Theorems · Theorem · commutative algebra

Polynomial.exists_min_eq_gaussNorm

∀ {R : Type u_1} {F : Type u_2} [inst : Semiring R] [inst_1 : FunLike F R ℝ] (v : F) {c : ℝ} [ZeroHomClass F R ℝ]
  [NonnegHomClass F R ℝ] (p : Polynomial R),
  0 ≤ c →
    ∃ i,
      Polynomial.gaussNorm v c p = v (p.coeff i) * c ^ i ∧ ∀ j < i, v (p.coeff j) * c ^ j < Polynomial.gaussNorm v c p

If v is a nonnegative function with v 0 = 0 and c is nonnegative, there exists a minimal index i such that the Gauss norm of p at c is attained at i.

Defined in

Mathlib.RingTheory.Polynomial.GaussNorm

Cited by

0 results in Mathlib

Foundations

Depth 120 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringFunLikeZeroHomClassNonnegHomClass