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.

def polynomial.div_X {R : Type u} [semiring R] (p : 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
@[simp]
theorem polynomial.coeff_div_X {R : Type u} {n : } [semiring R] {p : polynomial R} :
p.div_X.coeff n = p.coeff (n + 1)
theorem polynomial.div_X_mul_X_add {R : Type u} [semiring R] (p : polynomial R) :
@[simp]
theorem polynomial.div_X_C {R : Type u} [semiring R] (a : R) :
theorem polynomial.div_X_eq_zero_iff {R : Type u} [semiring R] {p : polynomial R} :
theorem polynomial.div_X_add {R : Type u} [semiring R] {p q : polynomial R} :
(p + q).div_X = p.div_X + q.div_X
theorem polynomial.degree_div_X_lt {R : Type u} [semiring R] {p : polynomial R} (hp0 : p 0) :
def polynomial.rec_on_horner {R : Type u} [semiring R] {M : polynomial RSort u_1} (p : polynomial R) :
M 0(Π (p : polynomial R) (a : R), p.coeff 0 = 0a 0M pM (p + polynomial.C a))(Π (p : polynomial R), p 0M pM (p * polynomial.X))M p

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

Equations
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 0P ((polynomial.C a) * polynomial.X)) (hX : ∀ {p : polynomial R}, 0 < p.degreeP pP (p * polynomial.X)) (hadd : ∀ {p : polynomial R} {a : R}, 0 < p.degreeP pP (p + polynomial.C a)) :
P p
theorem polynomial.X_dvd_iff {α : Type u} [comm_semiring α] {f : polynomial α} :
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) :
theorem polynomial.div_wf_lemma {R : Type u} [ring R] {p q : polynomial R} (h : q.degree p.degree p 0) (hq : q.monic) :
def polynomial.div_mod_by_monic_aux {R : Type u} [ring R] (p : polynomial R) {q : polynomial R} :

See div_by_monic.

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

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

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

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

Equations
theorem polynomial.degree_mod_by_monic_lt {R : Type u} [ring R] (p : polynomial R) {q : polynomial R} (hq : q.monic) (hq0 : q 0) :
@[simp]
theorem polynomial.zero_mod_by_monic {R : Type u} [ring R] (p : polynomial R) :
0 %ₘ p = 0
@[simp]
theorem polynomial.zero_div_by_monic {R : Type u} [ring R] (p : polynomial R) :
0 /ₘ p = 0
@[simp]
theorem polynomial.mod_by_monic_zero {R : Type u} [ring R] (p : polynomial R) :
p %ₘ 0 = p
@[simp]
theorem polynomial.div_by_monic_zero {R : Type u} [ring R] (p : polynomial R) :
p /ₘ 0 = 0
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
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
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
theorem polynomial.degree_mod_by_monic_le {R : Type u} [ring R] (p : polynomial R) {q : polynomial R} (hq : q.monic) :
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)
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
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
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) :
theorem polynomial.degree_div_by_monic_le {R : Type u} [comm_ring R] (p q : polynomial R) :
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) :
theorem polynomial.nat_degree_div_by_monic {R : Type u} [comm_ring R] (f : polynomial R) {g : polynomial R} (hg : g.monic) :
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
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) :
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) :
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) :
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
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) :
@[simp]
theorem polynomial.mod_by_monic_one {R : Type u} [comm_ring R] (p : polynomial R) :
p %ₘ 1 = 0
@[simp]
theorem polynomial.div_by_monic_one {R : Type u} [comm_ring R] (p : polynomial R) :
p /ₘ 1 = p
theorem polynomial.dvd_iff_is_root {R : Type u} {a : R} [comm_ring R] {p : polynomial R} :
theorem polynomial.eval₂_mod_by_monic_eq_self_of_root {R : Type u} {S : Type v} [comm_ring R] [comm_ring S] {f : R →+* S} {p q : polynomial R} (hq : q.monic) {x : S} (hx : polynomial.eval₂ f x q = 0) :
def polynomial.decidable_dvd_monic {R : Type u} [comm_ring R] {q : polynomial R} (p : polynomial R) (hq : q.monic) :

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

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

Equations
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 _)