# Documentation

Mathlib.Data.Polynomial.PartialFractions

# Partial fractions #

These results were formalised by the Xena Project, at the suggestion of Patrick Massot.

# The main theorem #

• div_eq_quo_add_sum_rem_div: General partial fraction decomposition theorem for polynomials over an integral domain R : If f, g₁, g₂, ..., gₙ ∈ R[X] and the gᵢs are all monic and pairwise coprime, then ∃ q, r₁, ..., rₙ ∈ R[X] such that f / g₁g₂...gₙ = q + r₁/g₁ + ... + rₙ/gₙ and for all i, deg(rᵢ) < deg(gᵢ).

• The result is formalized here in slightly more generality, using finsets. That is, if ι is an arbitrary index type, g denotes a map from ι to R[X], and if s is an arbitrary finite subset of ι, with g i monic for all i ∈ s and for all i,j ∈ s, i ≠ j → g i is coprime to g j, then we have ∃ q ∈ R[X] , r : ι → R[X] such that ∀ i ∈ s, deg(r i) < deg(g i) and f / ∏ g i = q + ∑ (r i) / (g i), where the product and sum are over s.

• The proof is done by proving the two-denominator case and then performing finset induction for an arbitrary (finite) number of denominators.

## Scope for Expansion #

• Proving uniqueness of the decomposition
theorem div_eq_quo_add_rem_div_add_rem_div (R : Type) [] [] (K : Type) [] [Algebra () K] [] (f : ) {g₁ : } {g₂ : } (hg₁ : ) (hg₂ : ) (hcoprime : IsCoprime g₁ g₂) :
q r₁ r₂, f / (g₁ * g₂) = q + r₁ / g₁ + r₂ / g₂

Let R be an integral domain and f, g₁, g₂ ∈ R[X]. Let g₁ and g₂ be monic and coprime. Then, ∃ q, r₁, r₂ ∈ R[X] such that f / g₁g₂ = q + r₁/g₁ + r₂/g₂ and deg(r₁) < deg(g₁) and deg(r₂) < deg(g₂).

theorem div_eq_quo_add_sum_rem_div (R : Type) [] [] (K : Type) [] [Algebra () K] [] (f : ) {ι : Type u_1} {g : ι} {s : } (hg : ∀ (i : ι), i sPolynomial.Monic (g i)) (hcop : Set.Pairwise s fun i j => IsCoprime (g i) (g j)) :
q r, (∀ (i : ι), i sPolynomial.degree (r i) < Polynomial.degree (g i)) (f / Finset.prod s fun i => ↑(g i)) = q + Finset.sum s fun i => ↑(r i) / ↑(g i)

Let R be an integral domain and f ∈ R[X]. Let s be a finite index set. Then, a fraction of the form f / ∏ (g i) can be rewritten as q + ∑ (r i) / (g i), where deg(r i) < deg(g i), provided that the g i are monic and pairwise coprime.