mathlib3 documentation

ring_theory.polynomial.dickson

Dickson polynomials #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

The (generalised) Dickson polynomials are a family of polynomials indexed by ℕ × ℕ, with coefficients in a commutative ring R depending on an element a∈R. More precisely, the they satisfy the recursion dickson k a (n + 2) = X * (dickson k a n + 1) - a * (dickson k a n) with starting values dickson k a 0 = 3 - k and dickson k a 1 = X. In the literature, dickson k a n is called the n-th Dickson polynomial of the k-th kind associated to the parameter a : R. They are closely related to the Chebyshev polynomials in the case that a=1. When a=0 they are just the family of monomials X ^ n.

Main definition #

polynomial.dickson: the generalised Dickson polynomials.

Main statements #

polynomial.dickson_one_one_mul, the (m * n)-th Dickson polynomial of the first kind for parameter 1 : R is the composition of the m-th and n-th Dickson polynomials of the first kind for 1 : R.
polynomial.dickson_one_one_char_p, for a prime number p, the p-th Dickson polynomial of the first kind associated to parameter 1 : R is congruent to X ^ p modulo p.

References #

R. Lidl, G. L. Mullen and G. Turnwald, Dickson polynomials

TODO #

Redefine dickson in terms of linear_recurrence.
Show that dickson 2 1 is equal to the characteristic polynomial of the adjacency matrix of a type A Dynkin diagram.
Prove that the adjacency matrices of simply laced Dynkin diagrams are precisely the adjacency matrices of simple connected graphs which annihilate dickson 2 1.

noncomputable def polynomial.dickson {R : Type u_1} [comm_ring R] (k : ℕ) (a : R) :

ℕ → polynomial R

dickson is the nthe (generalised) Dickson polynomial of the k-th kind associated to the element a ∈ R.

Equations

polynomial.dickson k a (n + 2) = polynomial.X * polynomial.dickson k a n.succ - ⇑polynomial.C a * polynomial.dickson k a n
polynomial.dickson k a 1 = polynomial.X
polynomial.dickson k a 0 = 3 - ↑k

@[simp]

theorem polynomial.dickson_zero {R : Type u_1} [comm_ring R] (k : ℕ) (a : R) :

polynomial.dickson k a 0 = 3 - ↑k

@[simp]

theorem polynomial.dickson_one {R : Type u_1} [comm_ring R] (k : ℕ) (a : R) :

polynomial.dickson k a 1 = polynomial.X

theorem polynomial.dickson_two {R : Type u_1} [comm_ring R] (k : ℕ) (a : R) :

polynomial.dickson k a 2 = polynomial.X ^ 2 - ⇑polynomial.C a * (3 - ↑k)

@[simp]

theorem polynomial.dickson_add_two {R : Type u_1} [comm_ring R] (k : ℕ) (a : R) (n : ℕ) :

polynomial.dickson k a (n + 2) = polynomial.X * polynomial.dickson k a (n + 1) - ⇑polynomial.C a * polynomial.dickson k a n

theorem polynomial.dickson_of_two_le {R : Type u_1} [comm_ring R] (k : ℕ) (a : R) {n : ℕ} (h : 2 ≤ n) :

polynomial.dickson k a n = polynomial.X * polynomial.dickson k a (n - 1) - ⇑polynomial.C a * polynomial.dickson k a (n - 2)

theorem polynomial.map_dickson {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] {k : ℕ} {a : R} (f : R →+* S) (n : ℕ) :

polynomial.map f (polynomial.dickson k a n) = polynomial.dickson k (⇑f a) n

@[simp]

theorem polynomial.dickson_two_zero {R : Type u_1} [comm_ring R] (n : ℕ) :

polynomial.dickson 2 0 n = polynomial.X ^ n

A Lambda structure on `ℤ[X]` #

Mathlib doesn't currently know what a Lambda ring is. But once it does, we can endow ℤ[X] with a Lambda structure in terms of the dickson 1 1 polynomials defined below. There is exactly one other Lambda structure on ℤ[X] in terms of binomial polynomials.

theorem polynomial.dickson_one_one_eval_add_inv {R : Type u_1} [comm_ring R] (x y : R) (h : x * y = 1) (n : ℕ) :

polynomial.eval (x + y) (polynomial.dickson 1 1 n) = x ^ n + y ^ n

theorem polynomial.dickson_one_one_eq_chebyshev_T (R : Type u_1) [comm_ring R] [invertible 2] (n : ℕ) :

polynomial.dickson 1 1 n = 2 * (polynomial.chebyshev.T R n).comp (⇑polynomial.C (⅟ 2) * polynomial.X)

theorem polynomial.chebyshev_T_eq_dickson_one_one (R : Type u_1) [comm_ring R] [invertible 2] (n : ℕ) :

polynomial.chebyshev.T R n = ⇑polynomial.C (⅟ 2) * (polynomial.dickson 1 1 n).comp (2 * polynomial.X)

theorem polynomial.dickson_one_one_mul (R : Type u_1) [comm_ring R] (m n : ℕ) :

polynomial.dickson 1 1 (m * n) = (polynomial.dickson 1 1 m).comp (polynomial.dickson 1 1 n)

The (m * n)-th Dickson polynomial of the first kind is the composition of the m-th and n-th.

theorem polynomial.dickson_one_one_comp_comm (R : Type u_1) [comm_ring R] (m n : ℕ) :

(polynomial.dickson 1 1 m).comp (polynomial.dickson 1 1 n) = (polynomial.dickson 1 1 n).comp (polynomial.dickson 1 1 m)

theorem polynomial.dickson_one_one_zmod_p (p : ℕ) [fact (nat.prime p)] :

polynomial.dickson 1 1 p = polynomial.X ^ p

theorem polynomial.dickson_one_one_char_p (R : Type u_1) [comm_ring R] (p : ℕ) [fact (nat.prime p)] [char_p R p] :

polynomial.dickson 1 1 p = polynomial.X ^ p