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data.polynomial.monomial

Univariate monomials #

Preparatory lemmas for degree_basic.

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theorem polynomial.monomial_one_eq_iff {R : Type u} [semiring R] [nontrivial R] {i j : ℕ} :
⇑(polynomial.monomial i) 1 = ⇑(polynomial.monomial j) 1 ↔ i = j
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theorem polynomial.smul_eq_C_mul {R : Type u} [semiring R] {p : polynomial R} (a : R) :
a • p = (⇑polynomial.C a) * p
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@[instance]
def polynomial.infinite {R : Type u} [semiring R] [nontrivial R] :
infinite (polynomial R)
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theorem polynomial.card_support_le_one_iff_monomial {R : Type u} [semiring R] {f : polynomial R} :
f.support.card ≤ 1 ↔ ∃ (n : ℕ) (a : R), f = ⇑(polynomial.monomial n) a
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theorem polynomial.ring_hom_ext {R : Type u} [semiring R] {S : Type u_1} [semiring S] {f g : polynomial R →+* S} (h₁ : ∀ (a : R), ⇑f (⇑polynomial.C a) = ⇑g (⇑polynomial.C a)) (h₂ : ⇑f polynomial.X = ⇑g polynomial.X) :
f = g
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@[ext]
theorem polynomial.ring_hom_ext' {R : Type u} [semiring R] {S : Type u_1} [semiring S] {f g : polynomial R →+* S} (h₁ : f.comp polynomial.C = g.comp polynomial.C) (h₂ : ⇑f polynomial.X = ⇑g polynomial.X) :
f = g