mathlib documentation

algebra.polynomial.big_operators

Lemmas for the interaction between polynomials and and . #

Recall that and are notation for finset.sum and finset.prod respectively.

Main results #

theorem polynomial.nat_degree_prod_le {R : Type u} {ι : Type w} (s : finset ι) [comm_semiring R] (f : ι → polynomial R) :
(∏ (i : ι) in s, f i).nat_degree ∑ (i : ι) in s, (f i).nat_degree
theorem polynomial.leading_coeff_prod' {R : Type u} {ι : Type w} (s : finset ι) [comm_semiring R] (f : ι → polynomial R) (h : ∏ (i : ι) in s, (f i).leading_coeff 0) :
(∏ (i : ι) in s, f i).leading_coeff = ∏ (i : ι) in s, (f i).leading_coeff

The leading coefficient of a product of polynomials is equal to the product of the leading coefficients, provided that this product is nonzero.

See polynomial.leading_coeff_prod (without the ') for a version for integral domains, where this condition is automatically satisfied.

theorem polynomial.nat_degree_prod' {R : Type u} {ι : Type w} (s : finset ι) [comm_semiring R] (f : ι → polynomial R) (h : ∏ (i : ι) in s, (f i).leading_coeff 0) :
(∏ (i : ι) in s, f i).nat_degree = ∑ (i : ι) in s, (f i).nat_degree

The degree of a product of polynomials is equal to the sum of the degrees, provided that the product of leading coefficients is nonzero.

See polynomial.nat_degree_prod (without the ') for a version for integral domains, where this condition is automatically satisfied.

theorem polynomial.nat_degree_prod_of_monic {R : Type u} {ι : Type w} (s : finset ι) [comm_semiring R] (f : ι → polynomial R) [nontrivial R] (h : ∀ (i : ι), i s(f i).monic) :
(∏ (i : ι) in s, f i).nat_degree = ∑ (i : ι) in s, (f i).nat_degree
theorem polynomial.coeff_zero_prod {R : Type u} {ι : Type w} (s : finset ι) [comm_semiring R] (f : ι → polynomial R) :
(∏ (i : ι) in s, f i).coeff 0 = ∏ (i : ι) in s, (f i).coeff 0
theorem polynomial.prod_X_sub_C_next_coeff {R : Type u} {ι : Type w} [comm_ring R] [nontrivial R] {s : finset ι} (f : ι → R) :
(∏ (i : ι) in s, (polynomial.X - polynomial.C (f i))).next_coeff = -∑ (i : ι) in s, f i
theorem polynomial.prod_X_sub_C_coeff_card_pred {R : Type u} {ι : Type w} [comm_ring R] [nontrivial R] (s : finset ι) (f : ι → R) (hs : 0 < s.card) :
(∏ (i : ι) in s, (polynomial.X - polynomial.C (f i))).coeff (s.card - 1) = -∑ (i : ι) in s, f i
theorem polynomial.nat_degree_prod {R : Type u} {ι : Type w} (s : finset ι) [comm_ring R] [no_zero_divisors R] (f : ι → polynomial R) [nontrivial R] (h : ∀ (i : ι), i sf i 0) :
(∏ (i : ι) in s, f i).nat_degree = ∑ (i : ι) in s, (f i).nat_degree

The degree of a product of polynomials is equal to the sum of the degrees.

See polynomial.nat_degree_prod' (with a ') for a version for commutative semirings, where additionally, the product of the leading coefficients must be nonzero.

theorem polynomial.degree_prod {R : Type u} {ι : Type w} (s : finset ι) [comm_ring R] [no_zero_divisors R] (f : ι → polynomial R) [nontrivial R] :
(∏ (i : ι) in s, f i).degree = ∑ (i : ι) in s, (f i).degree

The degree of a product of polynomials is equal to the sum of the degrees, where the degree of the zero polynomial is ⊥.

theorem polynomial.leading_coeff_prod {R : Type u} {ι : Type w} (s : finset ι) [comm_ring R] [no_zero_divisors R] (f : ι → polynomial R) :
(∏ (i : ι) in s, f i).leading_coeff = ∏ (i : ι) in s, (f i).leading_coeff

The leading coefficient of a product of polynomials is equal to the product of the leading coefficients.

See polynomial.leading_coeff_prod' (with a ') for a version for commutative semirings, where additionally, the product of the leading coefficients must be nonzero.