# Admissible absolute values on polynomials #

This file defines an admissible absolute value Polynomial.cardPowDegreeIsAdmissible which we use to show the class number of the ring of integers of a function field is finite.

## Main results #

• Polynomial.cardPowDegreeIsAdmissible shows cardPowDegree, mapping p : Polynomial 𝔽_q to q ^ degree p, is admissible
theorem Polynomial.exists_eq_polynomial {Fq : Type u_1} [Fintype Fq] [Semiring Fq] {d : } {m : } (hm : ^ d m) (b : ) (hb : ) (A : Fin ()) (hA : ∀ (i : Fin ()), ) :
i₀ i₁, i₀ i₁ A i₁ = A i₀

If A is a family of enough low-degree polynomials over a finite semiring, there is a pair of equal elements in A.

theorem Polynomial.exists_approx_polynomial_aux {Fq : Type u_1} [Fintype Fq] [Ring Fq] {d : } {m : } (hm : ^ d m) (b : ) (A : Fin ()) (hA : ∀ (i : Fin ()), ) :
i₀ i₁, i₀ i₁ Polynomial.degree (A i₁ - A i₀) < ↑()

If A is a family of enough low-degree polynomials over a finite ring, there is a pair of elements in A (with different indices but not necessarily distinct), such that their difference has small degree.

theorem Polynomial.exists_approx_polynomial {Fq : Type u_1} [Fintype Fq] [Field Fq] {b : } (hb : b 0) {ε : } (hε : 0 < ε) (A : Fin (Nat.succ ( ^ / Real.log ↑()⌉₊))) :
i₀ i₁, i₀ i₁ ↑(Polynomial.cardPowDegree (A i₁ % b - A i₀ % b)) < Polynomial.cardPowDegree b ε

If A is a family of enough low-degree polynomials over a finite field, there is a pair of elements in A (with different indices but not necessarily distinct), such that the difference of their remainders is close together.

theorem Polynomial.cardPowDegree_anti_archimedean {Fq : Type u_1} [Fintype Fq] [Field Fq] {x : } {y : } {z : } {a : } (hxy : Polynomial.cardPowDegree (x - y) < a) (hyz : Polynomial.cardPowDegree (y - z) < a) :
Polynomial.cardPowDegree (x - z) < a

If x is close to y and y is close to z, then x and z are at least as close.

theorem Polynomial.exists_partition_polynomial_aux {Fq : Type u_1} [Fintype Fq] [Field Fq] (n : ) {ε : } (hε : 0 < ε) {b : } (hb : b 0) (A : Fin n) :
t, ∀ (i₀ i₁ : Fin n), t i₀ = t i₁ ↑(Polynomial.cardPowDegree (A i₁ % b - A i₀ % b)) < Polynomial.cardPowDegree b ε

A slightly stronger version of exists_partition on which we perform induction on n: for all ε > 0, we can partition the remainders of any family of polynomials A into equivalence classes, where the equivalence(!) relation is "closer than ε".

theorem Polynomial.exists_partition_polynomial {Fq : Type u_1} [Fintype Fq] [Field Fq] (n : ) {ε : } (hε : 0 < ε) {b : } (hb : b 0) (A : Fin n) :
t, ∀ (i₀ i₁ : Fin n), t i₀ = t i₁↑(Polynomial.cardPowDegree (A i₁ % b - A i₀ % b)) < Polynomial.cardPowDegree b ε

For all ε > 0, we can partition the remainders of any family of polynomials A into classes, where all remainders in a class are close together.

noncomputable def Polynomial.cardPowDegreeIsAdmissible {Fq : Type u_1} [Fintype Fq] [Field Fq] :
fun p => Fintype.card Fq ^ degree p is an admissible absolute value. We set q ^ degree 0 = 0.