Degree of univariate polynomials

Main definitions

• Polynomial.degree: the degree of a polynomial, where 0 has degree ⊥
• Polynomial.natDegree: the degree of a polynomial, where 0 has degree 0
• Polynomial.leadingCoeff: the leading coefficient of a polynomial
• Polynomial.Monic: a polynomial is monic if its leading coefficient is 1
• Polynomial.nextCoeff: the next coefficient after the leading coefficient

Main results

• Polynomial.degree_eq_natDegree: the degree and natDegree coincide for nonzero polynomials

def Polynomial.degree {R : Type u} [Semiring R] (p : Polynomial R) : WithBot ℕ

degree p is the degree of the polynomial p, i.e. the largest X-exponent in p. degree p = some n when p ≠ 0 and n is the highest power of X that appears in p, otherwise degree 0 = ⊥.

Equations

• p.degree = p.support.max

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

natDegree p forces degree p to ℕ, by defining natDegree 0 = 0.

Equations

• p.natDegree = WithBot.unbotD 0 p.degree

def Polynomial.leadingCoeff {R : Type u} [Semiring R] (p : Polynomial R) : R

leadingCoeff p gives the coefficient of the highest power of X in p.

Equations

• p.leadingCoeff = p.coeff p.natDegree

def Polynomial.Monic {R : Type u} [Semiring R] (p : Polynomial R) : Prop

a polynomial is Monic if its leading coefficient is 1

Equations

• p.Monic = (p.leadingCoeff = 1)

theorem Polynomial.Monic.def {R : Type u} [Semiring R] {p : Polynomial R} : p.Monic ↔ p.leadingCoeff = 1

@[instance_reducible]

instance Polynomial.Monic.decidable {R : Type u} [Semiring R] {p : Polynomial R} [DecidableEq R] : Decidable p.Monic

Equations

• Polynomial.Monic.decidable = Polynomial.Monic.decidable._aux_1

@[simp]

theorem Polynomial.Monic.leadingCoeff {R : Type u} [Semiring R] {p : Polynomial R} (hp : p.Monic) : p.leadingCoeff = 1

theorem Polynomial.Monic.coeff_natDegree {R : Type u} [Semiring R] {p : Polynomial R} (hp : p.Monic) : p.coeff p.natDegree = 1

@[simp]

theorem Polynomial.degree_zero {R : Type u} [Semiring R] : degree 0 = ⊥

@[simp]

theorem Polynomial.natDegree_zero {R : Type u} [Semiring R] : natDegree 0 = 0

@[simp]

theorem Polynomial.coeff_natDegree {R : Type u} [Semiring R] {p : Polynomial R} : p.coeff p.natDegree = p.leadingCoeff

@[simp]

theorem Polynomial.degree_eq_bot {R : Type u} [Semiring R] {p : Polynomial R} : p.degree = ⊥ ↔ p = 0

theorem Polynomial.degree_ne_bot {R : Type u} [Semiring R] {p : Polynomial R} : p.degree ≠ ⊥ ↔ p ≠ 0

theorem Polynomial.degree_eq_natDegree {R : Type u} [Semiring R] {p : Polynomial R} (hp : p ≠ 0) : p.degree = ↑p.natDegree

theorem Polynomial.degree_eq_iff_natDegree_eq {R : Type u} [Semiring R] {p : Polynomial R} {n : ℕ} (hp : p ≠ 0) : p.degree = ↑n ↔ p.natDegree = n

theorem Polynomial.degree_eq_iff_natDegree_eq_of_pos {R : Type u} [Semiring R] {p : Polynomial R} {n : ℕ} (hn : 0 < n) : p.degree = ↑n ↔ p.natDegree = n

theorem Polynomial.degree_eq_iff_natDegree_eq_of_neZero {R : Type u} [Semiring R] {p : Polynomial R} {n : ℕ} [NeZero n] : p.degree = ↑n ↔ p.natDegree = n

theorem Polynomial.natDegree_eq_of_degree_eq_some {R : Type u} [Semiring R] {p : Polynomial R} {n : ℕ} (h : p.degree = ↑n) : p.natDegree = n

theorem Polynomial.degree_ne_of_natDegree_ne {R : Type u} [Semiring R] {p : Polynomial R} {n : ℕ} : p.natDegree ≠ n → p.degree ≠ ↑n

@[simp]

theorem Polynomial.degree_le_natDegree {R : Type u} [Semiring R] {p : Polynomial R} : p.degree ≤ ↑p.natDegree

theorem Polynomial.natDegree_eq_of_degree_eq {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] {q : Polynomial S} (h : p.degree = q.degree) : p.natDegree = q.natDegree

theorem Polynomial.le_degree_of_ne_zero {R : Type u} {n : ℕ} [Semiring R] {p : Polynomial R} (h : p.coeff n ≠ 0) : ↑n ≤ p.degree

theorem Polynomial.degree_mono {R : Type u} {S : Type v} [Semiring R] [Semiring S] {f : Polynomial R} {g : Polynomial S} (h : f.support ⊆ g.support) : f.degree ≤ g.degree

theorem Polynomial.degree_le_degree {R : Type u} [Semiring R] {p q : Polynomial R} (h : q.coeff p.natDegree ≠ 0) : p.degree ≤ q.degree

theorem Polynomial.natDegree_le_iff_degree_le {R : Type u} [Semiring R] {p : Polynomial R} {n : ℕ} : p.natDegree ≤ n ↔ p.degree ≤ ↑n

theorem Polynomial.natDegree_lt_iff_degree_lt {R : Type u} {n : ℕ} [Semiring R] {p : Polynomial R} (hp : p ≠ 0) : p.natDegree < n ↔ p.degree < ↑n

theorem Polynomial.natDegree_le_of_degree_le {R : Type u} [Semiring R] {p : Polynomial R} {n : ℕ} : p.degree ≤ ↑n → p.natDegree ≤ n

Alias of the reverse direction of Polynomial.natDegree_le_iff_degree_le.

theorem Polynomial.degree_le_of_natDegree_le {R : Type u} [Semiring R] {p : Polynomial R} {n : ℕ} : p.natDegree ≤ n → p.degree ≤ ↑n

Alias of the forward direction of Polynomial.natDegree_le_iff_degree_le.

theorem Polynomial.natDegree_le_natDegree {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] {q : Polynomial S} (hpq : p.degree ≤ q.degree) : p.natDegree ≤ q.natDegree

@[simp]

theorem Polynomial.degree_C {R : Type u} {a : R} [Semiring R] (ha : a ≠ 0) : (C a).degree = 0

theorem Polynomial.degree_C_le {R : Type u} {a : R} [Semiring R] : (C a).degree ≤ 0

theorem Polynomial.degree_C_lt {R : Type u} {a : R} [Semiring R] : (C a).degree < 1

theorem Polynomial.degree_one_le {R : Type u} [Semiring R] : degree 1 ≤ 0

@[simp]

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

@[simp]

theorem Polynomial.natDegree_one {R : Type u} [Semiring R] : natDegree 1 = 0

@[simp]

theorem Polynomial.natDegree_natCast {R : Type u} [Semiring R] (n : ℕ) : (↑n).natDegree = 0

@[simp]

theorem Polynomial.natDegree_ofNat {R : Type u} [Semiring R] (n : ℕ) [n.AtLeastTwo] : (OfNat.ofNat n).natDegree = 0

theorem Polynomial.degree_natCast_le {R : Type u} [Semiring R] (n : ℕ) : (↑n).degree ≤ 0

@[simp]

theorem Polynomial.degree_monomial {R : Type u} {a : R} [Semiring R] (n : ℕ) (ha : a ≠ 0) : ((monomial n) a).degree = ↑n

@[simp]

theorem Polynomial.degree_C_mul_X_pow {R : Type u} {a : R} [Semiring R] (n : ℕ) (ha : a ≠ 0) : (C a * X ^ n).degree = ↑n

theorem Polynomial.degree_C_mul_X {R : Type u} {a : R} [Semiring R] (ha : a ≠ 0) : (C a * X).degree = 1

theorem Polynomial.degree_monomial_le {R : Type u} [Semiring R] (n : ℕ) (a : R) : ((monomial n) a).degree ≤ ↑n

theorem Polynomial.degree_C_mul_X_pow_le {R : Type u} [Semiring R] (n : ℕ) (a : R) : (C a * X ^ n).degree ≤ ↑n

theorem Polynomial.degree_C_mul_X_le {R : Type u} [Semiring R] (a : R) : (C a * X).degree ≤ 1

@[simp]

theorem Polynomial.natDegree_C_mul_X_pow {R : Type u} [Semiring R] (n : ℕ) (a : R) (ha : a ≠ 0) : (C a * X ^ n).natDegree = n

@[simp]

theorem Polynomial.natDegree_C_mul_X {R : Type u} [Semiring R] (a : R) (ha : a ≠ 0) : (C a * X).natDegree = 1

@[simp]

theorem Polynomial.natDegree_monomial {R : Type u} [Semiring R] [DecidableEq R] (i : ℕ) (r : R) : ((monomial i) r).natDegree = if r = 0 then 0 else i

theorem Polynomial.natDegree_monomial_le {R : Type u} [Semiring R] (a : R) {m : ℕ} : ((monomial m) a).natDegree ≤ m

theorem Polynomial.natDegree_monomial_eq {R : Type u} [Semiring R] (i : ℕ) {r : R} (r0 : r ≠ 0) : ((monomial i) r).natDegree = i

theorem Polynomial.coeff_ne_zero_of_eq_degree {R : Type u} {n : ℕ} [Semiring R] {p : Polynomial R} (hn : p.degree = ↑n) : p.coeff n ≠ 0

theorem Polynomial.degree_X_pow_le {R : Type u} [Semiring R] (n : ℕ) : (X ^ n).degree ≤ ↑n

theorem Polynomial.degree_X_le {R : Type u} [Semiring R] : X.degree ≤ 1

theorem Polynomial.natDegree_X_le {R : Type u} [Semiring R] : X.natDegree ≤ 1

theorem Polynomial.withBotSucc_degree_eq_natDegree_add_one {R : Type u} [Semiring R] {p : Polynomial R} (h : p ≠ 0) : p.degree.succ = p.natDegree + 1

@[simp]

theorem Polynomial.degree_one {R : Type u} [Semiring R] [Nontrivial R] : degree 1 = 0

@[simp]

theorem Polynomial.degree_X {R : Type u} [Semiring R] [Nontrivial R] : X.degree = 1

@[simp]

theorem Polynomial.natDegree_X {R : Type u} [Semiring R] [Nontrivial R] : X.natDegree = 1

@[simp]

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

theorem Polynomial.degree_neg_le_of_le {R : Type u} [Ring R] {a : WithBot ℕ} {p : Polynomial R} (hp : p.degree ≤ a) : (-p).degree ≤ a

@[simp]

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

theorem Polynomial.natDegree_neg_le_of_le {R : Type u} {m : ℕ} [Ring R] {p : Polynomial R} (hp : p.natDegree ≤ m) : (-p).natDegree ≤ m

@[simp]

theorem Polynomial.natDegree_intCast {R : Type u} [Ring R] (n : ℤ) : (↑n).natDegree = 0

theorem Polynomial.degree_intCast_le {R : Type u} [Ring R] (n : ℤ) : (↑n).degree ≤ 0

@[simp]

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

def Polynomial.nextCoeff {R : Type u} [Semiring R] (p : Polynomial R) : R

The second-highest coefficient, or 0 for constants

Equations

• p.nextCoeff = if p.natDegree = 0 then 0 else p.coeff (p.natDegree - 1)

theorem Polynomial.nextCoeff_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} : p.nextCoeff = 0 ↔ p.natDegree = 0 ∨ 0 < p.natDegree ∧ p.coeff (p.natDegree - 1) = 0

theorem Polynomial.nextCoeff_ne_zero {R : Type u} [Semiring R] {p : Polynomial R} : p.nextCoeff ≠ 0 ↔ p.natDegree ≠ 0 ∧ p.coeff (p.natDegree - 1) ≠ 0

@[simp]

theorem Polynomial.nextCoeff_C_eq_zero {R : Type u} [Semiring R] (c : R) : (C c).nextCoeff = 0

theorem Polynomial.nextCoeff_of_natDegree_pos {R : Type u} [Semiring R] {p : Polynomial R} (hp : 0 < p.natDegree) : p.nextCoeff = p.coeff (p.natDegree - 1)

theorem Polynomial.degree_add_le {R : Type u} [Semiring R] (p q : Polynomial R) : (p + q).degree ≤ max p.degree q.degree

theorem Polynomial.degree_add_le_of_degree_le {R : Type u} [Semiring R] {p q : Polynomial R} {n : ℕ} (hp : p.degree ≤ ↑n) (hq : q.degree ≤ ↑n) : (p + q).degree ≤ ↑n

theorem Polynomial.degree_add_le_of_le {R : Type u} [Semiring R] {p q : Polynomial R} {a b : WithBot ℕ} (hp : p.degree ≤ a) (hq : q.degree ≤ b) : (p + q).degree ≤ max a b

theorem Polynomial.natDegree_add_le {R : Type u} [Semiring R] (p q : Polynomial R) : (p + q).natDegree ≤ max p.natDegree q.natDegree

theorem Polynomial.natDegree_add_le_of_degree_le {R : Type u} [Semiring R] {p q : Polynomial R} {n : ℕ} (hp : p.natDegree ≤ n) (hq : q.natDegree ≤ n) : (p + q).natDegree ≤ n

theorem Polynomial.natDegree_add_le_of_le {R : Type u} {n m : ℕ} [Semiring R] {p q : Polynomial R} (hp : p.natDegree ≤ m) (hq : q.natDegree ≤ n) : (p + q).natDegree ≤ max m n

@[simp]

theorem Polynomial.leadingCoeff_zero {R : Type u} [Semiring R] : leadingCoeff 0 = 0

@[simp]

theorem Polynomial.leadingCoeff_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} : p.leadingCoeff = 0 ↔ p = 0

theorem Polynomial.leadingCoeff_ne_zero {R : Type u} [Semiring R] {p : Polynomial R} : p.leadingCoeff ≠ 0 ↔ p ≠ 0

theorem Polynomial.leadingCoeff_eq_zero_iff_deg_eq_bot {R : Type u} [Semiring R] {p : Polynomial R} : p.leadingCoeff = 0 ↔ p.degree = ⊥

theorem Polynomial.natDegree_C_mul_X_pow_le {R : Type u} [Semiring R] (a : R) (n : ℕ) : (C a * X ^ n).natDegree ≤ n

theorem Polynomial.degree_erase_le {R : Type u} [Semiring R] (p : Polynomial R) (n : ℕ) : (erase n p).degree ≤ p.degree

theorem Polynomial.degree_erase_lt {R : Type u} [Semiring R] {p : Polynomial R} (hp : p ≠ 0) : (erase p.natDegree p).degree < p.degree

theorem Polynomial.degree_update_le {R : Type u} [Semiring R] (p : Polynomial R) (n : ℕ) (a : R) : (p.update n a).degree ≤ max p.degree ↑n

theorem Polynomial.degree_sum_le {R : Type u} [Semiring R] {ι : Type u_1} (s : Finset ι) (f : ι → Polynomial R) : (∑ i ∈ s, f i).degree ≤ s.sup fun (b : ι) => (f b).degree

theorem Polynomial.degree_mul_le {R : Type u} [Semiring R] (p q : Polynomial R) : (p * q).degree ≤ p.degree + q.degree

theorem Polynomial.degree_mul_le_of_le {R : Type u} [Semiring R] {p q : Polynomial R} {a b : WithBot ℕ} (hp : p.degree ≤ a) (hq : q.degree ≤ b) : (p * q).degree ≤ a + b

theorem Polynomial.degree_pow_le {R : Type u} [Semiring R] (p : Polynomial R) (n : ℕ) : (p ^ n).degree ≤ n • p.degree

theorem Polynomial.degree_pow_le_of_le {R : Type u} [Semiring R] {p : Polynomial R} {a : WithBot ℕ} (b : ℕ) (hp : p.degree ≤ a) : (p ^ b).degree ≤ ↑b * a

@[simp]

theorem Polynomial.leadingCoeff_monomial {R : Type u} [Semiring R] (a : R) (n : ℕ) : ((monomial n) a).leadingCoeff = a

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

theorem Polynomial.leadingCoeff_C_mul_X {R : Type u} [Semiring R] (a : R) : (C a * X).leadingCoeff = a

@[simp]

theorem Polynomial.leadingCoeff_C {R : Type u} [Semiring R] (a : R) : (C a).leadingCoeff = a

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

theorem Polynomial.leadingCoeff_X {R : Type u} [Semiring R] : X.leadingCoeff = 1

@[simp]

theorem Polynomial.monic_X_pow {R : Type u} [Semiring R] (n : ℕ) : (X ^ n).Monic

@[simp]

theorem Polynomial.monic_X {R : Type u} [Semiring R] : X.Monic

theorem Polynomial.leadingCoeff_one {R : Type u} [Semiring R] : leadingCoeff 1 = 1

@[simp]

theorem Polynomial.monic_one {R : Type u} [Semiring R] : Monic 1

theorem Polynomial.Monic.ne_zero {R : Type u} [Semiring R] [Nontrivial R] {p : Polynomial R} (hp : p.Monic) : p ≠ 0

theorem Polynomial.Monic.ne_zero_of_ne {R : Type u} [Semiring R] (h : 0 ≠ 1) {p : Polynomial R} (hp : p.Monic) : p ≠ 0

theorem Polynomial.Monic.ne_zero_of_C {R : Type u} [Semiring R] [Nontrivial R] {c : R} (hc : (C c).Monic) : c ≠ 0

theorem Polynomial.Monic.ne_zero_of_polynomial_ne {R : Type u} [Semiring R] {p q r : Polynomial R} (hp : p.Monic) (hne : q ≠ r) : p ≠ 0

theorem Polynomial.natDegree_mul_le {R : Type u} [Semiring R] {p q : Polynomial R} : (p * q).natDegree ≤ p.natDegree + q.natDegree

theorem Polynomial.natDegree_mul_le_of_le {R : Type u} {n m : ℕ} [Semiring R] {p q : Polynomial R} (hp : p.natDegree ≤ m) (hg : q.natDegree ≤ n) : (p * q).natDegree ≤ m + n

theorem Polynomial.natDegree_pow_le {R : Type u} [Semiring R] {p : Polynomial R} {n : ℕ} : (p ^ n).natDegree ≤ n * p.natDegree

theorem Polynomial.natDegree_pow_le_of_le {R : Type u} {m : ℕ} [Semiring R] {p : Polynomial R} (n : ℕ) (hp : p.natDegree ≤ m) : (p ^ n).natDegree ≤ n * m

theorem Polynomial.natDegree_eq_zero_iff_degree_le_zero {R : Type u} [Semiring R] {p : Polynomial R} : p.natDegree = 0 ↔ p.degree ≤ 0

theorem Polynomial.degree_zero_le {R : Type u} [Semiring R] : degree 0 ≤ 0

theorem Polynomial.degree_le_iff_coeff_zero {R : Type u} [Semiring R] (f : Polynomial R) (n : WithBot ℕ) : f.degree ≤ n ↔ ∀ (m : ℕ), n < ↑m → f.coeff m = 0

theorem Polynomial.degree_lt_iff_coeff_zero {R : Type u} [Semiring R] (f : Polynomial R) (n : ℕ) : f.degree < ↑n ↔ ∀ (m : ℕ), n ≤ m → f.coeff m = 0

theorem Polynomial.natDegree_pos_iff_degree_pos {R : Type u} [Semiring R] {p : Polynomial R} : 0 < p.natDegree ↔ 0 < p.degree

@[simp]

theorem Polynomial.degree_X_pow {R : Type u} [Semiring R] [Nontrivial R] (n : ℕ) : (X ^ n).degree = ↑n

@[simp]

theorem Polynomial.natDegree_X_pow {R : Type u} [Semiring R] [Nontrivial R] (n : ℕ) : (X ^ n).natDegree = n

theorem Polynomial.degree_sub_le {R : Type u} [Ring R] (p q : Polynomial R) : (p - q).degree ≤ max p.degree q.degree

theorem Polynomial.degree_sub_le_of_le {R : Type u} [Ring R] {p q : Polynomial R} {a b : WithBot ℕ} (hp : p.degree ≤ a) (hq : q.degree ≤ b) : (p - q).degree ≤ max a b

theorem Polynomial.natDegree_sub_le {R : Type u} [Ring R] (p q : Polynomial R) : (p - q).natDegree ≤ max p.natDegree q.natDegree

theorem Polynomial.natDegree_sub_le_of_le {R : Type u} {n m : ℕ} [Ring R] {p q : Polynomial R} (hp : p.natDegree ≤ m) (hq : q.natDegree ≤ n) : (p - q).natDegree ≤ max m n

theorem Polynomial.degree_sub_lt_left {R : Type u} [Ring R] {p q : Polynomial R} (hd : p.degree = q.degree) (hp0 : p ≠ 0) (hlc : p.leadingCoeff = q.leadingCoeff) : (p - q).degree < p.degree

@[deprecated Polynomial.degree_sub_lt_left (since := "2026-06-30")]

theorem Polynomial.degree_sub_lt {R : Type u} [Ring R] {p q : Polynomial R} (hd : p.degree = q.degree) (hp0 : p ≠ 0) (hlc : p.leadingCoeff = q.leadingCoeff) : (p - q).degree < p.degree

Alias of Polynomial.degree_sub_lt_left.

theorem Polynomial.degree_sub_lt_right {R : Type u} [Ring R] {p q : Polynomial R} (hd : p.degree = q.degree) (hq0 : q ≠ 0) (hlc : p.leadingCoeff = q.leadingCoeff) : (p - q).degree < q.degree

theorem Polynomial.degree_X_sub_C_le {R : Type u} [Ring R] (r : R) : (X - C r).degree ≤ 1

theorem Polynomial.natDegree_X_sub_C_le {R : Type u} [Ring R] (r : R) : (X - C r).natDegree ≤ 1