Linearized polynomials over finite fields revisited

• A linearized polynomial's rank equals the rank of its associated Dickson matrix, extending Dickson's permutation criterion.The paper also introduces an adjugate polynomial via the fact that the adjugate of a Dickson matrix is itself a Dickson matrix.Source: §1 · p. 3
• The two characterizations provide simpler approaches to establishing trace-form representations of linearized polynomials.The paper presents these representations as direct consequences of the composition-algebra isomorphism and also derives them using Dickson matrices.Source: §1 · p. 4
• The paper describes the structure of the subalgebra of linearized polynomials with coefficients in a subfield.It does so by constructing isomorphisms with subalgebras of the composition algebra and Dickson matrix algebra.Source: §1 · p. 4
• The algebra is isomorphic to both the composition algebra and the Dickson matrix algebra.The paper defines the Dickson matrices as matrices with a prescribed form.Source: §1 · pp. 3–4

2.1 The skew-polynomial ring approach

• The skew-polynomial representation gives a rank formula using the degree of a greatest common right divisor.For L, rk L = n − deg gcrd(φ−1(L(x)), x^n−1).Source: §2.1 · p. 6

2.2 The semi-linear group algebra approach

• The semi-linear group algebra approach defines an algebra homomorphism and establishes the characterization by comparing dimensions.The group algebra is formed from the Galois group of F_{q^n}/F_q.Source: §2.2 · pp. 6–7

2.3 The matrix algebra approach

• The matrix algebra approach identifies linearized polynomials with linear transformations of F_{q^n} over F_q.The paper also constructs the matrix for any linearized polynomial under any fixed basis, using its associated Dickson matrix.Source: §2.3 · pp. 7–8

3 Characterize Ln(Fqn) via the composition algebra

• The composition algebra E∨⊗K E is an associative, non-commutative K-algebra, and Λ identifies it with EndK(E) when E is finite-dimensional.Λ sends a simple tensor l ⊗ x to the map y ↦ l(y)x.Source: §3 · p. 8
• The paper recalls an isomorphism between Ln(Fqn) and End(Fqn), and says the theorem follows straightforwardly from the preceding discussions; the passage gives no further statement of that theorem.Source: §3 · p. 8
• Every L(x) ∈ Ln(Fqn) admits a trace representation as a sum of terms tr(ω_i x)θ_i.The paper notes that this representation is not unique.Source: §3 · p. 9

4 Characterize Ln(Fqn) via the Dickson matrix algebra and further

• The Dickson matrices form an Fq-algebra, but not an Fqn-algebra, and the paper establishes an Fq-algebra isomorphism with Ln(Fqn).The Dickson algebra is an Fq-subalgebra of Mn(Fqn).Source: §4 · p. 9
• For every L(x) ∈ Ln(Fqn), the rank and determinant of L equal the rank and determinant of its associated Dickson matrix.This generalizes Dickson’s criterion that a linearized polynomial is a permutation polynomial exactly when its Dickson matrix is nonsingular [8, Chap. 7].Source: §4 · pp. 11–12
• For a linearized permutation polynomial, the inverse of its Dickson matrix is the Dickson matrix of its composition inverse, providing a way to compute inverse polynomials.The nonsingular Dickson matrices form a group isomorphic to GLn(Fq).Source: §4 · p. 12
• The adjugate polynomial is the linearized polynomial associated to the adjugate of the Dickson matrix, with coefficients given by first-column cofactors.The definition is meaningful for polynomials of rank at least n−1.Source: §4 · p. 16
• When rk L = n−1, the adjugate polynomial has rank 1, and its image equals ker L while ker L⋆ equals Im L.The adjugate polynomial can also be used to obtain a defining parameter for the one-dimensional kernel.Source: §4 · p. 16

5 Representations of linearized polynomials

• A rank-k linearized polynomial has trace representations described by ordered sets of rank k, as stated in Theorem 5.1.The section reestablishes the representation result previously proved by Ling and Qu.Source: §5 · p. 18

5.1 Composition algebra approach to Theorem 5.1

• The composition-algebra approach derives the representations more simply and explains their existence through decomposition into single tensors.The authors say this approach gives the same result as Ling and Qu.Source: §5.1 · p. 21

5.2 Dickson matrix algebra approach to Theorem 5.1

• The Dickson-matrix approach connects the rank of a linearized polynomial to the rank of its associated matrix and yields a trace representation with k terms for rank k.The proof uses a rank-k matrix factorization into rank-one matrices and establishes that fewer than k trace terms cannot suffice.Source: §5.2 · pp. 21, 24

5.3 Further remarks on Theorem 5.1

• The representations also support composition into elementary linearized polynomials, while trace component functions correspond to basis component functions and their Fq-linear combinations.The paper describes these representations as convenient for some problems.Source: §5.3 · pp. 25–26

6 Characterizations of Ln(Fqm)

• The induced-transformation matrix BL has a block-circulant form, with its blocks arranged cyclically.The passage gives the block sequence B0, B1, …, Bt−1 and its cyclic shifts.Source: §6 · p. 27
• The authors recover a result first obtained by Brawley et al. through a matrix-theoretic approach.Source: §6 · p. 27
• Dickson matrices associated with Ln(Fqm) form a Fq-subalgebra isomorphic to Ln(Fqm), with block entries over Fqm subject to relations.The matrices are block circulant of type (t, m).Source: §6 · p. 28

7 Concluding remarks

• The paper proposes two new characterizations of the linearized polynomial algebra Ln(Fqn).Source: §7 · p. 28
• The characterizations provide simpler ways to establish recently proposed trace representations of linearized polynomials.Source: §7 · pp. 28–29
• The Dickson-matrix isomorphism can transform many linearized-polynomial problems into matrix-theoretic formats.The paper notes that Dickson matrices have been used to classify equivalent classes of polynomials in MPKC.Source: §7 · p. 29