Erase the leading term of a univariate polynomial

Definition

• eraseLead f: the polynomial f - leading term of f

eraseLead serves as reduction step in an induction, shaving off one monomial from a polynomial. The definition is set up so that it does not mention subtraction in the definition, and thus works for polynomials over semirings as well as rings.

def Polynomial.eraseLead {R : Type u_1} [Semiring R] (f : Polynomial R) : Polynomial R

eraseLead f for a polynomial f is the polynomial obtained by subtracting from f the leading term of f.

Equations

• f.eraseLead = Polynomial.erase f.natDegree f

theorem Polynomial.eraseLead_support {R : Type u_1} [Semiring R] (f : Polynomial R) : f.eraseLead.support = f.support.erase f.natDegree

theorem Polynomial.eraseLead_coeff {R : Type u_1} [Semiring R] {f : Polynomial R} (i : ℕ) : f.eraseLead.coeff i = if i = f.natDegree then 0 else f.coeff i

@[simp]

theorem Polynomial.eraseLead_coeff_natDegree {R : Type u_1} [Semiring R] {f : Polynomial R} : f.eraseLead.coeff f.natDegree = 0

theorem Polynomial.eraseLead_coeff_of_ne {R : Type u_1} [Semiring R] {f : Polynomial R} (i : ℕ) (hi : i ≠ f.natDegree) : f.eraseLead.coeff i = f.coeff i

@[simp]

theorem Polynomial.eraseLead_zero {R : Type u_1} [Semiring R] : eraseLead 0 = 0

@[simp]

theorem Polynomial.eraseLead_add_monomial_natDegree_leadingCoeff {R : Type u_1} [Semiring R] (f : Polynomial R) : f.eraseLead + (monomial f.natDegree) f.leadingCoeff = f

@[simp]

theorem Polynomial.eraseLead_add_C_mul_X_pow {R : Type u_1} [Semiring R] (f : Polynomial R) : f.eraseLead + C f.leadingCoeff * X ^ f.natDegree = f

@[simp]

theorem Polynomial.self_sub_monomial_natDegree_leadingCoeff {R : Type u_2} [Ring R] (f : Polynomial R) : f - (monomial f.natDegree) f.leadingCoeff = f.eraseLead

@[simp]

theorem Polynomial.self_sub_C_mul_X_pow {R : Type u_2} [Ring R] (f : Polynomial R) : f - C f.leadingCoeff * X ^ f.natDegree = f.eraseLead

theorem Polynomial.eraseLead_ne_zero {R : Type u_1} [Semiring R] {f : Polynomial R} (f0 : 2 ≤ f.support.card) : f.eraseLead ≠ 0

theorem Polynomial.lt_natDegree_of_mem_eraseLead_support {R : Type u_1} [Semiring R] {f : Polynomial R} {a : ℕ} (h : a ∈ f.eraseLead.support) : a < f.natDegree

theorem Polynomial.ne_natDegree_of_mem_eraseLead_support {R : Type u_1} [Semiring R] {f : Polynomial R} {a : ℕ} (h : a ∈ f.eraseLead.support) : a ≠ f.natDegree

theorem Polynomial.natDegree_notMem_eraseLead_support {R : Type u_1} [Semiring R] {f : Polynomial R} : f.natDegree ∉ f.eraseLead.support

@[deprecated Polynomial.natDegree_notMem_eraseLead_support (since := "2025-05-23")]

theorem Polynomial.natDegree_not_mem_eraseLead_support {R : Type u_1} [Semiring R] {f : Polynomial R} : f.natDegree ∉ f.eraseLead.support

Alias of Polynomial.natDegree_notMem_eraseLead_support.

theorem Polynomial.eraseLead_support_card_lt {R : Type u_1} [Semiring R] {f : Polynomial R} (h : f ≠ 0) : f.eraseLead.support.card < f.support.card

theorem Polynomial.card_support_eraseLead_add_one {R : Type u_1} [Semiring R] {f : Polynomial R} (h : f ≠ 0) : f.eraseLead.support.card + 1 = f.support.card

@[simp]

theorem Polynomial.card_support_eraseLead {R : Type u_1} [Semiring R] {f : Polynomial R} : f.eraseLead.support.card = f.support.card - 1

theorem Polynomial.card_support_eraseLead' {R : Type u_1} [Semiring R] {f : Polynomial R} {c : ℕ} (fc : f.support.card = c + 1) : f.eraseLead.support.card = c

theorem Polynomial.card_support_eq_one_of_eraseLead_eq_zero {R : Type u_1} [Semiring R] {f : Polynomial R} (h₀ : f ≠ 0) (h₁ : f.eraseLead = 0) : f.support.card = 1

theorem Polynomial.card_support_le_one_of_eraseLead_eq_zero {R : Type u_1} [Semiring R] {f : Polynomial R} (h : f.eraseLead = 0) : f.support.card ≤ 1

@[simp]

theorem Polynomial.eraseLead_monomial {R : Type u_1} [Semiring R] (i : ℕ) (r : R) : ((monomial i) r).eraseLead = 0

@[simp]

theorem Polynomial.eraseLead_C {R : Type u_1} [Semiring R] (r : R) : (C r).eraseLead = 0

@[simp]

theorem Polynomial.eraseLead_X {R : Type u_1} [Semiring R] : X.eraseLead = 0

@[simp]

theorem Polynomial.eraseLead_X_pow {R : Type u_1} [Semiring R] (n : ℕ) : (X ^ n).eraseLead = 0

@[simp]

theorem Polynomial.eraseLead_C_mul_X_pow {R : Type u_1} [Semiring R] (r : R) (n : ℕ) : (C r * X ^ n).eraseLead = 0

@[simp]

theorem Polynomial.eraseLead_C_mul_X {R : Type u_1} [Semiring R] (r : R) : (C r * X).eraseLead = 0

theorem Polynomial.eraseLead_add_of_degree_lt_left {R : Type u_1} [Semiring R] {p q : Polynomial R} (pq : q.degree < p.degree) : (p + q).eraseLead = p.eraseLead + q

theorem Polynomial.eraseLead_add_of_natDegree_lt_left {R : Type u_1} [Semiring R] {p q : Polynomial R} (pq : q.natDegree < p.natDegree) : (p + q).eraseLead = p.eraseLead + q

theorem Polynomial.eraseLead_add_of_degree_lt_right {R : Type u_1} [Semiring R] {p q : Polynomial R} (pq : p.degree < q.degree) : (p + q).eraseLead = p + q.eraseLead

theorem Polynomial.eraseLead_add_of_natDegree_lt_right {R : Type u_1} [Semiring R] {p q : Polynomial R} (pq : p.natDegree < q.natDegree) : (p + q).eraseLead = p + q.eraseLead

theorem Polynomial.eraseLead_degree_le {R : Type u_1} [Semiring R] {f : Polynomial R} : f.eraseLead.degree ≤ f.degree

theorem Polynomial.degree_eraseLead_lt {R : Type u_1} [Semiring R] {f : Polynomial R} (hf : f ≠ 0) : f.eraseLead.degree < f.degree

theorem Polynomial.eraseLead_natDegree_le_aux {R : Type u_1} [Semiring R] {f : Polynomial R} : f.eraseLead.natDegree ≤ f.natDegree

theorem Polynomial.eraseLead_natDegree_lt {R : Type u_1} [Semiring R] {f : Polynomial R} (f0 : 2 ≤ f.support.card) : f.eraseLead.natDegree < f.natDegree

theorem Polynomial.natDegree_pos_of_eraseLead_ne_zero {R : Type u_1} [Semiring R] {f : Polynomial R} (h : f.eraseLead ≠ 0) : 0 < f.natDegree

theorem Polynomial.eraseLead_natDegree_lt_or_eraseLead_eq_zero {R : Type u_1} [Semiring R] (f : Polynomial R) : f.eraseLead.natDegree < f.natDegree ∨ f.eraseLead = 0

theorem Polynomial.eraseLead_natDegree_le {R : Type u_1} [Semiring R] (f : Polynomial R) : f.eraseLead.natDegree ≤ f.natDegree - 1

theorem Polynomial.natDegree_eraseLead {R : Type u_1} [Semiring R] {f : Polynomial R} (h : f.nextCoeff ≠ 0) : f.eraseLead.natDegree = f.natDegree - 1

theorem Polynomial.natDegree_eraseLead_add_one {R : Type u_1} [Semiring R] {f : Polynomial R} (h : f.nextCoeff ≠ 0) : f.eraseLead.natDegree + 1 = f.natDegree

theorem Polynomial.natDegree_eraseLead_le_of_nextCoeff_eq_zero {R : Type u_1} [Semiring R] {f : Polynomial R} (h : f.nextCoeff = 0) : f.eraseLead.natDegree ≤ f.natDegree - 2

theorem Polynomial.two_le_natDegree_of_nextCoeff_eraseLead {R : Type u_1} [Semiring R] {f : Polynomial R} (hlead : f.eraseLead ≠ 0) (hnext : f.nextCoeff = 0) : 2 ≤ f.natDegree

theorem Polynomial.leadingCoeff_eraseLead_eq_nextCoeff {R : Type u_1} [Semiring R] {f : Polynomial R} (h : f.nextCoeff ≠ 0) : f.eraseLead.leadingCoeff = f.nextCoeff

theorem Polynomial.nextCoeff_eq_zero_of_eraseLead_eq_zero {R : Type u_1} [Semiring R] {f : Polynomial R} (h : f.eraseLead = 0) : f.nextCoeff = 0

theorem Polynomial.induction_with_natDegree_le {R : Type u_1} [Semiring R] (P : Polynomial R → Prop) (N : ℕ) (P_0 : P 0) (P_C_mul_pow : ∀ (n : ℕ) (r : R), r ≠ 0 → n ≤ N → P (C r * X ^ n)) (P_C_add : ∀ (f g : Polynomial R), f.natDegree < g.natDegree → g.natDegree ≤ N → P f → P g → P (f + g)) (f : Polynomial R) : f.natDegree ≤ N → P f

An induction lemma for polynomials. It takes a natural number N as a parameter, that is required to be at least as big as the nat_degree of the polynomial. This is useful to prove results where you want to change each term in a polynomial to something else depending on the nat_degree of the polynomial itself and not on the specific nat_degree of each term.

theorem Polynomial.mono_map_natDegree_eq {R : Type u_1} [Semiring R] {S : Type u_2} {F : Type u_3} [Semiring S] [FunLike F (Polynomial R) (Polynomial S)] [AddMonoidHomClass F (Polynomial R) (Polynomial S)] {φ : F} {p : Polynomial R} (k : ℕ) (fu : ℕ → ℕ) (fu0 : ∀ {n : ℕ}, n ≤ k → fu n = 0) (fc : ∀ {n m : ℕ}, k ≤ n → n < m → fu n < fu m) (φ_k : ∀ {f : Polynomial R}, f.natDegree < k → φ f = 0) (φ_mon_nat : ∀ (n : ℕ) (c : R), c ≠ 0 → (φ ((monomial n) c)).natDegree = fu n) : (φ p).natDegree = fu p.natDegree

Let φ : R[x] → S[x] be an additive map, k : ℕ a bound, and fu : ℕ → ℕ a "sufficiently monotone" map. Assume also that

• φ maps to 0 all monomials of degree less than k,
• φ maps each monomial m in R[x] to a polynomial φ m of degree fu (deg m). Then, φ maps each polynomial p in R[x] to a polynomial of degree fu (deg p).

theorem Polynomial.map_natDegree_eq_sub {R : Type u_1} [Semiring R] {S : Type u_2} {F : Type u_3} [Semiring S] [FunLike F (Polynomial R) (Polynomial S)] [AddMonoidHomClass F (Polynomial R) (Polynomial S)] {φ : F} {p : Polynomial R} {k : ℕ} (φ_k : ∀ (f : Polynomial R), f.natDegree < k → φ f = 0) (φ_mon : ∀ (n : ℕ) (c : R), c ≠ 0 → (φ ((monomial n) c)).natDegree = n - k) : (φ p).natDegree = p.natDegree - k

theorem Polynomial.map_natDegree_eq_natDegree {R : Type u_1} [Semiring R] {S : Type u_2} {F : Type u_3} [Semiring S] [FunLike F (Polynomial R) (Polynomial S)] [AddMonoidHomClass F (Polynomial R) (Polynomial S)] {φ : F} (p : Polynomial R) (φ_mon_nat : ∀ (n : ℕ) (c : R), c ≠ 0 → (φ ((monomial n) c)).natDegree = n) : (φ p).natDegree = p.natDegree

theorem Polynomial.card_support_eq' {R : Type u_1} [Semiring R] {n : ℕ} (k : Fin n → ℕ) (x : Fin n → R) (hk : Function.Injective k) (hx : ∀ (i : Fin n), x i ≠ 0) : (∑ i : Fin n, C (x i) * X ^ k i).support.card = n

theorem Polynomial.card_support_eq {R : Type u_1} [Semiring R] {f : Polynomial R} {n : ℕ} : f.support.card = n ↔ ∃ (k : Fin n → ℕ) (x : Fin n → R) (_ : StrictMono k) (_ : ∀ (i : Fin n), x i ≠ 0), f = ∑ i : Fin n, C (x i) * X ^ k i

theorem Polynomial.card_support_eq_one {R : Type u_1} [Semiring R] {f : Polynomial R} : f.support.card = 1 ↔ ∃ (k : ℕ) (x : R) (_ : x ≠ 0), f = C x * X ^ k

theorem Polynomial.card_support_eq_two {R : Type u_1} [Semiring R] {f : Polynomial R} : f.support.card = 2 ↔ ∃ (k : ℕ) (m : ℕ) (_ : k < m) (x : R) (y : R) (_ : x ≠ 0) (_ : y ≠ 0), f = C x * X ^ k + C y * X ^ m

theorem Polynomial.card_support_eq_three {R : Type u_1} [Semiring R] {f : Polynomial R} : f.support.card = 3 ↔ ∃ (k : ℕ) (m : ℕ) (n : ℕ) (_ : k < m) (_ : m < n) (x : R) (y : R) (z : R) (_ : x ≠ 0) (_ : y ≠ 0) (_ : z ≠ 0), f = C x * X ^ k + C y * X ^ m + C z * X ^ n