mathlib documentation

data.polynomial.div

Division of univariate polynomials

The main defs are div_by_monic and mod_by_monic. The compatibility between these is given by mod_by_monic_add_div. We also define root_multiplicity.

source
def polynomial.coeff_coe_to_fun {R : Type u} [semiring R] :
has_coe_to_fun (polynomial R)

The coercion turning a polynomial into the function which reports the coefficient of a given monomial X^n

Equations
source
theorem polynomial.apply_eq_coeff {R : Type u} {n : ℕ} [semiring R] {p : polynomial R} :
⇑p n = p.coeff n

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

div_X p return a polynomial q such that q * X + C (p.coeff 0) = p. It can be used in a semiring where the usual division algorithm is not possible

Equations
source
theorem polynomial.div_X_mul_X_add {R : Type u} [semiring R] (p : polynomial R) :
(p.div_X) * polynomial.X + ⇑polynomial.C (p.coeff 0) = p

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

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

source
theorem polynomial.div_X_add {R : Type u} [semiring R] {p q : polynomial R} :
(p + q).div_X = p.div_X + q.div_X

source
theorem polynomial.degree_div_X_lt {R : Type u} [semiring R] {p : polynomial R} (hp0 : p ≠ 0) :
p.div_X.degree < p.degree

source
def polynomial.rec_on_horner {R : Type u} [semiring R] {M : polynomial R → Sort u_1} (p : polynomial R) (a : M 0) (a_1 : Π (p : polynomial R) (a : R), p.coeff 0 = 0 → a ≠ 0 → M p → M (p + ⇑polynomial.C a)) (a_2 : Π (p : polynomial R), p ≠ 0 → M p → M (p * polynomial.X)) :
M p

An induction principle for polynomials, valued in Sort* instead of Prop.

Equations
source
theorem polynomial.degree_pos_induction_on {R : Type u} [semiring R] {P : polynomial R → Prop} (p : polynomial R) (h0 : 0 < p.degree) (hC : ∀ {a : R}, a ≠ 0 → P ((⇑polynomial.C a) * polynomial.X)) (hX : ∀ {p : polynomial R}, 0 < p.degree → P p → P (p * polynomial.X)) (hadd : ∀ {p : polynomial R} {a : R}, 0 < p.degree → P p → P (p + ⇑polynomial.C a)) :
P p

source
theorem polynomial.X_dvd_iff {α : Type u} [comm_semiring α] {f : polynomial α} :
polynomial.X ∣ f ↔ f.coeff 0 = 0

source
theorem polynomial.multiplicity_finite_of_degree_pos_of_monic {R : Type u} [comm_semiring R] {p q : polynomial R} (hp : 0 < p.degree) (hmp : p.monic) (hq : q ≠ 0) :
multiplicity.finite p q

source
theorem polynomial.div_wf_lemma {R : Type u} [ring R] {p q : polynomial R} (h : q.degree ≤ p.degree ∧ p ≠ 0) (hq : q.monic) :
(p - ((⇑polynomial.C p.leading_coeff) * polynomial.X ^ (p.nat_degree - q.nat_degree)) * q).degree < p.degree

source
def polynomial.div_mod_by_monic_aux {R : Type u} [ring R] (p : polynomial R) {q : polynomial R} (a : q.monic) :
polynomial R × polynomial R

See div_by_monic.

Equations
source
def polynomial.div_by_monic {R : Type u} [ring R] (p q : polynomial R) :
polynomial R

div_by_monic gives the quotient of p by a monic polynomial q.

Equations
source
def polynomial.mod_by_monic {R : Type u} [ring R] (p q : polynomial R) :
polynomial R

mod_by_monic gives the remainder of p by a monic polynomial q.

Equations
source
theorem polynomial.degree_mod_by_monic_lt {R : Type u} [ring R] (p : polynomial R) {q : polynomial R} (hq : q.monic) (hq0 : q ≠ 0) :
(p %ₘ q).degree < q.degree

source
@[simp]
theorem polynomial.zero_mod_by_monic {R : Type u} [ring R] (p : polynomial R) :
0 %ₘ p = 0

source
@[simp]
theorem polynomial.zero_div_by_monic {R : Type u} [ring R] (p : polynomial R) :
0 /ₘ p = 0

source
@[simp]
theorem polynomial.mod_by_monic_zero {R : Type u} [ring R] (p : polynomial R) :
p %ₘ 0 = p

source
@[simp]
theorem polynomial.div_by_monic_zero {R : Type u} [ring R] (p : polynomial R) :
p /ₘ 0 = 0

source
theorem polynomial.div_by_monic_eq_of_not_monic {R : Type u} [ring R] {q : polynomial R} (p : polynomial R) (hq : ¬q.monic) :
p /ₘ q = 0

source
theorem polynomial.mod_by_monic_eq_of_not_monic {R : Type u} [ring R] {q : polynomial R} (p : polynomial R) (hq : ¬q.monic) :
p %ₘ q = p

source
theorem polynomial.mod_by_monic_eq_self_iff {R : Type u} [ring R] {p q : polynomial R} (hq : q.monic) (hq0 : q ≠ 0) :
p %ₘ q = p ↔ p.degree < q.degree

source
theorem polynomial.degree_mod_by_monic_le {R : Type u} [ring R] (p : polynomial R) {q : polynomial R} (hq : q.monic) :
(p %ₘ q).degree ≤ q.degree

source
theorem polynomial.mod_by_monic_eq_sub_mul_div {R : Type u} [comm_ring R] (p : polynomial R) {q : polynomial R} (hq : q.monic) :
p %ₘ q = p - q * (p /ₘ q)

source
theorem polynomial.mod_by_monic_add_div {R : Type u} [comm_ring R] (p : polynomial R) {q : polynomial R} (hq : q.monic) :
p %ₘ q + q * (p /ₘ q) = p

source
theorem polynomial.div_by_monic_eq_zero_iff {R : Type u} [comm_ring R] {p q : polynomial R} (hq : q.monic) (hq0 : q ≠ 0) :
p /ₘ q = 0 ↔ p.degree < q.degree

source
theorem polynomial.degree_add_div_by_monic {R : Type u} [comm_ring R] {p q : polynomial R} (hq : q.monic) (h : q.degree ≤ p.degree) :
q.degree + (p /ₘ q).degree = p.degree

source
theorem polynomial.degree_div_by_monic_le {R : Type u} [comm_ring R] (p q : polynomial R) :
(p /ₘ q).degree ≤ p.degree

source
theorem polynomial.degree_div_by_monic_lt {R : Type u} [comm_ring R] (p : polynomial R) {q : polynomial R} (hq : q.monic) (hp0 : p ≠ 0) (h0q : 0 < q.degree) :
(p /ₘ q).degree < p.degree

source
theorem polynomial.nat_degree_div_by_monic {R : Type u} [comm_ring R] (f : polynomial R) {g : polynomial R} (hg : g.monic) :
(f /ₘ g).nat_degree = f.nat_degree - g.nat_degree

source
theorem polynomial.div_mod_by_monic_unique {R : Type u} [comm_ring R] {f g : polynomial R} (q r : polynomial R) (hg : g.monic) (h : r + g * q = f ∧ r.degree < g.degree) :
f /ₘ g = q ∧ f %ₘ g = r

source
theorem polynomial.map_mod_div_by_monic {R : Type u} {S : Type v} [comm_ring R] {p q : polynomial R} [comm_ring S] (f : R →+* S) (hq : q.monic) :
polynomial.map f (p /ₘ q) = polynomial.map f p /ₘ polynomial.map f q ∧ polynomial.map f (p %ₘ q) = polynomial.map f p %ₘ polynomial.map f q

source
theorem polynomial.map_div_by_monic {R : Type u} {S : Type v} [comm_ring R] {p q : polynomial R} [comm_ring S] (f : R →+* S) (hq : q.monic) :
polynomial.map f (p /ₘ q) = polynomial.map f p /ₘ polynomial.map f q

source
theorem polynomial.map_mod_by_monic {R : Type u} {S : Type v} [comm_ring R] {p q : polynomial R} [comm_ring S] (f : R →+* S) (hq : q.monic) :
polynomial.map f (p %ₘ q) = polynomial.map f p %ₘ polynomial.map f q

source
theorem polynomial.dvd_iff_mod_by_monic_eq_zero {R : Type u} [comm_ring R] {p q : polynomial R} (hq : q.monic) :
p %ₘ q = 0 ↔ q ∣ p

source
theorem polynomial.map_dvd_map {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] (f : R →+* S) (hf : function.injective ⇑f) {x y : polynomial R} (hx : x.monic) :
polynomial.map f x ∣ polynomial.map f y ↔ x ∣ y

source
@[simp]
theorem polynomial.mod_by_monic_one {R : Type u} [comm_ring R] (p : polynomial R) :
p %ₘ 1 = 0

source
@[simp]
theorem polynomial.div_by_monic_one {R : Type u} [comm_ring R] (p : polynomial R) :
p /ₘ 1 = p

source
@[simp]
theorem polynomial.mod_by_monic_X_sub_C_eq_C_eval {R : Type u} [comm_ring R] (p : polynomial R) (a : R) :
p %ₘ (polynomial.X - ⇑polynomial.C a) = ⇑polynomial.C (polynomial.eval a p)

source
theorem polynomial.mul_div_by_monic_eq_iff_is_root {R : Type u} {a : R} [comm_ring R] {p : polynomial R} :
(polynomial.X - ⇑polynomial.C a) * (p /ₘ (polynomial.X - ⇑polynomial.C a)) = p ↔ p.is_root a

source
theorem polynomial.dvd_iff_is_root {R : Type u} {a : R} [comm_ring R] {p : polynomial R} :
polynomial.X - ⇑polynomial.C a ∣ p ↔ p.is_root a

source
theorem polynomial.mod_by_monic_X {R : Type u} [comm_ring R] (p : polynomial R) :
p %ₘ polynomial.X = ⇑polynomial.C (polynomial.eval 0 p)

source
def polynomial.decidable_dvd_monic {R : Type u} [comm_ring R] {q : polynomial R} (p : polynomial R) (hq : q.monic) :
decidable (q ∣ p)

An algorithm for deciding polynomial divisibility. The algorithm is "compute p %ₘ q and compare to 0". Seepolynomial.mod_by_monicfor the algorithm that computes%ₘ`.

Equations
source
theorem polynomial.multiplicity_X_sub_C_finite {R : Type u} [comm_ring R] {p : polynomial R} (a : R) (h0 : p ≠ 0) :
multiplicity.finite (polynomial.X - ⇑polynomial.C a) p

source
def polynomial.root_multiplicity {R : Type u} [comm_ring R] (a : R) (p : polynomial R) :
ℕ

The largest power of X - C a which divides p. This is computable via the divisibility algorithm decidable_dvd_monic.

Equations
source
theorem polynomial.root_multiplicity_eq_multiplicity {R : Type u} [comm_ring R] (p : polynomial R) (a : R) :
polynomial.root_multiplicity a p = dite (p = 0) (λ (h0 : p = 0), 0) (λ (h0 : ¬p = 0), (multiplicity (polynomial.X - ⇑polynomial.C a) p).get _)

source
theorem polynomial.pow_root_multiplicity_dvd {R : Type u} [comm_ring R] (p : polynomial R) (a : R) :
(polynomial.X - ⇑polynomial.C a) ^ polynomial.root_multiplicity a p ∣ p

source
theorem polynomial.div_by_monic_mul_pow_root_multiplicity_eq {R : Type u} [comm_ring R] (p : polynomial R) (a : R) :
(p /ₘ (polynomial.X - ⇑polynomial.C a) ^ polynomial.root_multiplicity a p) * (polynomial.X - ⇑polynomial.C a) ^ polynomial.root_multiplicity a p = p

source
theorem polynomial.eval_div_by_monic_pow_root_multiplicity_ne_zero {R : Type u} [comm_ring R] {p : polynomial R} (a : R) (hp : p ≠ 0) :
polynomial.eval a (p /ₘ (polynomial.X - ⇑polynomial.C a) ^ polynomial.root_multiplicity a p) ≠ 0