mathlib documentation

data.polynomial.degree.basic

Theory of univariate polynomials

The definitions include degree, monic, leading_coeff

Results include

source
def polynomial.degree {R : Type u} [semiring R] (p : polynomial R) :
with_bot 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
source
theorem polynomial.degree_lt_wf {R : Type u} [semiring R] :
well_founded (λ (p q : polynomial R), p.degree < q.degree)

source
@[instance]
def polynomial.has_well_founded {R : Type u} [semiring R] :
has_well_founded (polynomial R)

Equations
source
def polynomial.nat_degree {R : Type u} [semiring R] (p : polynomial R) :
ℕ

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

Equations
source
def polynomial.leading_coeff {R : Type u} [semiring R] (p : polynomial R) :
R

leading_coeff p gives the coefficient of the highest power of X in p

Equations
source
def polynomial.monic {R : Type u} [semiring R] (p : polynomial R) :
Prop

a polynomial is monic if its leading coefficient is 1

Equations
source
theorem polynomial.monic.def {R : Type u} [semiring R] {p : polynomial R} :
p.monic ↔ p.leading_coeff = 1

source
@[instance]
def polynomial.monic.decidable {R : Type u} [semiring R] {p : polynomial R} [decidable_eq R] :
decidable p.monic

Equations
source
@[simp]
theorem polynomial.monic.leading_coeff {R : Type u} [semiring R] {p : polynomial R} (hp : p.monic) :
p.leading_coeff = 1

source
@[simp]
theorem polynomial.degree_zero {R : Type u} [semiring R] :
0.degree = ⊥

source
@[simp]
theorem polynomial.nat_degree_zero {R : Type u} [semiring R] :
0.nat_degree = 0

source
theorem polynomial.degree_eq_bot {R : Type u} [semiring R] {p : polynomial R} :
p.degree = ⊥ ↔ p = 0

source
theorem polynomial.degree_eq_nat_degree {R : Type u} [semiring R] {p : polynomial R} (hp : p ≠ 0) :
p.degree = ↑(p.nat_degree)

source
theorem polynomial.degree_eq_iff_nat_degree_eq {R : Type u} [semiring R] {p : polynomial R} {n : ℕ} (hp : p ≠ 0) :
p.degree = ↑n ↔ p.nat_degree = n

source
theorem polynomial.degree_eq_iff_nat_degree_eq_of_pos {R : Type u} [semiring R] {p : polynomial R} {n : ℕ} (hn : 0 < n) :
p.degree = ↑n ↔ p.nat_degree = n

source
theorem polynomial.nat_degree_eq_of_degree_eq_some {R : Type u} [semiring R] {p : polynomial R} {n : ℕ} (h : p.degree = ↑n) :
p.nat_degree = n

source
@[simp]
theorem polynomial.degree_le_nat_degree {R : Type u} [semiring R] {p : polynomial R} :
p.degree ≤ ↑(p.nat_degree)

source
theorem polynomial.nat_degree_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.nat_degree = q.nat_degree

source
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

source
theorem polynomial.le_nat_degree_of_ne_zero {R : Type u} {n : ℕ} [semiring R] {p : polynomial R} (h : p.coeff n ≠ 0) :
n ≤ p.nat_degree

source
theorem polynomial.degree_le_degree {R : Type u} [semiring R] {p q : polynomial R} (h : q.coeff p.nat_degree ≠ 0) :
p.degree ≤ q.degree

source
theorem polynomial.degree_ne_of_nat_degree_ne {R : Type u} [semiring R] {p : polynomial R} {n : ℕ} (a : p.nat_degree ≠ n) :
p.degree ≠ ↑n

source
theorem polynomial.nat_degree_le_of_degree_le {R : Type u} [semiring R] {p : polynomial R} {n : ℕ} (H : p.degree ≤ ↑n) :
p.nat_degree ≤ n

source
theorem polynomial.nat_degree_le_nat_degree {R : Type u} [semiring R] {p q : polynomial R} (hpq : p.degree ≤ q.degree) :
p.nat_degree ≤ q.nat_degree

source
@[simp]
theorem polynomial.degree_C {R : Type u} {a : R} [semiring R] (ha : a ≠ 0) :
(⇑polynomial.C a).degree = 0

source
theorem polynomial.degree_C_le {R : Type u} {a : R} [semiring R] :
(⇑polynomial.C a).degree ≤ 0

source
theorem polynomial.degree_one_le {R : Type u} [semiring R] :
1.degree ≤ 0

source
@[simp]
theorem polynomial.nat_degree_C {R : Type u} [semiring R] (a : R) :
(⇑polynomial.C a).nat_degree = 0

source
@[simp]
theorem polynomial.nat_degree_one {R : Type u} [semiring R] :
1.nat_degree = 0

source
@[simp]
theorem polynomial.nat_degree_nat_cast {R : Type u} [semiring R] (n : ℕ) :
↑n.nat_degree = 0

source
@[simp]
theorem polynomial.degree_monomial {R : Type u} {a : R} [semiring R] (n : ℕ) (ha : a ≠ 0) :
((⇑polynomial.C a) * polynomial.X ^ n).degree = ↑n

source
theorem polynomial.degree_monomial_le {R : Type u} [semiring R] (n : ℕ) (a : R) :
((⇑polynomial.C a) * polynomial.X ^ n).degree ≤ ↑n

source
theorem polynomial.coeff_eq_zero_of_degree_lt {R : Type u} {n : ℕ} [semiring R] {p : polynomial R} (h : p.degree < ↑n) :
p.coeff n = 0

source
theorem polynomial.coeff_eq_zero_of_nat_degree_lt {R : Type u} [semiring R] {p : polynomial R} {n : ℕ} (h : p.nat_degree < n) :
p.coeff n = 0

source
@[simp]
theorem polynomial.coeff_nat_degree_succ_eq_zero {R : Type u} [semiring R] {p : polynomial R} :
p.coeff (p.nat_degree + 1) = 0

source
theorem polynomial.ite_le_nat_degree_coeff {R : Type u} [semiring R] (p : polynomial R) (n : ℕ) (I : decidable (n < 1 + p.nat_degree)) :
ite (n < 1 + p.nat_degree) (p.coeff n) 0 = p.coeff n

source
theorem polynomial.as_sum {R : Type u} [semiring R] (p : polynomial R) :
p = ∑ (i : ℕ) in finset.range (p.nat_degree + 1), (⇑polynomial.C (p.coeff i)) * polynomial.X ^ i

source
theorem polynomial.sum_over_range' {R : Type u} {S : Type v} [semiring R] [add_comm_monoid S] (p : polynomial R) {f : ℕ → R → S} (h : ∀ (n : ℕ), f n 0 = 0) (n : ℕ) (w : p.nat_degree < n) :
finsupp.sum p f = ∑ (a : ℕ) in finset.range n, f a (p.coeff a)

We can reexpress a sum over p.support as a sum over range n, for any n satisfying p.nat_degree < n.

source
theorem polynomial.sum_over_range {R : Type u} {S : Type v} [semiring R] [add_comm_monoid S] (p : polynomial R) {f : ℕ → R → S} (h : ∀ (n : ℕ), f n 0 = 0) :
finsupp.sum p f = ∑ (a : ℕ) in finset.range (p.nat_degree + 1), f a (p.coeff a)

We can reexpress a sum over p.support as a sum over range (p.nat_degree + 1).

source
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

source
theorem polynomial.eq_X_add_C_of_degree_le_one {R : Type u} [semiring R] {p : polynomial R} (h : p.degree ≤ 1) :
p = (⇑polynomial.C (p.coeff 1)) * polynomial.X + ⇑polynomial.C (p.coeff 0)

source
theorem polynomial.eq_X_add_C_of_degree_eq_one {R : Type u} [semiring R] {p : polynomial R} (h : p.degree = 1) :
p = (⇑polynomial.C p.leading_coeff) * polynomial.X + ⇑polynomial.C (p.coeff 0)

source
theorem polynomial.degree_C_mul_X_pow_le {R : Type u} [semiring R] (r : R) (n : ℕ) :
((⇑polynomial.C r) * polynomial.X ^ n).degree ≤ ↑n

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

source
theorem polynomial.degree_X_le {R : Type u} [semiring R] :
polynomial.X.degree ≤ 1

source
theorem polynomial.nat_degree_X_le {R : Type u} [semiring R] :
polynomial.X.nat_degree ≤ 1

source
@[simp]
theorem polynomial.degree_one {R : Type u} [semiring R] [nontrivial R] :
1.degree = 0

source
@[simp]
theorem polynomial.degree_X {R : Type u} [semiring R] [nontrivial R] :
polynomial.X.degree = 1

source
@[simp]
theorem polynomial.nat_degree_X {R : Type u} [semiring R] [nontrivial R] :
polynomial.X.nat_degree = 1

source
theorem polynomial.coeff_mul_X_sub_C {R : Type u} [ring R] {p : polynomial R} {r : R} {a : ℕ} :
(p * (polynomial.X - ⇑polynomial.C r)).coeff (a + 1) = p.coeff a - (p.coeff (a + 1)) * r

source
@[simp]
theorem polynomial.C_eq_int_cast {R : Type u} [ring R] (n : ℤ) :
⇑polynomial.C ↑n = ↑n

source
@[simp]
theorem polynomial.degree_neg {R : Type u} [ring R] (p : polynomial R) :
(-p).degree = p.degree

source
@[simp]
theorem polynomial.nat_degree_neg {R : Type u} [ring R] (p : polynomial R) :
(-p).nat_degree = p.nat_degree

source
@[simp]
theorem polynomial.nat_degree_int_cast {R : Type u} [ring R] (n : ℤ) :
↑n.nat_degree = 0

source
def polynomial.next_coeff {R : Type u} [semiring R] (p : polynomial R) :
R

The second-highest coefficient, or 0 for constants

Equations
source
@[simp]
theorem polynomial.next_coeff_C_eq_zero {R : Type u} [semiring R] (c : R) :
(⇑polynomial.C c).next_coeff = 0

source
theorem polynomial.next_coeff_of_pos_nat_degree {R : Type u} [semiring R] (p : polynomial R) (hp : 0 < p.nat_degree) :
p.next_coeff = p.coeff (p.nat_degree - 1)

source
theorem polynomial.coeff_nat_degree_eq_zero_of_degree_lt {R : Type u} [semiring R] {p q : polynomial R} (h : p.degree < q.degree) :
p.coeff q.nat_degree = 0

source
theorem polynomial.ne_zero_of_degree_gt {R : Type u} [semiring R] {p : polynomial R} {n : with_bot ℕ} (h : n < p.degree) :
p ≠ 0

source
theorem polynomial.eq_C_of_degree_le_zero {R : Type u} [semiring R] {p : polynomial R} (h : p.degree ≤ 0) :
p = ⇑polynomial.C (p.coeff 0)

source
theorem polynomial.eq_C_of_degree_eq_zero {R : Type u} [semiring R] {p : polynomial R} (h : p.degree = 0) :
p = ⇑polynomial.C (p.coeff 0)

source
theorem polynomial.degree_le_zero_iff {R : Type u} [semiring R] {p : polynomial R} :
p.degree ≤ 0 ↔ p = ⇑polynomial.C (p.coeff 0)

source
theorem polynomial.degree_add_le {R : Type u} [semiring R] (p q : polynomial R) :
(p + q).degree ≤ max p.degree q.degree

source
@[simp]
theorem polynomial.leading_coeff_zero {R : Type u} [semiring R] :
0.leading_coeff = 0

source
@[simp]
theorem polynomial.leading_coeff_eq_zero {R : Type u} [semiring R] {p : polynomial R} :
p.leading_coeff = 0 ↔ p = 0

source
theorem polynomial.leading_coeff_eq_zero_iff_deg_eq_bot {R : Type u} [semiring R] {p : polynomial R} :
p.leading_coeff = 0 ↔ p.degree = ⊥

source
theorem polynomial.degree_add_eq_of_degree_lt {R : Type u} [semiring R] {p q : polynomial R} (h : p.degree < q.degree) :
(p + q).degree = q.degree

source
theorem polynomial.degree_add_C {R : Type u} {a : R} [semiring R] {p : polynomial R} (hp : 0 < p.degree) :
(p + ⇑polynomial.C a).degree = p.degree

source
theorem polynomial.degree_add_eq_of_leading_coeff_add_ne_zero {R : Type u} [semiring R] {p q : polynomial R} (h : p.leading_coeff + q.leading_coeff ≠ 0) :
(p + q).degree = max p.degree q.degree

source
theorem polynomial.degree_erase_le {R : Type u} [semiring R] (p : polynomial R) (n : ℕ) :
polynomial.degree (finsupp.erase n p) ≤ p.degree

source
theorem polynomial.degree_erase_lt {R : Type u} [semiring R] {p : polynomial R} (hp : p ≠ 0) :
polynomial.degree (finsupp.erase p.nat_degree p) < p.degree

source
theorem polynomial.degree_sum_le {R : Type u} [semiring R] {ι : Type u_1} (s : finset ι) (f : ι → polynomial R) :
(∑ (i : ι) in s, f i).degree ≤ s.sup (λ (b : ι), (f b).degree)

source
theorem polynomial.degree_mul_le {R : Type u} [semiring R] (p q : polynomial R) :
(p * q).degree ≤ p.degree + q.degree

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

source
@[simp]
theorem polynomial.leading_coeff_monomial {R : Type u} [semiring R] (a : R) (n : ℕ) :
((⇑polynomial.C a) * polynomial.X ^ n).leading_coeff = a

source
@[simp]
theorem polynomial.leading_coeff_C {R : Type u} [semiring R] (a : R) :
(⇑polynomial.C a).leading_coeff = a

source
@[simp]
theorem polynomial.leading_coeff_X {R : Type u} [semiring R] :
polynomial.X.leading_coeff = 1

source
@[simp]
theorem polynomial.monic_X {R : Type u} [semiring R] :
polynomial.X.monic

source
@[simp]
theorem polynomial.leading_coeff_one {R : Type u} [semiring R] :
1.leading_coeff = 1

source
@[simp]
theorem polynomial.monic_one {R : Type u} [semiring R] :
1.monic

source
theorem polynomial.monic.ne_zero_of_zero_ne_one {R : Type u} [semiring R] (h : 0 ≠ 1) {p : polynomial R} (hp : p.monic) :
p ≠ 0

source
theorem polynomial.monic.ne_zero {R : Type u_1} [semiring R] [nontrivial R] {p : polynomial R} (hp : p.monic) :
p ≠ 0

source
theorem polynomial.leading_coeff_add_of_degree_lt {R : Type u} [semiring R] {p q : polynomial R} (h : p.degree < q.degree) :
(p + q).leading_coeff = q.leading_coeff

source
theorem polynomial.leading_coeff_add_of_degree_eq {R : Type u} [semiring R] {p q : polynomial R} (h : p.degree = q.degree) (hlc : p.leading_coeff + q.leading_coeff ≠ 0) :
(p + q).leading_coeff = p.leading_coeff + q.leading_coeff

source
@[simp]
theorem polynomial.coeff_mul_degree_add_degree {R : Type u} [semiring R] (p q : polynomial R) :
(p * q).coeff (p.nat_degree + q.nat_degree) = (p.leading_coeff) * q.leading_coeff

source
theorem polynomial.degree_mul' {R : Type u} [semiring R] {p q : polynomial R} (h : (p.leading_coeff) * q.leading_coeff ≠ 0) :
(p * q).degree = p.degree + q.degree

source
theorem polynomial.nat_degree_mul' {R : Type u} [semiring R] {p q : polynomial R} (h : (p.leading_coeff) * q.leading_coeff ≠ 0) :
(p * q).nat_degree = p.nat_degree + q.nat_degree

source
theorem polynomial.leading_coeff_mul' {R : Type u} [semiring R] {p q : polynomial R} (h : (p.leading_coeff) * q.leading_coeff ≠ 0) :
(p * q).leading_coeff = (p.leading_coeff) * q.leading_coeff

source
theorem polynomial.leading_coeff_pow' {R : Type u} {n : ℕ} [semiring R] {p : polynomial R} (a : p.leading_coeff ^ n ≠ 0) :
(p ^ n).leading_coeff = p.leading_coeff ^ n

source
theorem polynomial.degree_pow' {R : Type u} [semiring R] {p : polynomial R} {n : ℕ} (a : p.leading_coeff ^ n ≠ 0) :
(p ^ n).degree = n •ℕ p.degree

source
theorem polynomial.nat_degree_pow' {R : Type u} [semiring R] {p : polynomial R} {n : ℕ} (h : p.leading_coeff ^ n ≠ 0) :
(p ^ n).nat_degree = n * p.nat_degree

source
@[simp]
theorem polynomial.leading_coeff_X_pow {R : Type u} [semiring R] (n : ℕ) :
(polynomial.X ^ n).leading_coeff = 1

source
theorem polynomial.leading_coeff_mul_X_pow {R : Type u} [semiring R] {p : polynomial R} {n : ℕ} :
(p * polynomial.X ^ n).leading_coeff = p.leading_coeff

source
theorem polynomial.nat_degree_mul_le {R : Type u} [semiring R] {p q : polynomial R} :
(p * q).nat_degree ≤ p.nat_degree + q.nat_degree

source
theorem polynomial.subsingleton_of_monic_zero {R : Type u} [semiring R] (h : 0.monic) :
(∀ (p q : polynomial R), p = q) ∧ ∀ (a b : R), a = b

source
theorem polynomial.zero_le_degree_iff {R : Type u} [semiring R] {p : polynomial R} :
0 ≤ p.degree ↔ p ≠ 0

source
theorem polynomial.degree_nonneg_iff_ne_zero {R : Type u} [semiring R] {p : polynomial R} :
0 ≤ p.degree ↔ p ≠ 0

source
theorem polynomial.nat_degree_eq_zero_iff_degree_le_zero {R : Type u} [semiring R] {p : polynomial R} :
p.nat_degree = 0 ↔ p.degree ≤ 0

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

source
theorem polynomial.degree_lt_degree_mul_X {R : Type u} [semiring R] {p : polynomial R} (hp : p ≠ 0) :
p.degree < (p * polynomial.X).degree

source
theorem polynomial.eq_C_of_nat_degree_le_zero {R : Type u} [semiring R] {p : polynomial R} (h : p.nat_degree ≤ 0) :
p = ⇑polynomial.C (p.coeff 0)

source
theorem polynomial.nat_degree_pos_iff_degree_pos {R : Type u} [semiring R] {p : polynomial R} :
0 < p.nat_degree ↔ 0 < p.degree

source
@[simp]
theorem polynomial.degree_X_pow {R : Type u} [semiring R] [nontrivial R] (n : ℕ) :
(polynomial.X ^ n).degree = ↑n

source
theorem polynomial.not_is_unit_X {R : Type u} [semiring R] [nontrivial R] :
¬is_unit polynomial.X

source
theorem polynomial.degree_sub_le {R : Type u} [ring R] (p q : polynomial R) :
(p - q).degree ≤ max p.degree q.degree

source
theorem polynomial.degree_sub_lt {R : Type u} [ring R] {p q : polynomial R} (hd : p.degree = q.degree) (hp0 : p ≠ 0) (hlc : p.leading_coeff = q.leading_coeff) :
(p - q).degree < p.degree

source
theorem polynomial.nat_degree_X_sub_C_le {R : Type u} [ring R] {r : R} :
(polynomial.X - ⇑polynomial.C r).nat_degree ≤ 1

source
@[simp]
theorem polynomial.degree_X_sub_C {R : Type u} [nontrivial R] [ring R] (a : R) :
(polynomial.X - ⇑polynomial.C a).degree = 1

source
@[simp]
theorem polynomial.nat_degree_X_sub_C {R : Type u} [nontrivial R] [ring R] (x : R) :
(polynomial.X - ⇑polynomial.C x).nat_degree = 1

source
@[simp]
theorem polynomial.next_coeff_X_sub_C {R : Type u} [nontrivial R] [ring R] (c : R) :
(polynomial.X - ⇑polynomial.C c).next_coeff = -c

source
theorem polynomial.degree_X_pow_sub_C {R : Type u} [nontrivial R] [ring R] {n : ℕ} (hn : 0 < n) (a : R) :
(polynomial.X ^ n - ⇑polynomial.C a).degree = ↑n

source
theorem polynomial.X_pow_sub_C_ne_zero {R : Type u} [nontrivial R] [ring R] {n : ℕ} (hn : 0 < n) (a : R) :
polynomial.X ^ n - ⇑polynomial.C a ≠ 0

source
theorem polynomial.X_sub_C_ne_zero {R : Type u} [nontrivial R] [ring R] (r : R) :
polynomial.X - ⇑polynomial.C r ≠ 0

source
theorem polynomial.nat_degree_X_pow_sub_C {R : Type u} [nontrivial R] [ring R] {n : ℕ} (hn : 0 < n) {r : R} :
(polynomial.X ^ n - ⇑polynomial.C r).nat_degree = n

source
@[simp]
theorem polynomial.degree_mul {R : Type u} [integral_domain R] {p q : polynomial R} :
(p * q).degree = p.degree + q.degree

source
@[simp]
theorem polynomial.degree_pow {R : Type u} [integral_domain R] (p : polynomial R) (n : ℕ) :
(p ^ n).degree = n •ℕ p.degree

source
@[simp]
theorem polynomial.leading_coeff_mul {R : Type u} [integral_domain R] (p q : polynomial R) :
(p * q).leading_coeff = (p.leading_coeff) * q.leading_coeff

source
@[simp]
theorem polynomial.leading_coeff_X_add_C {R : Type u} [integral_domain R] (a b : R) (ha : a ≠ 0) :
((⇑polynomial.C a) * polynomial.X + ⇑polynomial.C b).leading_coeff = a

source
def polynomial.leading_coeff_hom {R : Type u} [integral_domain R] :
polynomial R →* R

polynomial.leading_coeff bundled as a monoid_hom when R is an integral_domain, and thus leading_coeff is multiplicative

Equations
source
@[simp]
theorem polynomial.leading_coeff_hom_apply {R : Type u} [integral_domain R] (p : polynomial R) :
⇑polynomial.leading_coeff_hom p = p.leading_coeff

source
@[simp]
theorem polynomial.leading_coeff_pow {R : Type u} [integral_domain R] (p : polynomial R) (n : ℕ) :
(p ^ n).leading_coeff = p.leading_coeff ^ n