mathlib documentation

data.polynomial.degree.definitions

Theory of univariate polynomials #

The definitions include degree, monic, leading_coeff

Results include

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

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
theorem polynomial.degree_lt_wf {R : Type u} [semiring R] :
well_founded (λ (p q : polynomial R), p.degree < q.degree)
@[protected, instance]
Equations
def polynomial.nat_degree {R : Type u} [semiring R] (p : polynomial R) :

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

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

a polynomial is monic if its leading coefficient is 1

Equations
Instances for polynomial.monic
theorem polynomial.monic.def {R : Type u} [semiring R] {p : polynomial R} :
@[protected, instance]
Equations
@[simp]
theorem polynomial.monic.leading_coeff {R : Type u} [semiring R] {p : polynomial R} (hp : p.monic) :
theorem polynomial.monic.coeff_nat_degree {R : Type u} [semiring R] {p : polynomial R} (hp : p.monic) :
@[simp]
theorem polynomial.degree_zero {R : Type u} [semiring R] :
@[simp]
theorem polynomial.nat_degree_zero {R : Type u} [semiring R] :
theorem polynomial.degree_eq_bot {R : Type u} [semiring R] {p : polynomial R} :
p.degree = p = 0
theorem polynomial.degree_eq_nat_degree {R : Type u} [semiring R] {p : polynomial R} (hp : p 0) :
theorem polynomial.degree_eq_iff_nat_degree_eq {R : Type u} [semiring R] {p : polynomial R} {n : } (hp : p 0) :
theorem polynomial.degree_eq_iff_nat_degree_eq_of_pos {R : Type u} [semiring R] {p : polynomial R} {n : } (hn : 0 < n) :
@[simp]
theorem polynomial.le_degree_of_ne_zero {R : Type u} {n : } [semiring R] {p : polynomial R} (h : p.coeff n 0) :
theorem polynomial.le_nat_degree_of_ne_zero {R : Type u} {n : } [semiring R] {p : polynomial R} (h : p.coeff n 0) :
theorem polynomial.degree_eq_of_le_of_coeff_ne_zero {R : Type u} {n : } [semiring R] {p : polynomial R} (pn : p.degree n) (p1 : p.coeff n 0) :
theorem polynomial.nat_degree_eq_of_le_of_coeff_ne_zero {R : Type u} {n : } [semiring R] {p : polynomial R} (pn : p.nat_degree n) (p1 : p.coeff n 0) :
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) :
theorem polynomial.supp_subset_range {R : Type u} {m : } [semiring R] {p : polynomial R} (h : p.nat_degree < m) :
theorem polynomial.degree_le_degree {R : Type u} [semiring R] {p q : polynomial R} (h : q.coeff p.nat_degree 0) :
theorem polynomial.nat_degree_lt_iff_degree_lt {R : Type u} {n : } [semiring R] {p : polynomial R} (hp : p 0) :

Alias of the forward direction of polynomial.nat_degree_le_iff_degree_le.

Alias of the reverse direction of polynomial.nat_degree_le_iff_degree_le.

theorem polynomial.nat_degree_le_nat_degree {R : Type u} {S : Type v} [semiring R] {p : polynomial R} [semiring S] {q : polynomial S} (hpq : p.degree q.degree) :
@[simp]
theorem polynomial.degree_C {R : Type u} {a : R} [semiring R] (ha : a 0) :
theorem polynomial.degree_C_le {R : Type u} {a : R} [semiring R] :
theorem polynomial.degree_C_lt {R : Type u} {a : R} [semiring R] :
theorem polynomial.degree_one_le {R : Type u} [semiring R] :
@[simp]
theorem polynomial.nat_degree_C {R : Type u} [semiring R] (a : R) :
@[simp]
theorem polynomial.nat_degree_one {R : Type u} [semiring R] :
@[simp]
theorem polynomial.nat_degree_nat_cast {R : Type u} [semiring R] (n : ) :
@[simp]
theorem polynomial.degree_monomial {R : Type u} {a : R} [semiring R] (n : ) (ha : a 0) :
@[simp]
theorem polynomial.degree_C_mul_X_pow {R : Type u} {a : R} [semiring R] (n : ) (ha : a 0) :
theorem polynomial.degree_C_mul_X {R : Type u} {a : R} [semiring R] (ha : a 0) :
theorem polynomial.degree_monomial_le {R : Type u} [semiring R] (n : ) (a : R) :
@[simp]
theorem polynomial.nat_degree_C_mul_X_pow {R : Type u} [semiring R] (n : ) (a : R) (ha : a 0) :
@[simp]
theorem polynomial.nat_degree_C_mul_X {R : Type u} [semiring R] (a : R) (ha : a 0) :
@[simp]
theorem polynomial.nat_degree_monomial {R : Type u} [semiring R] [decidable_eq R] (i : ) (r : R) :
theorem polynomial.nat_degree_monomial_eq {R : Type u} [semiring R] (i : ) {r : R} (r0 : r 0) :
theorem polynomial.coeff_eq_zero_of_degree_lt {R : Type u} {n : } [semiring R] {p : polynomial R} (h : p.degree < n) :
p.coeff n = 0
theorem polynomial.coeff_eq_zero_of_nat_degree_lt {R : Type u} [semiring R] {p : polynomial R} {n : } (h : p.nat_degree < n) :
p.coeff n = 0
theorem polynomial.ext_iff_nat_degree_le {R : Type u} [semiring R] {p q : polynomial R} {n : } (hp : p.nat_degree n) (hq : q.nat_degree n) :
p = q (i : ), i n p.coeff i = q.coeff i
theorem polynomial.ext_iff_degree_le {R : Type u} [semiring R] {p q : polynomial R} {n : } (hp : p.degree n) (hq : q.degree n) :
p = q (i : ), i n p.coeff i = q.coeff i
@[simp]
theorem polynomial.ite_le_nat_degree_coeff {R : Type u} [semiring R] (p : polynomial R) (n : ) (I : decidable (n < 1 + p.nat_degree)) :
ite (n < 1 + p.nat_degree) (p.coeff n) 0 = p.coeff n
theorem polynomial.as_sum_support {R : Type u} [semiring R] (p : polynomial R) :
p = p.support.sum (λ (i : ), (polynomial.monomial i) (p.coeff i))
theorem polynomial.sum_over_range' {R : Type u} {S : Type v} [semiring R] [add_comm_monoid S] (p : polynomial R) {f : R S} (h : (n : ), f n 0 = 0) (n : ) (w : p.nat_degree < n) :
p.sum f = (finset.range n).sum (λ (a : ), 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.

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

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

theorem polynomial.sum_fin {R : Type u} {S : Type v} [semiring R] [add_comm_monoid S] (f : R S) (hf : (i : ), f i 0 = 0) {n : } {p : polynomial R} (hn : p.degree < n) :
finset.univ.sum (λ (i : fin n), f i (p.coeff i)) = p.sum f
theorem polynomial.as_sum_range' {R : Type u} [semiring R] (p : polynomial R) (n : ) (w : p.nat_degree < n) :
p = (finset.range n).sum (λ (i : ), (polynomial.monomial i) (p.coeff i))
theorem polynomial.as_sum_range {R : Type u} [semiring R] (p : polynomial R) :
p = (finset.range (p.nat_degree + 1)).sum (λ (i : ), (polynomial.monomial i) (p.coeff i))
theorem polynomial.coeff_ne_zero_of_eq_degree {R : Type u} {n : } [semiring R] {p : polynomial R} (hn : p.degree = n) :
p.coeff n 0
theorem polynomial.mem_support_C_mul_X_pow {R : Type u} [semiring R] {n a : } {c : R} (h : a (polynomial.C c * polynomial.X ^ n).support) :
a = n
@[simp]
theorem polynomial.degree_one {R : Type u} [semiring R] [nontrivial R] :
1.degree = 0
@[simp]
theorem polynomial.coeff_mul_X_sub_C {R : Type u} [ring R] {p : polynomial R} {r : R} {a : } :
(p * (polynomial.X - polynomial.C r)).coeff (a + 1) = p.coeff a - p.coeff (a + 1) * r
@[simp]
theorem polynomial.degree_neg {R : Type u} [ring R] (p : polynomial R) :
@[simp]
theorem polynomial.nat_degree_neg {R : Type u} [ring R] (p : polynomial R) :
@[simp]
theorem polynomial.nat_degree_int_cast {R : Type u} [ring R] (n : ) :
@[simp]
def polynomial.next_coeff {R : Type u} [semiring R] (p : polynomial R) :
R

The second-highest coefficient, or 0 for constants

Equations
@[simp]
theorem polynomial.ne_zero_of_degree_gt {R : Type u} [semiring R] {p : polynomial R} {n : with_bot } (h : n < p.degree) :
p 0
theorem polynomial.ne_zero_of_degree_ge_degree {R : Type u} [semiring R] {p q : polynomial R} (hpq : p.degree q.degree) (hp : p 0) :
q 0
theorem polynomial.ne_zero_of_nat_degree_gt {R : Type u} [semiring R] {p : polynomial R} {n : } (h : n < p.nat_degree) :
p 0
theorem polynomial.eq_C_of_degree_eq_zero {R : Type u} [semiring R] {p : polynomial R} (h : p.degree = 0) :
theorem polynomial.degree_add_le_of_degree_le {R : Type u} [semiring R] {p q : polynomial R} {n : } (hp : p.degree n) (hq : q.degree n) :
(p + q).degree n
theorem polynomial.nat_degree_add_le_of_degree_le {R : Type u} [semiring R] {p q : polynomial R} {n : } (hp : p.nat_degree n) (hq : q.nat_degree n) :
(p + q).nat_degree n
@[simp]
@[simp]
theorem polynomial.leading_coeff_eq_zero {R : Type u} [semiring R] {p : polynomial R} :
theorem polynomial.degree_add_C {R : Type u} {a : R} [semiring R] {p : polynomial R} (hp : 0 < p.degree) :
theorem polynomial.degree_update_le {R : Type u} [semiring R] (p : polynomial R) (n : ) (a : R) :
theorem polynomial.degree_sum_le {R : Type u} [semiring R] {ι : Type u_1} (s : finset ι) (f : ι polynomial R) :
(s.sum (λ (i : ι), f i)).degree s.sup (λ (b : ι), (f b).degree)
theorem polynomial.degree_mul_le {R : Type u} [semiring R] (p q : polynomial R) :
theorem polynomial.degree_pow_le {R : Type u} [semiring R] (p : polynomial R) (n : ) :
(p ^ n).degree n p.degree
@[simp]
@[simp]
@[simp]
@[simp]
theorem polynomial.monic_X_pow {R : Type u} [semiring R] (n : ) :
@[simp]
@[simp]
@[simp]
theorem polynomial.monic_one {R : Type u} [semiring R] :
theorem polynomial.monic.ne_zero {R : Type u_1} [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_of_nat_degree_le_of_coeff_eq_one {R : Type u} [semiring R] {p : polynomial R} (n : ) (pn : p.nat_degree n) (p1 : p.coeff n = 1) :
theorem polynomial.monic_of_degree_le_of_coeff_eq_one {R : Type u} [semiring R] {p : polynomial R} (n : ) (pn : p.degree n) (p1 : p.coeff n = 1) :
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.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
theorem polynomial.monic.degree_mul {R : Type u} [semiring R] {p q : polynomial R} (hq : q.monic) :
(p * q).degree = p.degree + q.degree
theorem polynomial.degree_pow' {R : Type u} [semiring R] {p : polynomial R} {n : } :
theorem polynomial.nat_degree_pow' {R : Type u} [semiring R] {p : polynomial R} {n : } (h : p.leading_coeff ^ n 0) :
theorem polynomial.nat_degree_pow_le {R : Type u} [semiring R] {p : polynomial R} {n : } :
@[simp]
theorem polynomial.coeff_pow_mul_nat_degree {R : Type u} [semiring R] (p : polynomial R) (n : ) :
(p ^ n).coeff (n * p.nat_degree) = p.leading_coeff ^ n
theorem polynomial.zero_le_degree_iff {R : Type u} [semiring R] {p : polynomial R} :
0 p.degree p 0
theorem polynomial.degree_le_iff_coeff_zero {R : Type u} [semiring R] (f : polynomial R) (n : with_bot ) :
f.degree n (m : ), n < m f.coeff m = 0
theorem polynomial.degree_lt_iff_coeff_zero {R : Type u} [semiring R] (f : polynomial R) (n : ) :
f.degree < n (m : ), n m f.coeff m = 0
theorem polynomial.degree_smul_le {R : Type u} [semiring R] (a : R) (p : polynomial R) :
theorem polynomial.nat_degree_smul_le {R : Type u} [semiring R] (a : R) (p : polynomial R) :
theorem polynomial.degree_lt_degree_mul_X {R : Type u} [semiring R] {p : polynomial R} (hp : p 0) :
theorem polynomial.ne_zero_of_coe_le_degree {R : Type u} {n : } [semiring R] {p : polynomial R} (hdeg : n p.degree) :
p 0
theorem polynomial.le_nat_degree_of_coe_le_degree {R : Type u} {n : } [semiring R] {p : polynomial R} (hdeg : n p.degree) :
theorem polynomial.degree_sum_fin_lt {R : Type u} [semiring R] {n : } (f : fin n R) :
@[simp]
theorem polynomial.degree_linear {R : Type u} {a b : R} [semiring R] (ha : a 0) :
@[simp]
@[simp]
theorem polynomial.degree_X_pow {R : Type u} [semiring R] [nontrivial R] (n : ) :
@[simp]
@[simp]
theorem polynomial.degree_mul_X {R : Type u} [semiring R] [nontrivial R] {p : polynomial R} :
@[simp]
theorem polynomial.degree_mul_X_pow {R : Type u} {n : } [semiring R] [nontrivial R] {p : polynomial R} :
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
theorem polynomial.degree_sub_eq_left_of_degree_lt {R : Type u} [ring R] {p q : polynomial R} (h : q.degree < p.degree) :
(p - q).degree = p.degree
theorem polynomial.degree_sub_eq_right_of_degree_lt {R : Type u} [ring R] {p q : polynomial R} (h : p.degree < q.degree) :
(p - q).degree = q.degree
@[simp]
theorem polynomial.degree_X_pow_add_C {R : Type u} [nontrivial R] [semiring R] {n : } (hn : 0 < n) (a : R) :
theorem polynomial.X_pow_add_C_ne_zero {R : Type u} [nontrivial R] [semiring R] {n : } (hn : 0 < n) (a : R) :
theorem polynomial.zero_nmem_multiset_map_X_add_C {R : Type u} [nontrivial R] [semiring R] {α : Type u_1} (m : multiset α) (f : α R) :
0 multiset.map (λ (a : α), polynomial.X + polynomial.C (f a)) m
@[simp]
theorem polynomial.leading_coeff_X_pow_add_C {R : Type u} [semiring R] {n : } (hn : 0 < n) {r : R} :
@[simp]
theorem polynomial.leading_coeff_X_pow_add_one {R : Type u} [semiring R] {n : } (hn : 0 < n) :
@[simp]
@[simp]
theorem polynomial.leading_coeff_X_pow_sub_C {R : Type u} [ring R] {n : } (hn : 0 < n) {r : R} :
@[simp]
theorem polynomial.leading_coeff_X_pow_sub_one {R : Type u} [ring R] {n : } (hn : 0 < n) :
@[simp]
theorem polynomial.degree_X_pow_sub_C {R : Type u} [ring R] [nontrivial R] {n : } (hn : 0 < n) (a : R) :
theorem polynomial.X_pow_sub_C_ne_zero {R : Type u} [ring R] [nontrivial R] {n : } (hn : 0 < n) (a : R) :
theorem polynomial.zero_nmem_multiset_map_X_sub_C {R : Type u} [ring R] [nontrivial R] {α : Type u_1} (m : multiset α) (f : α R) :
0 multiset.map (λ (a : α), polynomial.X - polynomial.C (f a)) m
@[simp]
theorem polynomial.degree_mul {R : Type u} [semiring R] [no_zero_divisors R] {p q : polynomial R} :
(p * q).degree = p.degree + q.degree

degree as a monoid homomorphism between R[X] and multiplicative (with_bot ℕ). This is useful to prove results about multiplication and degree.

Equations
@[simp]
theorem polynomial.degree_pow {R : Type u} [semiring R] [no_zero_divisors R] [nontrivial R] (p : polynomial R) (n : ) :
(p ^ n).degree = n p.degree

polynomial.leading_coeff bundled as a monoid_hom when R has no_zero_divisors, and thus leading_coeff is multiplicative

Equations
@[simp]