Theory of univariate polynomials

This file defines Polynomial R, the type of univariate polynomials over the semiring R, builds a semiring structure on it, and gives basic definitions that are expanded in other files in this directory.

Main definitions

• monomial n a is the polynomial a X^n. Note that monomial n is defined as an R-linear map.
• C a is the constant polynomial a. Note that C is defined as a ring homomorphism.
• X is the polynomial X, i.e., monomial 1 1.
• p.sum f is ∑ n ∈ p.support, f n (p.coeff n), i.e., one sums the values of functions applied to coefficients of the polynomial p.
• p.erase n is the polynomial p in which one removes the c X^n term.

There are often two natural variants of lemmas involving sums, depending on whether one acts on the polynomials, or on the function. The naming convention is that one adds index when acting on the polynomials. For instance,

• sum_add_index states that (p + q).sum f = p.sum f + q.sum f;
• sum_add states that p.sum (fun n x ↦ f n x + g n x) = p.sum f + p.sum g.
• Notation to refer to Polynomial R, as R[X] or R[t].

Implementation

Polynomials are defined using R[ℕ], where R is a semiring. The variable X commutes with every polynomial p: lemma X_mul proves the identity X * p = p * X. The relationship to R[ℕ] is through a structure to make polynomials irreducible from the point of view of the kernel. Most operations are irreducible since Lean can not compute anyway with AddMonoidAlgebra. There are two exceptions that we make semireducible:

• The zero polynomial, so that its coefficients are definitionally equal to 0.
• The scalar action, to permit typeclass search to unfold it to resolve potential instance diamonds.

The raw implementation of the equivalence between R[X] and R[ℕ] is done through ofFinsupp and toFinsupp (or, equivalently, rcases p when p is a polynomial gives an element q of R[ℕ], and conversely ⟨q⟩ gives back p). The equivalence is also registered as a ring equiv in Polynomial.toFinsuppIso. These should in general not be used once the basic API for polynomials is constructed.

structure Polynomial (R : Type u_1) [Semiring R] : Type u_1

Polynomial R is the type of univariate polynomials over R.

Polynomials should be seen as (semi-)rings with the additional constructor X. The embedding from R is called C.

• ofFinsupp :: (
• toFinsupp : AddMonoidAlgebra R ℕ
• )

def Polynomial.«term_[X]» : Lean.TrailingParserDescr

Polynomial R is the type of univariate polynomials over R.

Polynomials should be seen as (semi-)rings with the additional constructor X. The embedding from R is called C.

Equations

• Polynomial.«term_[X]» = Lean.ParserDescr.trailingNode `Polynomial.term_[X] 9000 0 (Lean.ParserDescr.symbol "[X]")

theorem Polynomial.forall_iff_forall_finsupp {R : Type u} [Semiring R] (P : Polynomial R → Prop) : (∀ (p : Polynomial R), P p) ↔ ∀ (q : AddMonoidAlgebra R ℕ), P { toFinsupp := q }

theorem Polynomial.exists_iff_exists_finsupp {R : Type u} [Semiring R] (P : Polynomial R → Prop) : (∃ (p : Polynomial R), P p) ↔ ∃ (q : AddMonoidAlgebra R ℕ), P { toFinsupp := q }

@[simp]

theorem Polynomial.eta {R : Type u} [Semiring R] (f : Polynomial R) : { toFinsupp := f.toFinsupp } = f

Conversions to and from AddMonoidAlgebra

Since R[X] is not defeq to R[ℕ], but instead is a structure wrapping it, we have to copy across all the arithmetic operators manually, along with the lemmas about how they unfold around Polynomial.ofFinsupp and Polynomial.toFinsupp.

instance Polynomial.zero {R : Type u} [Semiring R] : Zero (Polynomial R)

Equations

• Polynomial.zero = { zero := { toFinsupp := 0 } }

instance Polynomial.one {R : Type u} [Semiring R] : One (Polynomial R)

Equations

• Polynomial.one = { one := { toFinsupp := 1 } }

instance Polynomial.add' {R : Type u} [Semiring R] : Add (Polynomial R)

Equations

• Polynomial.add' = { add := Polynomial.add }

instance Polynomial.neg' {R : Type u} [Ring R] : Neg (Polynomial R)

Equations

• Polynomial.neg' = { neg := Polynomial.neg }

instance Polynomial.sub {R : Type u} [Ring R] : Sub (Polynomial R)

Equations

• Polynomial.sub = { sub := fun (a b : Polynomial R) => a + -b }

instance Polynomial.mul' {R : Type u} [Semiring R] : Mul (Polynomial R)

Equations

• Polynomial.mul' = { mul := Polynomial.mul }

@[simp]

theorem Polynomial.add_eq_add {R : Type u} [Semiring R] {p : Polynomial R} {q : Polynomial R} : Polynomial.add p q = p + q

@[simp]

theorem Polynomial.mul_eq_mul {R : Type u} [Semiring R] {p : Polynomial R} {q : Polynomial R} : Polynomial.mul p q = p * q

instance Polynomial.smulZeroClass {R : Type u} [Semiring R] {S : Type u_1} [SMulZeroClass S R] : SMulZeroClass S (Polynomial R)

Equations

• Polynomial.smulZeroClass = SMulZeroClass.mk ⋯

@[instance 1]

instance Polynomial.pow {R : Type u} [Semiring R] : Pow (Polynomial R) ℕ

Equations

• Polynomial.pow = { pow := fun (p : Polynomial R) (n : ℕ) => npowRec n p }

@[simp]

theorem Polynomial.ofFinsupp_zero {R : Type u} [Semiring R] : { toFinsupp := 0 } = 0

@[simp]

theorem Polynomial.ofFinsupp_one {R : Type u} [Semiring R] : { toFinsupp := 1 } = 1

@[simp]

theorem Polynomial.ofFinsupp_add {R : Type u} [Semiring R] {a : AddMonoidAlgebra R ℕ} {b : AddMonoidAlgebra R ℕ} : { toFinsupp := a + b } = { toFinsupp := a } + { toFinsupp := b }

@[simp]

theorem Polynomial.ofFinsupp_neg {R : Type u} [Ring R] {a : AddMonoidAlgebra R ℕ} : { toFinsupp := -a } = -{ toFinsupp := a }

@[simp]

theorem Polynomial.ofFinsupp_sub {R : Type u} [Ring R] {a : AddMonoidAlgebra R ℕ} {b : AddMonoidAlgebra R ℕ} : { toFinsupp := a - b } = { toFinsupp := a } - { toFinsupp := b }

@[simp]

theorem Polynomial.ofFinsupp_mul {R : Type u} [Semiring R] (a : AddMonoidAlgebra R ℕ) (b : AddMonoidAlgebra R ℕ) : { toFinsupp := a * b } = { toFinsupp := a } * { toFinsupp := b }

@[simp]

theorem Polynomial.ofFinsupp_smul {R : Type u} [Semiring R] {S : Type u_1} [SMulZeroClass S R] (a : S) (b : AddMonoidAlgebra R ℕ) : { toFinsupp := a • b } = a • { toFinsupp := b }

@[simp]

theorem Polynomial.ofFinsupp_pow {R : Type u} [Semiring R] (a : AddMonoidAlgebra R ℕ) (n : ℕ) : { toFinsupp := a ^ n } = { toFinsupp := a } ^ n

@[simp]

theorem Polynomial.toFinsupp_zero {R : Type u} [Semiring R] : Polynomial.toFinsupp 0 = 0

@[simp]

theorem Polynomial.toFinsupp_one {R : Type u} [Semiring R] : Polynomial.toFinsupp 1 = 1

@[simp]

theorem Polynomial.toFinsupp_add {R : Type u} [Semiring R] (a : Polynomial R) (b : Polynomial R) : (a + b).toFinsupp = a.toFinsupp + b.toFinsupp

@[simp]

theorem Polynomial.toFinsupp_neg {R : Type u} [Ring R] (a : Polynomial R) : (-a).toFinsupp = -a.toFinsupp

@[simp]

theorem Polynomial.toFinsupp_sub {R : Type u} [Ring R] (a : Polynomial R) (b : Polynomial R) : (a - b).toFinsupp = a.toFinsupp - b.toFinsupp

@[simp]

theorem Polynomial.toFinsupp_mul {R : Type u} [Semiring R] (a : Polynomial R) (b : Polynomial R) : (a * b).toFinsupp = a.toFinsupp * b.toFinsupp

@[simp]

theorem Polynomial.toFinsupp_smul {R : Type u} [Semiring R] {S : Type u_1} [SMulZeroClass S R] (a : S) (b : Polynomial R) : (a • b).toFinsupp = a • b.toFinsupp

@[simp]

theorem Polynomial.toFinsupp_pow {R : Type u} [Semiring R] (a : Polynomial R) (n : ℕ) : (a ^ n).toFinsupp = a.toFinsupp ^ n

theorem IsSMulRegular.polynomial {R : Type u} [Semiring R] {S : Type u_1} [Monoid S] [DistribMulAction S R] {a : S} (ha : IsSMulRegular R a) : IsSMulRegular (Polynomial R) a

theorem Polynomial.toFinsupp_injective {R : Type u} [Semiring R] : Function.Injective Polynomial.toFinsupp

@[simp]

theorem Polynomial.toFinsupp_inj {R : Type u} [Semiring R] {a : Polynomial R} {b : Polynomial R} : a.toFinsupp = b.toFinsupp ↔ a = b

@[simp]

theorem Polynomial.toFinsupp_eq_zero {R : Type u} [Semiring R] {a : Polynomial R} : a.toFinsupp = 0 ↔ a = 0

@[simp]

theorem Polynomial.toFinsupp_eq_one {R : Type u} [Semiring R] {a : Polynomial R} : a.toFinsupp = 1 ↔ a = 1

theorem Polynomial.ofFinsupp_inj {R : Type u} [Semiring R] {a : AddMonoidAlgebra R ℕ} {b : AddMonoidAlgebra R ℕ} : { toFinsupp := a } = { toFinsupp := b } ↔ a = b

A more convenient spelling of Polynomial.ofFinsupp.injEq in terms of Iff.

@[simp]

theorem Polynomial.ofFinsupp_eq_zero {R : Type u} [Semiring R] {a : AddMonoidAlgebra R ℕ} : { toFinsupp := a } = 0 ↔ a = 0

@[simp]

theorem Polynomial.ofFinsupp_eq_one {R : Type u} [Semiring R] {a : AddMonoidAlgebra R ℕ} : { toFinsupp := a } = 1 ↔ a = 1

instance Polynomial.inhabited {R : Type u} [Semiring R] : Inhabited (Polynomial R)

Equations

• Polynomial.inhabited = { default := 0 }

instance Polynomial.instNatCast {R : Type u} [Semiring R] : NatCast (Polynomial R)

Equations

• Polynomial.instNatCast = { natCast := fun (n : ℕ) => { toFinsupp := ↑n } }

instance Polynomial.semiring {R : Type u} [Semiring R] : Semiring (Polynomial R)

Equations

• Polynomial.semiring = let __src := Function.Injective.semiring Polynomial.toFinsupp ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯; Semiring.mk ⋯ ⋯ ⋯ ⋯ (fun (n : ℕ) (x : Polynomial R) => x ^ n) ⋯ ⋯

instance Polynomial.distribSMul {R : Type u} [Semiring R] {S : Type u_1} [DistribSMul S R] : DistribSMul S (Polynomial R)

Equations

• Polynomial.distribSMul = let __src := Function.Injective.distribSMul { toFun := Polynomial.toFinsupp, map_zero' := ⋯, map_add' := ⋯ } ⋯ ⋯; DistribSMul.mk ⋯

instance Polynomial.distribMulAction {R : Type u} [Semiring R] {S : Type u_1} [Monoid S] [DistribMulAction S R] : DistribMulAction S (Polynomial R)

Equations

• Polynomial.distribMulAction = let __src := Function.Injective.distribMulAction { toFun := Polynomial.toFinsupp, map_zero' := ⋯, map_add' := ⋯ } ⋯ ⋯; DistribMulAction.mk ⋯ ⋯

instance Polynomial.faithfulSMul {R : Type u} [Semiring R] {S : Type u_1} [SMulZeroClass S R] [FaithfulSMul S R] : FaithfulSMul S (Polynomial R)

Equations

• ⋯ = ⋯

instance Polynomial.module {R : Type u} [Semiring R] {S : Type u_1} [Semiring S] [Module S R] : Module S (Polynomial R)

Equations

• Polynomial.module = let __src := Function.Injective.module S { toFun := Polynomial.toFinsupp, map_zero' := ⋯, map_add' := ⋯ } ⋯ ⋯; Module.mk ⋯ ⋯

instance Polynomial.smulCommClass {R : Type u} [Semiring R] {S₁ : Type u_1} {S₂ : Type u_2} [SMulZeroClass S₁ R] [SMulZeroClass S₂ R] [SMulCommClass S₁ S₂ R] : SMulCommClass S₁ S₂ (Polynomial R)

Equations

• ⋯ = ⋯

instance Polynomial.isScalarTower {R : Type u} [Semiring R] {S₁ : Type u_1} {S₂ : Type u_2} [SMul S₁ S₂] [SMulZeroClass S₁ R] [SMulZeroClass S₂ R] [IsScalarTower S₁ S₂ R] : IsScalarTower S₁ S₂ (Polynomial R)

Equations

• ⋯ = ⋯

instance Polynomial.isScalarTower_right {α : Type u_1} {K : Type u_2} [Semiring K] [DistribSMul α K] [IsScalarTower α K K] : IsScalarTower α (Polynomial K) (Polynomial K)

Equations

• ⋯ = ⋯

instance Polynomial.isCentralScalar {R : Type u} [Semiring R] {S : Type u_1} [SMulZeroClass S R] [SMulZeroClass Sᵐᵒᵖ R] [IsCentralScalar S R] : IsCentralScalar S (Polynomial R)

Equations

• ⋯ = ⋯

instance Polynomial.unique {R : Type u} [Semiring R] [Subsingleton R] : Unique (Polynomial R)

Equations

• Polynomial.unique = let __src := Polynomial.inhabited; { toInhabited := __src, uniq := ⋯ }

@[simp]

theorem Polynomial.toFinsuppIso_symm_apply (R : Type u) [Semiring R] (toFinsupp : AddMonoidAlgebra R ℕ) : (Polynomial.toFinsuppIso R).symm toFinsupp = { toFinsupp := toFinsupp }

@[simp]

theorem Polynomial.toFinsuppIso_apply (R : Type u) [Semiring R] (self : Polynomial R) : (Polynomial.toFinsuppIso R) self = self.toFinsupp

def Polynomial.toFinsuppIso (R : Type u) [Semiring R] : Polynomial R ≃+* AddMonoidAlgebra R ℕ

Ring isomorphism between R[X] and R[ℕ]. This is just an implementation detail, but it can be useful to transfer results from Finsupp to polynomials.

Equations

• Polynomial.toFinsuppIso R = { toFun := Polynomial.toFinsupp, invFun := Polynomial.ofFinsupp, left_inv := ⋯, right_inv := ⋯, map_mul' := ⋯, map_add' := ⋯ }

instance Polynomial.instDecidableEq (R : Type u) [Semiring R] [DecidableEq R] : DecidableEq (Polynomial R)

Equations

• Polynomial.instDecidableEq R = (Polynomial.toFinsuppIso R).decidableEq

theorem Polynomial.ofFinsupp_sum {R : Type u} [Semiring R] {ι : Type u_1} (s : Finset ι) (f : ι → AddMonoidAlgebra R ℕ) : { toFinsupp := ∑ i ∈ s, f i } = ∑ i ∈ s, { toFinsupp := f i }

theorem Polynomial.toFinsupp_sum {R : Type u} [Semiring R] {ι : Type u_1} (s : Finset ι) (f : ι → Polynomial R) : (∑ i ∈ s, f i).toFinsupp = ∑ i ∈ s, (f i).toFinsupp

def Polynomial.support {R : Type u} [Semiring R] : Polynomial R → Finset ℕ

The set of all n such that X^n has a non-zero coefficient.

Equations

• x.support = match x with | { toFinsupp := p } => p.support

@[simp]

theorem Polynomial.support_ofFinsupp {R : Type u} [Semiring R] (p : AddMonoidAlgebra R ℕ) : { toFinsupp := p }.support = p.support

theorem Polynomial.support_toFinsupp {R : Type u} [Semiring R] (p : Polynomial R) : p.toFinsupp.support = p.support

@[simp]

theorem Polynomial.support_zero {R : Type u} [Semiring R] : Polynomial.support 0 = ∅

@[simp]

theorem Polynomial.support_eq_empty {R : Type u} [Semiring R] {p : Polynomial R} : p.support = ∅ ↔ p = 0

@[simp]

theorem Polynomial.support_nonempty {R : Type u} [Semiring R] {p : Polynomial R} : p.support.Nonempty ↔ p ≠ 0

theorem Polynomial.card_support_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} : p.support.card = 0 ↔ p = 0

def Polynomial.monomial {R : Type u} [Semiring R] (n : ℕ) : R →ₗ[R] Polynomial R

monomial s a is the monomial a * X^s

Equations

• Polynomial.monomial n = { toFun := fun (t : R) => { toFinsupp := Finsupp.single n t }, map_add' := ⋯, map_smul' := ⋯ }

@[simp]

theorem Polynomial.toFinsupp_monomial {R : Type u} [Semiring R] (n : ℕ) (r : R) : ((Polynomial.monomial n) r).toFinsupp = Finsupp.single n r

@[simp]

theorem Polynomial.ofFinsupp_single {R : Type u} [Semiring R] (n : ℕ) (r : R) : { toFinsupp := Finsupp.single n r } = (Polynomial.monomial n) r

theorem Polynomial.monomial_zero_right {R : Type u} [Semiring R] (n : ℕ) : (Polynomial.monomial n) 0 = 0

theorem Polynomial.monomial_zero_one {R : Type u} [Semiring R] : (Polynomial.monomial 0) 1 = 1

theorem Polynomial.monomial_add {R : Type u} [Semiring R] (n : ℕ) (r : R) (s : R) : (Polynomial.monomial n) (r + s) = (Polynomial.monomial n) r + (Polynomial.monomial n) s

theorem Polynomial.monomial_mul_monomial {R : Type u} [Semiring R] (n : ℕ) (m : ℕ) (r : R) (s : R) : (Polynomial.monomial n) r * (Polynomial.monomial m) s = (Polynomial.monomial (n + m)) (r * s)

@[simp]

theorem Polynomial.monomial_pow {R : Type u} [Semiring R] (n : ℕ) (r : R) (k : ℕ) : (Polynomial.monomial n) r ^ k = (Polynomial.monomial (n * k)) (r ^ k)

theorem Polynomial.smul_monomial {R : Type u} [Semiring R] {S : Type u_1} [SMulZeroClass S R] (a : S) (n : ℕ) (b : R) : a • (Polynomial.monomial n) b = (Polynomial.monomial n) (a • b)

theorem Polynomial.monomial_injective {R : Type u} [Semiring R] (n : ℕ) : Function.Injective ⇑(Polynomial.monomial n)

@[simp]

theorem Polynomial.monomial_eq_zero_iff {R : Type u} [Semiring R] (t : R) (n : ℕ) : (Polynomial.monomial n) t = 0 ↔ t = 0

theorem Polynomial.support_add {R : Type u} [Semiring R] {p : Polynomial R} {q : Polynomial R} : (p + q).support ⊆ p.support ∪ q.support

def Polynomial.C {R : Type u} [Semiring R] : R →+* Polynomial R

C a is the constant polynomial a. C is provided as a ring homomorphism.

Equations

• Polynomial.C = let __src := Polynomial.monomial 0; { toFun := __src.toFun, map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add' := ⋯ }

@[simp]

theorem Polynomial.monomial_zero_left {R : Type u} [Semiring R] (a : R) : (Polynomial.monomial 0) a = Polynomial.C a

@[simp]

theorem Polynomial.toFinsupp_C {R : Type u} [Semiring R] (a : R) : (Polynomial.C a).toFinsupp = AddMonoidAlgebra.single 0 a

theorem Polynomial.C_0 {R : Type u} [Semiring R] : Polynomial.C 0 = 0

theorem Polynomial.C_1 {R : Type u} [Semiring R] : Polynomial.C 1 = 1

theorem Polynomial.C_mul {R : Type u} {a : R} {b : R} [Semiring R] : Polynomial.C (a * b) = Polynomial.C a * Polynomial.C b

theorem Polynomial.C_add {R : Type u} {a : R} {b : R} [Semiring R] : Polynomial.C (a + b) = Polynomial.C a + Polynomial.C b

@[simp]

theorem Polynomial.smul_C {R : Type u} [Semiring R] {S : Type u_1} [SMulZeroClass S R] (s : S) (r : R) : s • Polynomial.C r = Polynomial.C (s • r)

theorem Polynomial.C_pow {R : Type u} {a : R} {n : ℕ} [Semiring R] : Polynomial.C (a ^ n) = Polynomial.C a ^ n

theorem Polynomial.C_eq_natCast {R : Type u} [Semiring R] (n : ℕ) : Polynomial.C ↑n = ↑n

@[deprecated Polynomial.C_eq_natCast]

theorem Polynomial.C_eq_nat_cast {R : Type u} [Semiring R] (n : ℕ) : Polynomial.C ↑n = ↑n

Alias of Polynomial.C_eq_natCast.

@[simp]

theorem Polynomial.C_mul_monomial {R : Type u} {a : R} {b : R} {n : ℕ} [Semiring R] : Polynomial.C a * (Polynomial.monomial n) b = (Polynomial.monomial n) (a * b)

@[simp]

theorem Polynomial.monomial_mul_C {R : Type u} {a : R} {b : R} {n : ℕ} [Semiring R] : (Polynomial.monomial n) a * Polynomial.C b = (Polynomial.monomial n) (a * b)

def Polynomial.X {R : Type u} [Semiring R] : Polynomial R

X is the polynomial variable (aka indeterminate).

Equations

• Polynomial.X = (Polynomial.monomial 1) 1

theorem Polynomial.monomial_one_one_eq_X {R : Type u} [Semiring R] : (Polynomial.monomial 1) 1 = Polynomial.X

theorem Polynomial.monomial_one_right_eq_X_pow {R : Type u} [Semiring R] (n : ℕ) : (Polynomial.monomial n) 1 = Polynomial.X ^ n

@[simp]

theorem Polynomial.toFinsupp_X {R : Type u} [Semiring R] : Polynomial.X.toFinsupp = Finsupp.single 1 1

theorem Polynomial.X_mul {R : Type u} [Semiring R] {p : Polynomial R} : Polynomial.X * p = p * Polynomial.X

X commutes with everything, even when the coefficients are noncommutative.

theorem Polynomial.X_pow_mul {R : Type u} [Semiring R] {p : Polynomial R} {n : ℕ} : Polynomial.X ^ n * p = p * Polynomial.X ^ n

@[simp]

theorem Polynomial.X_mul_C {R : Type u} [Semiring R] (r : R) : Polynomial.X * Polynomial.C r = Polynomial.C r * Polynomial.X

Prefer putting constants to the left of X.

This lemma is the loop-avoiding simp version of Polynomial.X_mul.

@[simp]

theorem Polynomial.X_pow_mul_C {R : Type u} [Semiring R] (r : R) (n : ℕ) : Polynomial.X ^ n * Polynomial.C r = Polynomial.C r * Polynomial.X ^ n

Prefer putting constants to the left of X ^ n.

This lemma is the loop-avoiding simp version of X_pow_mul.

theorem Polynomial.X_pow_mul_assoc {R : Type u} [Semiring R] {p : Polynomial R} {q : Polynomial R} {n : ℕ} : p * Polynomial.X ^ n * q = p * q * Polynomial.X ^ n

@[simp]

theorem Polynomial.X_pow_mul_assoc_C {R : Type u} [Semiring R] {p : Polynomial R} {n : ℕ} (r : R) : p * Polynomial.X ^ n * Polynomial.C r = p * Polynomial.C r * Polynomial.X ^ n

Prefer putting constants to the left of X ^ n.

This lemma is the loop-avoiding simp version of X_pow_mul_assoc.

theorem Polynomial.commute_X {R : Type u} [Semiring R] (p : Polynomial R) : Commute Polynomial.X p

theorem Polynomial.commute_X_pow {R : Type u} [Semiring R] (p : Polynomial R) (n : ℕ) : Commute (Polynomial.X ^ n) p

@[simp]

theorem Polynomial.monomial_mul_X {R : Type u} [Semiring R] (n : ℕ) (r : R) : (Polynomial.monomial n) r * Polynomial.X = (Polynomial.monomial (n + 1)) r

@[simp]

theorem Polynomial.monomial_mul_X_pow {R : Type u} [Semiring R] (n : ℕ) (r : R) (k : ℕ) : (Polynomial.monomial n) r * Polynomial.X ^ k = (Polynomial.monomial (n + k)) r

@[simp]

theorem Polynomial.X_mul_monomial {R : Type u} [Semiring R] (n : ℕ) (r : R) : Polynomial.X * (Polynomial.monomial n) r = (Polynomial.monomial (n + 1)) r

@[simp]

theorem Polynomial.X_pow_mul_monomial {R : Type u} [Semiring R] (k : ℕ) (n : ℕ) (r : R) : Polynomial.X ^ k * (Polynomial.monomial n) r = (Polynomial.monomial (n + k)) r

def Polynomial.coeff {R : Type u} [Semiring R] : Polynomial R → ℕ → R

coeff p n (often denoted p.coeff n) is the coefficient of X^n in p.

Equations

• x.coeff = match (motive := Polynomial R → ℕ → R) x with | { toFinsupp := p } => ⇑p

@[simp]

theorem Polynomial.coeff_ofFinsupp {R : Type u} [Semiring R] (p : AddMonoidAlgebra R ℕ) : { toFinsupp := p }.coeff = ⇑p

theorem Polynomial.coeff_injective {R : Type u} [Semiring R] : Function.Injective Polynomial.coeff

@[simp]

theorem Polynomial.coeff_inj {R : Type u} [Semiring R] {p : Polynomial R} {q : Polynomial R} : p.coeff = q.coeff ↔ p = q

theorem Polynomial.toFinsupp_apply {R : Type u} [Semiring R] (f : Polynomial R) (i : ℕ) : f.toFinsupp i = f.coeff i

theorem Polynomial.coeff_monomial {R : Type u} {a : R} {m : ℕ} {n : ℕ} [Semiring R] : ((Polynomial.monomial n) a).coeff m = if n = m then a else 0

@[simp]

theorem Polynomial.coeff_zero {R : Type u} [Semiring R] (n : ℕ) : Polynomial.coeff 0 n = 0

theorem Polynomial.coeff_one {R : Type u} [Semiring R] {n : ℕ} : Polynomial.coeff 1 n = if n = 0 then 1 else 0

@[simp]

theorem Polynomial.coeff_one_zero {R : Type u} [Semiring R] : Polynomial.coeff 1 0 = 1

@[simp]

theorem Polynomial.coeff_X_one {R : Type u} [Semiring R] : Polynomial.X.coeff 1 = 1

@[simp]

theorem Polynomial.coeff_X_zero {R : Type u} [Semiring R] : Polynomial.X.coeff 0 = 0

@[simp]

theorem Polynomial.coeff_monomial_succ {R : Type u} {a : R} {n : ℕ} [Semiring R] : ((Polynomial.monomial (n + 1)) a).coeff 0 = 0

theorem Polynomial.coeff_X {R : Type u} {n : ℕ} [Semiring R] : Polynomial.X.coeff n = if 1 = n then 1 else 0

theorem Polynomial.coeff_X_of_ne_one {R : Type u} [Semiring R] {n : ℕ} (hn : n ≠ 1) : Polynomial.X.coeff n = 0

@[simp]

theorem Polynomial.mem_support_iff {R : Type u} {n : ℕ} [Semiring R] {p : Polynomial R} : n ∈ p.support ↔ p.coeff n ≠ 0

theorem Polynomial.not_mem_support_iff {R : Type u} {n : ℕ} [Semiring R] {p : Polynomial R} : n ∉ p.support ↔ p.coeff n = 0

theorem Polynomial.coeff_C {R : Type u} {a : R} {n : ℕ} [Semiring R] : (Polynomial.C a).coeff n = if n = 0 then a else 0

@[simp]

theorem Polynomial.coeff_C_zero {R : Type u} {a : R} [Semiring R] : (Polynomial.C a).coeff 0 = a

theorem Polynomial.coeff_C_ne_zero {R : Type u} {a : R} {n : ℕ} [Semiring R] (h : n ≠ 0) : (Polynomial.C a).coeff n = 0

@[simp]

theorem Polynomial.coeff_C_succ {R : Type u} [Semiring R] {r : R} {n : ℕ} : (Polynomial.C r).coeff (n + 1) = 0

@[simp]

theorem Polynomial.coeff_natCast_ite {R : Type u} {m : ℕ} {n : ℕ} [Semiring R] : (↑m).coeff n = ↑(if n = 0 then m else 0)

@[deprecated Polynomial.coeff_natCast_ite]

theorem Polynomial.coeff_nat_cast_ite {R : Type u} {m : ℕ} {n : ℕ} [Semiring R] : (↑m).coeff n = ↑(if n = 0 then m else 0)

Alias of Polynomial.coeff_natCast_ite.

@[simp]

theorem Polynomial.coeff_ofNat_zero {R : Type u} [Semiring R] (a : ℕ) [a.AtLeastTwo] : (OfNat.ofNat a).coeff 0 = OfNat.ofNat a

@[simp]

theorem Polynomial.coeff_ofNat_succ {R : Type u} [Semiring R] (a : ℕ) (n : ℕ) [h : a.AtLeastTwo] : (OfNat.ofNat a).coeff (n + 1) = 0

theorem Polynomial.C_mul_X_pow_eq_monomial {R : Type u} {a : R} [Semiring R] {n : ℕ} : Polynomial.C a * Polynomial.X ^ n = (Polynomial.monomial n) a

@[simp]

theorem Polynomial.toFinsupp_C_mul_X_pow {R : Type u} [Semiring R] (a : R) (n : ℕ) : (Polynomial.C a * Polynomial.X ^ n).toFinsupp = Finsupp.single n a

theorem Polynomial.C_mul_X_eq_monomial {R : Type u} {a : R} [Semiring R] : Polynomial.C a * Polynomial.X = (Polynomial.monomial 1) a

@[simp]

theorem Polynomial.toFinsupp_C_mul_X {R : Type u} [Semiring R] (a : R) : (Polynomial.C a * Polynomial.X).toFinsupp = Finsupp.single 1 a

theorem Polynomial.C_injective {R : Type u} [Semiring R] : Function.Injective ⇑Polynomial.C

@[simp]

theorem Polynomial.C_inj {R : Type u} {a : R} {b : R} [Semiring R] : Polynomial.C a = Polynomial.C b ↔ a = b

@[simp]

theorem Polynomial.C_eq_zero {R : Type u} {a : R} [Semiring R] : Polynomial.C a = 0 ↔ a = 0

theorem Polynomial.C_ne_zero {R : Type u} {a : R} [Semiring R] : Polynomial.C a ≠ 0 ↔ a ≠ 0

theorem Polynomial.subsingleton_iff_subsingleton {R : Type u} [Semiring R] : Subsingleton (Polynomial R) ↔ Subsingleton R

theorem Polynomial.Nontrivial.of_polynomial_ne {R : Type u} [Semiring R] {p : Polynomial R} {q : Polynomial R} (h : p ≠ q) : Nontrivial R

theorem Polynomial.forall_eq_iff_forall_eq {R : Type u} [Semiring R] : (∀ (f g : Polynomial R), f = g) ↔ ∀ (a b : R), a = b

theorem Polynomial.ext_iff {R : Type u} [Semiring R] {p : Polynomial R} {q : Polynomial R} : p = q ↔ ∀ (n : ℕ), p.coeff n = q.coeff n

theorem Polynomial.ext {R : Type u} [Semiring R] {p : Polynomial R} {q : Polynomial R} : (∀ (n : ℕ), p.coeff n = q.coeff n) → p = q

theorem Polynomial.addSubmonoid_closure_setOf_eq_monomial {R : Type u} [Semiring R] : AddSubmonoid.closure {p : Polynomial R | ∃ (n : ℕ) (a : R), p = (Polynomial.monomial n) a} = ⊤

Monomials generate the additive monoid of polynomials.

theorem Polynomial.addHom_ext {R : Type u} [Semiring R] {M : Type u_1} [AddMonoid M] {f : Polynomial R →+ M} {g : Polynomial R →+ M} (h : ∀ (n : ℕ) (a : R), f ((Polynomial.monomial n) a) = g ((Polynomial.monomial n) a)) : f = g

theorem Polynomial.addHom_ext' {R : Type u} [Semiring R] {M : Type u_1} [AddMonoid M] {f : Polynomial R →+ M} {g : Polynomial R →+ M} (h : ∀ (n : ℕ), f.comp (Polynomial.monomial n).toAddMonoidHom = g.comp (Polynomial.monomial n).toAddMonoidHom) : f = g

theorem Polynomial.lhom_ext' {R : Type u} [Semiring R] {M : Type u_1} [AddCommMonoid M] [Module R M] {f : Polynomial R →ₗ[R] M} {g : Polynomial R →ₗ[R] M} (h : ∀ (n : ℕ), f ∘ₗ Polynomial.monomial n = g ∘ₗ Polynomial.monomial n) : f = g

theorem Polynomial.eq_zero_of_eq_zero {R : Type u} [Semiring R] (h : 0 = 1) (p : Polynomial R) : p = 0

theorem Polynomial.support_monomial {R : Type u} [Semiring R] (n : ℕ) {a : R} (H : a ≠ 0) : ((Polynomial.monomial n) a).support = {n}

theorem Polynomial.support_monomial' {R : Type u} [Semiring R] (n : ℕ) (a : R) : ((Polynomial.monomial n) a).support ⊆ {n}

theorem Polynomial.support_C_mul_X {R : Type u} [Semiring R] {c : R} (h : c ≠ 0) : (Polynomial.C c * Polynomial.X).support = {1}

theorem Polynomial.support_C_mul_X' {R : Type u} [Semiring R] (c : R) : (Polynomial.C c * Polynomial.X).support ⊆ {1}

theorem Polynomial.support_C_mul_X_pow {R : Type u} [Semiring R] (n : ℕ) {c : R} (h : c ≠ 0) : (Polynomial.C c * Polynomial.X ^ n).support = {n}

theorem Polynomial.support_C_mul_X_pow' {R : Type u} [Semiring R] (n : ℕ) (c : R) : (Polynomial.C c * Polynomial.X ^ n).support ⊆ {n}

theorem Polynomial.support_binomial' {R : Type u} [Semiring R] (k : ℕ) (m : ℕ) (x : R) (y : R) : (Polynomial.C x * Polynomial.X ^ k + Polynomial.C y * Polynomial.X ^ m).support ⊆ {k, m}

theorem Polynomial.support_trinomial' {R : Type u} [Semiring R] (k : ℕ) (m : ℕ) (n : ℕ) (x : R) (y : R) (z : R) : (Polynomial.C x * Polynomial.X ^ k + Polynomial.C y * Polynomial.X ^ m + Polynomial.C z * Polynomial.X ^ n).support ⊆ {k, m, n}

theorem Polynomial.X_pow_eq_monomial {R : Type u} [Semiring R] (n : ℕ) : Polynomial.X ^ n = (Polynomial.monomial n) 1

@[simp]

theorem Polynomial.toFinsupp_X_pow {R : Type u} [Semiring R] (n : ℕ) : (Polynomial.X ^ n).toFinsupp = Finsupp.single n 1

theorem Polynomial.smul_X_eq_monomial {R : Type u} {a : R} [Semiring R] {n : ℕ} : a • Polynomial.X ^ n = (Polynomial.monomial n) a

theorem Polynomial.support_X_pow {R : Type u} [Semiring R] (H : ¬1 = 0) (n : ℕ) : (Polynomial.X ^ n).support = {n}

theorem Polynomial.support_X_empty {R : Type u} [Semiring R] (H : 1 = 0) : Polynomial.X.support = ∅

theorem Polynomial.support_X {R : Type u} [Semiring R] (H : ¬1 = 0) : Polynomial.X.support = {1}

theorem Polynomial.monomial_left_inj {R : Type u} [Semiring R] {a : R} (ha : a ≠ 0) {i : ℕ} {j : ℕ} : (Polynomial.monomial i) a = (Polynomial.monomial j) a ↔ i = j

theorem Polynomial.binomial_eq_binomial {R : Type u} [Semiring R] {k : ℕ} {l : ℕ} {m : ℕ} {n : ℕ} {u : R} {v : R} (hu : u ≠ 0) (hv : v ≠ 0) : Polynomial.C u * Polynomial.X ^ k + Polynomial.C v * Polynomial.X ^ l = Polynomial.C u * Polynomial.X ^ m + Polynomial.C v * Polynomial.X ^ n ↔ k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u + v = 0 ∧ k = l ∧ m = n

theorem Polynomial.natCast_mul {R : Type u} [Semiring R] (n : ℕ) (p : Polynomial R) : ↑n * p = n • p

@[deprecated Polynomial.natCast_mul]

theorem Polynomial.nat_cast_mul {R : Type u} [Semiring R] (n : ℕ) (p : Polynomial R) : ↑n * p = n • p

Alias of Polynomial.natCast_mul.

def Polynomial.sum {R : Type u} [Semiring R] {S : Type u_1} [AddCommMonoid S] (p : Polynomial R) (f : ℕ → R → S) : S

Summing the values of a function applied to the coefficients of a polynomial

Equations

• p.sum f = ∑ n ∈ p.support, f n (p.coeff n)

theorem Polynomial.sum_def {R : Type u} [Semiring R] {S : Type u_1} [AddCommMonoid S] (p : Polynomial R) (f : ℕ → R → S) : p.sum f = ∑ n ∈ p.support, f n (p.coeff n)

theorem Polynomial.sum_eq_of_subset {R : Type u} [Semiring R] {S : Type u_1} [AddCommMonoid S] {p : Polynomial R} (f : ℕ → R → S) (hf : ∀ (i : ℕ), f i 0 = 0) {s : Finset ℕ} (hs : p.support ⊆ s) : p.sum f = ∑ n ∈ s, f n (p.coeff n)

theorem Polynomial.mul_eq_sum_sum {R : Type u} [Semiring R] {p : Polynomial R} {q : Polynomial R} : p * q = ∑ i ∈ p.support, q.sum fun (j : ℕ) (a : R) => (Polynomial.monomial (i + j)) (p.coeff i * a)

Expressing the product of two polynomials as a double sum.

@[simp]

theorem Polynomial.sum_zero_index {R : Type u} [Semiring R] {S : Type u_1} [AddCommMonoid S] (f : ℕ → R → S) : Polynomial.sum 0 f = 0

@[simp]

theorem Polynomial.sum_monomial_index {R : Type u} [Semiring R] {S : Type u_1} [AddCommMonoid S] {n : ℕ} (a : R) (f : ℕ → R → S) (hf : f n 0 = 0) : ((Polynomial.monomial n) a).sum f = f n a

@[simp]

theorem Polynomial.sum_C_index {R : Type u} [Semiring R] {a : R} {β : Type u_1} [AddCommMonoid β] {f : ℕ → R → β} (h : f 0 0 = 0) : (Polynomial.C a).sum f = f 0 a

@[simp]

theorem Polynomial.sum_X_index {R : Type u} [Semiring R] {S : Type u_1} [AddCommMonoid S] {f : ℕ → R → S} (hf : f 1 0 = 0) : Polynomial.X.sum f = f 1 1

theorem Polynomial.sum_add_index {R : Type u} [Semiring R] {S : Type u_1} [AddCommMonoid S] (p : Polynomial R) (q : Polynomial R) (f : ℕ → R → S) (hf : ∀ (i : ℕ), f i 0 = 0) (h_add : ∀ (a : ℕ) (b₁ b₂ : R), f a (b₁ + b₂) = f a b₁ + f a b₂) : (p + q).sum f = p.sum f + q.sum f

theorem Polynomial.sum_add' {R : Type u} [Semiring R] {S : Type u_1} [AddCommMonoid S] (p : Polynomial R) (f : ℕ → R → S) (g : ℕ → R → S) : p.sum (f + g) = p.sum f + p.sum g

theorem Polynomial.sum_add {R : Type u} [Semiring R] {S : Type u_1} [AddCommMonoid S] (p : Polynomial R) (f : ℕ → R → S) (g : ℕ → R → S) : (p.sum fun (n : ℕ) (x : R) => f n x + g n x) = p.sum f + p.sum g

theorem Polynomial.sum_smul_index {R : Type u} [Semiring R] {S : Type u_1} [AddCommMonoid S] (p : Polynomial R) (b : R) (f : ℕ → R → S) (hf : ∀ (i : ℕ), f i 0 = 0) : (b • p).sum f = p.sum fun (n : ℕ) (a : R) => f n (b * a)

@[simp]

theorem Polynomial.sum_monomial_eq {R : Type u} [Semiring R] (p : Polynomial R) : (p.sum fun (n : ℕ) (a : R) => (Polynomial.monomial n) a) = p

theorem Polynomial.sum_C_mul_X_pow_eq {R : Type u} [Semiring R] (p : Polynomial R) : (p.sum fun (n : ℕ) (a : R) => Polynomial.C a * Polynomial.X ^ n) = p

@[irreducible]

def Polynomial.erase {R : Type u_1} [Semiring R] (n : ℕ) : Polynomial R → Polynomial R

erase p n is the polynomial p in which the X^n term has been erased.

Equations

• @Polynomial.erase = Polynomial.wrapped✝.1

theorem Polynomial.erase_def {R : Type u_1} [Semiring R] (n : ℕ) : ∀ (x : Polynomial R), Polynomial.erase n x = match x with | { toFinsupp := p } => { toFinsupp := Finsupp.erase n p }

@[simp]

theorem Polynomial.toFinsupp_erase {R : Type u} [Semiring R] (p : Polynomial R) (n : ℕ) : (Polynomial.erase n p).toFinsupp = Finsupp.erase n p.toFinsupp

@[simp]

theorem Polynomial.ofFinsupp_erase {R : Type u} [Semiring R] (p : AddMonoidAlgebra R ℕ) (n : ℕ) : { toFinsupp := Finsupp.erase n p } = Polynomial.erase n { toFinsupp := p }

@[simp]

theorem Polynomial.support_erase {R : Type u} [Semiring R] (p : Polynomial R) (n : ℕ) : (Polynomial.erase n p).support = p.support.erase n

theorem Polynomial.monomial_add_erase {R : Type u} [Semiring R] (p : Polynomial R) (n : ℕ) : (Polynomial.monomial n) (p.coeff n) + Polynomial.erase n p = p

theorem Polynomial.coeff_erase {R : Type u} [Semiring R] (p : Polynomial R) (n : ℕ) (i : ℕ) : (Polynomial.erase n p).coeff i = if i = n then 0 else p.coeff i

@[simp]

theorem Polynomial.erase_zero {R : Type u} [Semiring R] (n : ℕ) : Polynomial.erase n 0 = 0

@[simp]

theorem Polynomial.erase_monomial {R : Type u} [Semiring R] {n : ℕ} {a : R} : Polynomial.erase n ((Polynomial.monomial n) a) = 0

@[simp]

theorem Polynomial.erase_same {R : Type u} [Semiring R] (p : Polynomial R) (n : ℕ) : (Polynomial.erase n p).coeff n = 0

@[simp]

theorem Polynomial.erase_ne {R : Type u} [Semiring R] (p : Polynomial R) (n : ℕ) (i : ℕ) (h : i ≠ n) : (Polynomial.erase n p).coeff i = p.coeff i

def Polynomial.update {R : Type u} [Semiring R] (p : Polynomial R) (n : ℕ) (a : R) : Polynomial R

Replace the coefficient of a p : R[X] at a given degree n : ℕ by a given value a : R. If a = 0, this is equal to p.erase n If p.natDegree < n and a ≠ 0, this increases the degree to n.

Equations

• p.update n a = { toFinsupp := Finsupp.update p.toFinsupp n a }

theorem Polynomial.coeff_update {R : Type u} [Semiring R] (p : Polynomial R) (n : ℕ) (a : R) : (p.update n a).coeff = Function.update p.coeff n a

theorem Polynomial.coeff_update_apply {R : Type u} [Semiring R] (p : Polynomial R) (n : ℕ) (a : R) (i : ℕ) : (p.update n a).coeff i = if i = n then a else p.coeff i

@[simp]

theorem Polynomial.coeff_update_same {R : Type u} [Semiring R] (p : Polynomial R) (n : ℕ) (a : R) : (p.update n a).coeff n = a

theorem Polynomial.coeff_update_ne {R : Type u} [Semiring R] (p : Polynomial R) {n : ℕ} (a : R) {i : ℕ} (h : i ≠ n) : (p.update n a).coeff i = p.coeff i

@[simp]

theorem Polynomial.update_zero_eq_erase {R : Type u} [Semiring R] (p : Polynomial R) (n : ℕ) : p.update n 0 = Polynomial.erase n p

theorem Polynomial.support_update {R : Type u} [Semiring R] (p : Polynomial R) (n : ℕ) (a : R) [Decidable (a = 0)] : (p.update n a).support = if a = 0 then p.support.erase n else insert n p.support

theorem Polynomial.support_update_zero {R : Type u} [Semiring R] (p : Polynomial R) (n : ℕ) : (p.update n 0).support = p.support.erase n

theorem Polynomial.support_update_ne_zero {R : Type u} [Semiring R] (p : Polynomial R) (n : ℕ) {a : R} (ha : a ≠ 0) : (p.update n a).support = insert n p.support

instance Polynomial.commSemiring {R : Type u} [CommSemiring R] : CommSemiring (Polynomial R)

Equations

• Polynomial.commSemiring = let __src := Function.Injective.commSemigroup Polynomial.toFinsupp ⋯ ⋯; CommSemiring.mk ⋯

instance Polynomial.instIntCast {R : Type u} [Ring R] : IntCast (Polynomial R)

Equations

• Polynomial.instIntCast = { intCast := fun (n : ℤ) => { toFinsupp := ↑n } }

instance Polynomial.ring {R : Type u} [Ring R] : Ring (Polynomial R)

Equations

• Polynomial.ring = let __src := Function.Injective.ring Polynomial.toFinsupp ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯; Ring.mk ⋯ (fun (x : ℤ) (x_1 : Polynomial R) => x • x_1) ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

@[simp]

theorem Polynomial.coeff_neg {R : Type u} [Ring R] (p : Polynomial R) (n : ℕ) : (-p).coeff n = -p.coeff n

@[simp]

theorem Polynomial.coeff_sub {R : Type u} [Ring R] (p : Polynomial R) (q : Polynomial R) (n : ℕ) : (p - q).coeff n = p.coeff n - q.coeff n

theorem Polynomial.monomial_neg {R : Type u} [Ring R] (n : ℕ) (a : R) : (Polynomial.monomial n) (-a) = -(Polynomial.monomial n) a

theorem Polynomial.monomial_sub {R : Type u} {a : R} {b : R} [Ring R] (n : ℕ) : (Polynomial.monomial n) (a - b) = (Polynomial.monomial n) a - (Polynomial.monomial n) b

@[simp]

theorem Polynomial.support_neg {R : Type u} [Ring R] {p : Polynomial R} : (-p).support = p.support

theorem Polynomial.C_eq_intCast {R : Type u} [Ring R] (n : ℤ) : Polynomial.C ↑n = ↑n

@[deprecated Polynomial.C_eq_intCast]

theorem Polynomial.C_eq_int_cast {R : Type u} [Ring R] (n : ℤ) : Polynomial.C ↑n = ↑n

Alias of Polynomial.C_eq_intCast.

theorem Polynomial.C_neg {R : Type u} {a : R} [Ring R] : Polynomial.C (-a) = -Polynomial.C a

theorem Polynomial.C_sub {R : Type u} {a : R} {b : R} [Ring R] : Polynomial.C (a - b) = Polynomial.C a - Polynomial.C b

instance Polynomial.commRing {R : Type u} [CommRing R] : CommRing (Polynomial R)

Equations

• Polynomial.commRing = CommRing.mk ⋯

instance Polynomial.nontrivial {R : Type u} [Semiring R] [Nontrivial R] : Nontrivial (Polynomial R)

Equations

• ⋯ = ⋯

@[simp]

theorem Polynomial.X_ne_zero {R : Type u} [Semiring R] [Nontrivial R] : Polynomial.X ≠ 0

theorem Polynomial.nnqsmul_eq_C_mul {R : Type u} [DivisionSemiring R] (q : ℚ≥0) (f : Polynomial R) : q • f = Polynomial.C ↑q * f

theorem Polynomial.qsmul_eq_C_mul {R : Type u} [DivisionRing R] (a : ℚ) (f : Polynomial R) : a • f = Polynomial.C ↑a * f

@[simp]

theorem Polynomial.nontrivial_iff {R : Type u} [Semiring R] : Nontrivial (Polynomial R) ↔ Nontrivial R

instance Polynomial.repr {R : Type u} [Semiring R] [Repr R] [DecidableEq R] : Repr (Polynomial R)

Equations

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