Expand a polynomial by a factor of p, so ∑ aₙ xⁿ becomes ∑ aₙ xⁿᵖ.

Main definitions

• Polynomial.expand R p f: expand the polynomial f with coefficients in a commutative semiring R by a factor of p, so expand R p (∑ aₙ xⁿ) is ∑ aₙ xⁿᵖ.
• Polynomial.contract p f: the opposite of expand, so it sends ∑ aₙ xⁿᵖ to ∑ aₙ xⁿ.

noncomputable def Polynomial.expand (R : Type u) [CommSemiring R] (p : ℕ) : Polynomial R →ₐ[R] Polynomial R

Expand the polynomial by a factor of p, so ∑ aₙ xⁿ becomes ∑ aₙ xⁿᵖ.

Equations

• Polynomial.expand R p = { toRingHom := Polynomial.eval₂RingHom Polynomial.C (Polynomial.X ^ p), commutes' := ⋯ }

theorem Polynomial.coe_expand (R : Type u) [CommSemiring R] (p : ℕ) : ⇑(expand R p) = eval₂ C (X ^ p)

theorem Polynomial.expand_eq_comp_X_pow {R : Type u} [CommSemiring R] (p : ℕ) {f : Polynomial R} : (expand R p) f = f.comp (X ^ p)

theorem Polynomial.expand_eq_sum {R : Type u} [CommSemiring R] (p : ℕ) {f : Polynomial R} : (expand R p) f = f.sum fun (e : ℕ) (a : R) => C a * (X ^ p) ^ e

@[simp]

theorem Polynomial.expand_C {R : Type u} [CommSemiring R] (p : ℕ) (r : R) : (expand R p) (C r) = C r

@[simp]

theorem Polynomial.expand_X {R : Type u} [CommSemiring R] (p : ℕ) : (expand R p) X = X ^ p

@[simp]

theorem Polynomial.expand_monomial {R : Type u} [CommSemiring R] (p q : ℕ) (r : R) : (expand R p) ((monomial q) r) = (monomial (q * p)) r

theorem Polynomial.expand_expand {R : Type u} [CommSemiring R] (p q : ℕ) (f : Polynomial R) : (expand R p) ((expand R q) f) = (expand R (p * q)) f

theorem Polynomial.expand_mul {R : Type u} [CommSemiring R] (p q : ℕ) (f : Polynomial R) : (expand R (p * q)) f = (expand R p) ((expand R q) f)

@[simp]

theorem Polynomial.expand_zero {R : Type u} [CommSemiring R] (f : Polynomial R) : (expand R 0) f = C (eval 1 f)

@[simp]

theorem Polynomial.expand_one {R : Type u} [CommSemiring R] (f : Polynomial R) : (expand R 1) f = f

theorem Polynomial.expand_pow {R : Type u} [CommSemiring R] (p q : ℕ) (f : Polynomial R) : (expand R (p ^ q)) f = (⇑(expand R p))^[q] f

theorem Polynomial.derivative_expand {R : Type u} [CommSemiring R] (p : ℕ) (f : Polynomial R) : derivative ((expand R p) f) = (expand R p) (derivative f) * (↑p * X ^ (p - 1))

theorem Polynomial.coeff_expand {R : Type u} [CommSemiring R] {p : ℕ} (hp : 0 < p) (f : Polynomial R) (n : ℕ) : ((expand R p) f).coeff n = if p ∣ n then f.coeff (n / p) else 0

@[simp]

theorem Polynomial.coeff_expand_mul {R : Type u} [CommSemiring R] {p : ℕ} (hp : 0 < p) (f : Polynomial R) (n : ℕ) : ((expand R p) f).coeff (n * p) = f.coeff n

@[simp]

theorem Polynomial.coeff_expand_mul' {R : Type u} [CommSemiring R] {p : ℕ} (hp : 0 < p) (f : Polynomial R) (n : ℕ) : ((expand R p) f).coeff (p * n) = f.coeff n

theorem Polynomial.expand_injective {R : Type u} [CommSemiring R] {n : ℕ} (hn : 0 < n) : Function.Injective ⇑(expand R n)

Expansion is injective.

theorem Polynomial.expand_inj {R : Type u} [CommSemiring R] {p : ℕ} (hp : 0 < p) {f g : Polynomial R} : (expand R p) f = (expand R p) g ↔ f = g

theorem Polynomial.expand_eq_zero {R : Type u} [CommSemiring R] {p : ℕ} (hp : 0 < p) {f : Polynomial R} : (expand R p) f = 0 ↔ f = 0

theorem Polynomial.expand_ne_zero {R : Type u} [CommSemiring R] {p : ℕ} (hp : 0 < p) {f : Polynomial R} : (expand R p) f ≠ 0 ↔ f ≠ 0

theorem Polynomial.expand_eq_C {R : Type u} [CommSemiring R] {p : ℕ} (hp : 0 < p) {f : Polynomial R} {r : R} : (expand R p) f = C r ↔ f = C r

theorem Polynomial.natDegree_expand {R : Type u} [CommSemiring R] (p : ℕ) (f : Polynomial R) : ((expand R p) f).natDegree = f.natDegree * p

theorem Polynomial.leadingCoeff_expand {R : Type u} [CommSemiring R] {p : ℕ} {f : Polynomial R} (hp : 0 < p) : ((expand R p) f).leadingCoeff = f.leadingCoeff

theorem Polynomial.monic_expand_iff {R : Type u} [CommSemiring R] {p : ℕ} {f : Polynomial R} (hp : 0 < p) : ((expand R p) f).Monic ↔ f.Monic

theorem Polynomial.Monic.expand {R : Type u} [CommSemiring R] {p : ℕ} {f : Polynomial R} (hp : 0 < p) : f.Monic → ((Polynomial.expand R p) f).Monic

Alias of the reverse direction of Polynomial.monic_expand_iff.

theorem Polynomial.map_expand {R : Type u} [CommSemiring R] {S : Type v} [CommSemiring S] {p : ℕ} {f : R →+* S} {q : Polynomial R} : map f ((expand R p) q) = (expand S p) (map f q)

@[simp]

theorem Polynomial.expand_eval {R : Type u} [CommSemiring R] (p : ℕ) (P : Polynomial R) (r : R) : eval r ((expand R p) P) = eval (r ^ p) P

@[simp]

theorem Polynomial.expand_aeval {R : Type u} [CommSemiring R] {A : Type u_1} [Semiring A] [Algebra R A] (p : ℕ) (P : Polynomial R) (r : A) : (aeval r) ((expand R p) P) = (aeval (r ^ p)) P

noncomputable def Polynomial.contract {R : Type u} [CommSemiring R] (p : ℕ) (f : Polynomial R) : Polynomial R

The opposite of expand: sends ∑ aₙ xⁿᵖ to ∑ aₙ xⁿ.

Equations

• Polynomial.contract p f = ∑ n ∈ Finset.range (f.natDegree + 1), (Polynomial.monomial n) (f.coeff (n * p))

theorem Polynomial.coeff_contract {R : Type u} [CommSemiring R] {p : ℕ} (hp : p ≠ 0) (f : Polynomial R) (n : ℕ) : (contract p f).coeff n = f.coeff (n * p)

theorem Polynomial.map_contract {R : Type u} [CommSemiring R] {S : Type v} [CommSemiring S] {p : ℕ} (hp : p ≠ 0) {f : R →+* S} {q : Polynomial R} : map f (contract p q) = contract p (map f q)

theorem Polynomial.contract_expand {R : Type u} [CommSemiring R] (p : ℕ) {f : Polynomial R} (hp : p ≠ 0) : contract p ((expand R p) f) = f

theorem Polynomial.contract_one {R : Type u} [CommSemiring R] {f : Polynomial R} : contract 1 f = f

@[simp]

theorem Polynomial.contract_C {R : Type u} [CommSemiring R] (p : ℕ) (r : R) : contract p (C r) = C r

theorem Polynomial.contract_add {R : Type u} [CommSemiring R] {p : ℕ} (hp : p ≠ 0) (f g : Polynomial R) : contract p (f + g) = contract p f + contract p g

theorem Polynomial.contract_mul_expand {R : Type u} [CommSemiring R] {p : ℕ} (hp : p ≠ 0) (f g : Polynomial R) : contract p (f * (expand R p) g) = contract p f * g

@[simp]

theorem Polynomial.isCoprime_expand {R : Type u} [CommSemiring R] {f g : Polynomial R} {p : ℕ} (hp : p ≠ 0) : IsCoprime ((expand R p) f) ((expand R p) g) ↔ IsCoprime f g

theorem Polynomial.expand_contract {R : Type u} [CommSemiring R] (p : ℕ) [CharP R p] [NoZeroDivisors R] {f : Polynomial R} (hf : derivative f = 0) (hp : p ≠ 0) : (expand R p) (contract p f) = f

theorem Polynomial.expand_contract' {R : Type u} [CommSemiring R] (p : ℕ) [ExpChar R p] [NoZeroDivisors R] {f : Polynomial R} (hf : derivative f = 0) : (expand R p) (contract p f) = f

theorem Polynomial.expand_char {R : Type u} [CommSemiring R] (p : ℕ) [ExpChar R p] (f : Polynomial R) : map (frobenius R p) ((expand R p) f) = f ^ p

theorem Polynomial.map_expand_pow_char {R : Type u} [CommSemiring R] (p : ℕ) [ExpChar R p] (f : Polynomial R) (n : ℕ) : map (frobenius R p ^ n) ((expand R (p ^ n)) f) = f ^ p ^ n

theorem Polynomial.rootMultiplicity_expand_pow {R : Type u} [CommRing R] {p n : ℕ} [ExpChar R p] {f : Polynomial R} {r : R} : rootMultiplicity r ((expand R (p ^ n)) f) = p ^ n * rootMultiplicity (r ^ p ^ n) f

theorem Polynomial.rootMultiplicity_expand {R : Type u} [CommRing R] {p : ℕ} [ExpChar R p] {f : Polynomial R} {r : R} : rootMultiplicity r ((expand R p) f) = p * rootMultiplicity (r ^ p) f

theorem Polynomial.isLocalHom_expand (R : Type u) [CommRing R] [IsDomain R] {p : ℕ} (hp : 0 < p) : IsLocalHom (expand R p)

@[deprecated Polynomial.isLocalHom_expand (since := "2024-10-10")]

theorem Polynomial.isLocalRingHom_expand (R : Type u) [CommRing R] [IsDomain R] {p : ℕ} (hp : 0 < p) : IsLocalHom (expand R p)

Alias of Polynomial.isLocalHom_expand.

theorem Polynomial.of_irreducible_expand {R : Type u} [CommRing R] [IsDomain R] {p : ℕ} (hp : p ≠ 0) {f : Polynomial R} (hf : Irreducible ((expand R p) f)) : Irreducible f

theorem Polynomial.of_irreducible_expand_pow {R : Type u} [CommRing R] [IsDomain R] {p : ℕ} (hp : p ≠ 0) {f : Polynomial R} {n : ℕ} : Irreducible ((expand R (p ^ n)) f) → Irreducible f