Laurent Series

Main Definitions

• Defines LaurentSeries as an abbreviation for HahnSeries ℤ.
• Defines hasseDeriv of a Laurent series with coefficients in a module over a ring.
• Provides a coercion PowerSeries R into LaurentSeries R given by HahnSeries.ofPowerSeries.
• Defines LaurentSeries.powerSeriesPart
• Defines the localization map LaurentSeries.of_powerSeries_localization which evaluates to HahnSeries.ofPowerSeries.
• Embedding of rational functions into Laurent series, provided as a coercion, utilizing the underlying RatFunc.coeAlgHom.

Main Results

• Basic properties of Hasse derivatives

@[reducible, inline]

abbrev LaurentSeries (R : Type u) [Zero R] : Type u

A LaurentSeries is implemented as a HahnSeries with value group ℤ.

Equations

• LaurentSeries R = HahnSeries ℤ R

@[simp]

theorem LaurentSeries.hasseDeriv_apply (R : Type u_2) {V : Type u_3} [AddCommGroup V] [Semiring R] [Module R V] (k : ℕ) (f : LaurentSeries V) : (LaurentSeries.hasseDeriv R k) f = HahnSeries.ofSuppBddBelow (fun (n : ℤ) => Ring.choose (n + ↑k) k • f.coeff (n + ↑k)) ⋯

def LaurentSeries.hasseDeriv (R : Type u_2) {V : Type u_3} [AddCommGroup V] [Semiring R] [Module R V] (k : ℕ) : LaurentSeries V →ₗ[R] LaurentSeries V

The Hasse derivative of Laurent series, as a linear map.

Equations

• One or more equations did not get rendered due to their size.

theorem LaurentSeries.hasseDeriv_coeff {R : Type u_1} [Semiring R] {V : Type u_2} [AddCommGroup V] [Module R V] (k : ℕ) (f : LaurentSeries V) (n : ℤ) : ((LaurentSeries.hasseDeriv R k) f).coeff n = Ring.choose (n + ↑k) k • f.coeff (n + ↑k)

instance LaurentSeries.instCoePowerSeries {R : Type u_1} [Semiring R] : Coe (PowerSeries R) (LaurentSeries R)

Equations

• LaurentSeries.instCoePowerSeries = { coe := ⇑(HahnSeries.ofPowerSeries ℤ R) }

@[simp]

theorem LaurentSeries.coeff_coe_powerSeries {R : Type u_1} [Semiring R] (x : PowerSeries R) (n : ℕ) : ((HahnSeries.ofPowerSeries ℤ R) x).coeff ↑n = (PowerSeries.coeff R n) x

def LaurentSeries.powerSeriesPart {R : Type u_1} [Semiring R] (x : LaurentSeries R) : PowerSeries R

This is a power series that can be multiplied by an integer power of X to give our Laurent series. If the Laurent series is nonzero, powerSeriesPart has a nonzero constant term.

Equations

• x.powerSeriesPart = PowerSeries.mk fun (n : ℕ) => x.coeff (HahnSeries.order x + ↑n)

@[simp]

theorem LaurentSeries.powerSeriesPart_coeff {R : Type u_1} [Semiring R] (x : LaurentSeries R) (n : ℕ) : (PowerSeries.coeff R n) x.powerSeriesPart = x.coeff (HahnSeries.order x + ↑n)

@[simp]

theorem LaurentSeries.powerSeriesPart_zero {R : Type u_1} [Semiring R] : LaurentSeries.powerSeriesPart 0 = 0

@[simp]

theorem LaurentSeries.powerSeriesPart_eq_zero {R : Type u_1} [Semiring R] (x : LaurentSeries R) : x.powerSeriesPart = 0 ↔ x = 0

@[simp]

theorem LaurentSeries.single_order_mul_powerSeriesPart {R : Type u_1} [Semiring R] (x : LaurentSeries R) : (HahnSeries.single (HahnSeries.order x)) 1 * (HahnSeries.ofPowerSeries ℤ R) x.powerSeriesPart = x

theorem LaurentSeries.ofPowerSeries_powerSeriesPart {R : Type u_1} [Semiring R] (x : LaurentSeries R) : (HahnSeries.ofPowerSeries ℤ R) x.powerSeriesPart = (HahnSeries.single (-HahnSeries.order x)) 1 * x

instance LaurentSeries.instAlgebraPowerSeries {R : Type u_1} [CommSemiring R] : Algebra (PowerSeries R) (LaurentSeries R)

Equations

• LaurentSeries.instAlgebraPowerSeries = (HahnSeries.ofPowerSeries ℤ R).toAlgebra

@[simp]

theorem LaurentSeries.coe_algebraMap {R : Type u_1} [CommSemiring R] : ⇑(algebraMap (PowerSeries R) (LaurentSeries R)) = ⇑(HahnSeries.ofPowerSeries ℤ R)

instance LaurentSeries.of_powerSeries_localization {R : Type u_1} [CommRing R] : IsLocalization (Submonoid.powers PowerSeries.X) (LaurentSeries R)

The localization map from power series to Laurent series.

Equations

• ⋯ = ⋯

instance LaurentSeries.instIsFractionRingPowerSeries {K : Type u_2} [Field K] : IsFractionRing (PowerSeries K) (LaurentSeries K)

Equations

• ⋯ = ⋯

theorem PowerSeries.coe_zero {R : Type u_1} [Semiring R] : (HahnSeries.ofPowerSeries ℤ R) 0 = 0

theorem PowerSeries.coe_one {R : Type u_1} [Semiring R] : (HahnSeries.ofPowerSeries ℤ R) 1 = 1

theorem PowerSeries.coe_add {R : Type u_1} [Semiring R] (f : PowerSeries R) (g : PowerSeries R) : (HahnSeries.ofPowerSeries ℤ R) (f + g) = (HahnSeries.ofPowerSeries ℤ R) f + (HahnSeries.ofPowerSeries ℤ R) g

theorem PowerSeries.coe_sub {R' : Type u_2} [Ring R'] (f' : PowerSeries R') (g' : PowerSeries R') : (HahnSeries.ofPowerSeries ℤ R') (f' - g') = (HahnSeries.ofPowerSeries ℤ R') f' - (HahnSeries.ofPowerSeries ℤ R') g'

theorem PowerSeries.coe_neg {R' : Type u_2} [Ring R'] (f' : PowerSeries R') : (HahnSeries.ofPowerSeries ℤ R') (-f') = -(HahnSeries.ofPowerSeries ℤ R') f'

theorem PowerSeries.coe_mul {R : Type u_1} [Semiring R] (f : PowerSeries R) (g : PowerSeries R) : (HahnSeries.ofPowerSeries ℤ R) (f * g) = (HahnSeries.ofPowerSeries ℤ R) f * (HahnSeries.ofPowerSeries ℤ R) g

theorem PowerSeries.coeff_coe {R : Type u_1} [Semiring R] (f : PowerSeries R) (i : ℤ) : ((HahnSeries.ofPowerSeries ℤ R) f).coeff i = if i < 0 then 0 else (PowerSeries.coeff R i.natAbs) f

theorem PowerSeries.coe_C {R : Type u_1} [Semiring R] (r : R) : (HahnSeries.ofPowerSeries ℤ R) ((PowerSeries.C R) r) = HahnSeries.C r

theorem PowerSeries.coe_X {R : Type u_1} [Semiring R] : (HahnSeries.ofPowerSeries ℤ R) PowerSeries.X = (HahnSeries.single 1) 1

@[simp]

theorem PowerSeries.coe_smul {R : Type u_1} [Semiring R] {S : Type u_3} [Semiring S] [Module R S] (r : R) (x : PowerSeries S) : (HahnSeries.ofPowerSeries ℤ S) (r • x) = r • (HahnSeries.ofPowerSeries ℤ S) x

theorem PowerSeries.coe_pow {R : Type u_1} [Semiring R] (f : PowerSeries R) (n : ℕ) : (HahnSeries.ofPowerSeries ℤ R) (f ^ n) = (HahnSeries.ofPowerSeries ℤ R) f ^ n

def RatFunc.coeAlgHom (F : Type u) [Field F] : RatFunc F →ₐ[Polynomial F] LaurentSeries F

The coercion RatFunc F → LaurentSeries F as bundled alg hom.

Equations

• RatFunc.coeAlgHom F = RatFunc.liftAlgHom (Algebra.ofId (Polynomial F) (LaurentSeries F)) ⋯

def RatFunc.coeToLaurentSeries_fun {F : Type u} [Field F] : RatFunc F → LaurentSeries F

The coercion RatFunc F → LaurentSeries F as a function.

This is the implementation of coeToLaurentSeries.

Equations

• RatFunc.coeToLaurentSeries_fun = ⇑(RatFunc.coeAlgHom F)

instance RatFunc.coeToLaurentSeries {F : Type u} [Field F] : Coe (RatFunc F) (LaurentSeries F)

Equations

• RatFunc.coeToLaurentSeries = { coe := RatFunc.coeToLaurentSeries_fun }

theorem RatFunc.coe_def {F : Type u} [Field F] (f : RatFunc F) : ↑f = (RatFunc.coeAlgHom F) f

theorem RatFunc.coe_num_denom {F : Type u} [Field F] (f : RatFunc F) : ↑f = (HahnSeries.ofPowerSeries ℤ F) ↑f.num / (HahnSeries.ofPowerSeries ℤ F) ↑f.denom

theorem RatFunc.coe_injective {F : Type u} [Field F] : Function.Injective RatFunc.coeToLaurentSeries_fun

@[simp]

theorem RatFunc.coe_apply {F : Type u} [Field F] (f : RatFunc F) : (RatFunc.coeAlgHom F) f = ↑f

theorem RatFunc.coe_coe {F : Type u} [Field F] (P : Polynomial F) : (HahnSeries.ofPowerSeries ℤ F) ↑P = ↑↑P

@[simp]

theorem RatFunc.coe_zero {F : Type u} [Field F] : ↑0 = 0

theorem RatFunc.coe_ne_zero {F : Type u} [Field F] {f : Polynomial F} (hf : f ≠ 0) : ↑f ≠ 0

@[simp]

theorem RatFunc.coe_one {F : Type u} [Field F] : ↑1 = 1

@[simp]

theorem RatFunc.coe_add {F : Type u} [Field F] (f : RatFunc F) (g : RatFunc F) : ↑(f + g) = ↑f + ↑g

@[simp]

theorem RatFunc.coe_sub {F : Type u} [Field F] (f : RatFunc F) (g : RatFunc F) : ↑(f - g) = ↑f - ↑g

@[simp]

theorem RatFunc.coe_neg {F : Type u} [Field F] (f : RatFunc F) : ↑(-f) = -↑f

@[simp]

theorem RatFunc.coe_mul {F : Type u} [Field F] (f : RatFunc F) (g : RatFunc F) : ↑(f * g) = ↑f * ↑g

@[simp]

theorem RatFunc.coe_pow {F : Type u} [Field F] (f : RatFunc F) (n : ℕ) : ↑(f ^ n) = ↑f ^ n

@[simp]

theorem RatFunc.coe_div {F : Type u} [Field F] (f : RatFunc F) (g : RatFunc F) : ↑(f / g) = ↑f / ↑g

@[simp]

theorem RatFunc.coe_C {F : Type u} [Field F] (r : F) : ↑(RatFunc.C r) = HahnSeries.C r

@[simp]

theorem RatFunc.coe_smul {F : Type u} [Field F] (f : RatFunc F) (r : F) : ↑(r • f) = r • ↑f

@[simp]

theorem RatFunc.coe_X {F : Type u} [Field F] : ↑RatFunc.X = (HahnSeries.single 1) 1

theorem RatFunc.single_one_eq_pow {R : Type u_2} [Ring R] (n : ℕ) : (HahnSeries.single ↑n) 1 = (HahnSeries.single 1) 1 ^ n

theorem RatFunc.single_inv {F : Type u} [Field F] (d : ℤ) {α : F} (hα : α ≠ 0) : (HahnSeries.single (-d)) α⁻¹ = ((HahnSeries.single d) α)⁻¹

theorem RatFunc.single_zpow {F : Type u} [Field F] (n : ℤ) : (HahnSeries.single n) 1 = (HahnSeries.single 1) 1 ^ n

instance RatFunc.instAlgebraLaurentSeries {F : Type u} [Field F] : Algebra (RatFunc F) (LaurentSeries F)

Equations

• RatFunc.instAlgebraLaurentSeries = (RatFunc.coeAlgHom F).toAlgebra

theorem RatFunc.algebraMap_apply_div {F : Type u} [Field F] (p : Polynomial F) (q : Polynomial F) : (algebraMap (RatFunc F) (LaurentSeries F)) ((algebraMap (Polynomial F) (RatFunc F)) p / (algebraMap (Polynomial F) (RatFunc F)) q) = (algebraMap (Polynomial F) (LaurentSeries F)) p / (algebraMap (Polynomial F) (LaurentSeries F)) q

instance RatFunc.instIsScalarTowerPolynomialLaurentSeries {F : Type u} [Field F] : IsScalarTower (Polynomial F) (RatFunc F) (LaurentSeries F)

Equations

• ⋯ = ⋯