mark Giesbrecht acknowledges the supportof the Natural Sciences and Engineering Research Council of Canada (NSERC).Cette recherche a été financée par le Conseil de recherches en sciences naturelleset en génie du Canada (CRSNG).
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