Theorems · Theorem · commutative algebra

Multiset.prod_X_add_C_eq_sum_esymm

• 1000+ list: Vieta's formulas

∀ {R : Type u_1} [inst : CommSemiring R] (s : Multiset R),
  (Multiset.map (fun r => Polynomial.X + Polynomial.C r) s).prod =
    ∑ j ∈ Finset.range (s.card + 1), Polynomial.C (s.esymm j) * Polynomial.X ^ (s.card - j)

A sum version of Vieta's formula for Multiset: the product of the linear terms X + λ where λ runs through a multiset s is equal to a linear combination of the symmetric functions esymm s of the λ's .

Defined in

Mathlib.RingTheory.Polynomial.Vieta

Cited by

3 results in Mathlib

Foundations

Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiring