Theorems · Theorem · field theory

Polynomial.roots_countP_pos_le_signVariations

• #100 of the 100 theorems: Descartes Rule of Signs

∀ {R : Type u_1} [inst : CommRing R] [inst_1 : LinearOrder R] [inst_2 : IsStrictOrderedRing R] (P : Polynomial R),
  Multiset.countP (fun x => 0 < x) P.roots ≤ P.signVariations

Descartes' Rule of Signs: the number of positive roots is at most the number of sign variations.

Defined in

Mathlib.Algebra.Polynomial.RuleOfSigns

Cited by

0 results in Mathlib

Foundations

Depth 134 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingLinearOrderIsStrictOrderedRing