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.

Main Results

• Basic properties of Hasse derivatives

@[inline, reducible]

abbrev LaurentSeries (R : Type u_1) [Zero R] : Type u_1

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

Equations

• LaurentSeries.instCoePowerSeriesLaurentSeriesToZeroToMonoidWithZero = { 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

• LaurentSeries.powerSeriesPart x = 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) (LaurentSeries.powerSeriesPart x) = 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) : LaurentSeries.powerSeriesPart x = 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) (LaurentSeries.powerSeriesPart x) = x

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

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

Equations

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

@[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.instIsFractionRingPowerSeriesInstCommRingPowerSeriesToCommRingToEuclideanDomainLaurentSeriesToZeroToCommMonoidWithZeroToCommGroupWithZeroToSemifieldInstCommRingHahnSeriesToPartialOrderToOrderedAddCommMonoidToZeroToCommMonoidWithZeroToCommSemiringIntToOrderedCancelAddCommMonoidToStrictOrderedSemiringToLinearOrderedSemiringToLinearOrderedCommSemiringInstLinearOrderedCommRingInstAlgebraPowerSeriesLaurentSeriesToZeroToCommMonoidWithZeroInstCommSemiringPowerSeriesInstSemiringHahnSeriesToPartialOrderToOrderedAddCommMonoidToZeroToMonoidWithZeroIntToOrderedCancelAddCommMonoidToStrictOrderedSemiringToLinearOrderedSemiringToLinearOrderedCommSemiringInstLinearOrderedCommRingToSemiringToCommSemiring {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 (Int.natAbs i)) 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