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number_theory.class_number.admissible_card_pow_degree

Admissible absolute values on polynomials #

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

Main results #

theorem polynomial.exists_eq_polynomial {Fq : Type u_1} [fintype Fq] [semiring Fq] {d m : } (hm : fintype.card Fq ^ d m) (b : polynomial Fq) (hb : b.nat_degree d) (A : fin m.succ polynomial Fq) (hA : (i : fin m.succ), (A i).degree < b.degree) :
(i₀ i₁ : fin m.succ), 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 : fintype.card Fq ^ d m) (b : polynomial Fq) (A : fin m.succ polynomial Fq) (hA : (i : fin m.succ), (A i).degree < b.degree) :
(i₀ i₁ : fin m.succ), i₀ i₁ (A i₁ - A i₀).degree < (b.nat_degree - d)

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 : polynomial Fq} (hb : b 0) {ε : } (hε : 0 < ε) (A : fin (fintype.card Fq ^ -real.log ε / real.log (fintype.card Fq)⌉₊).succ polynomial Fq) :

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.

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 : polynomial Fq} (hb : b 0) (A : fin n polynomial Fq) :
(t : fin n fin (fintype.card Fq ^ -real.log ε / real.log (fintype.card Fq)⌉₊)), (i₀ i₁ : fin n), t i₀ = t i₁ (polynomial.card_pow_degree (A i₁ % b - A i₀ % b)) < polynomial.card_pow_degree 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 : polynomial Fq} (hb : b 0) (A : fin n polynomial Fq) :
(t : fin n fin (fintype.card Fq ^ -real.log ε / real.log (fintype.card Fq)⌉₊)), (i₀ i₁ : fin n), t i₀ = t i₁ (polynomial.card_pow_degree (A i₁ % b - A i₀ % b)) < polynomial.card_pow_degree 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.

λ p, fintype.card Fq ^ degree p is an admissible absolute value. We set q ^ degree 0 = 0.

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