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
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).
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.