Interlacing from a monomial chain

Let $T$ be a real-linear operator on polynomials of degree at most $D$. Suppose that $T$ preserves nonnegative coefficients, sends every nonzero PF polynomial in the degree box to a nonzero real-rooted polynomial, and its monomial images form the oriented chain

T(1) \ll T(X) \ll \dotsb \ll T(X^D).

Then $T$ preserves oriented interlacing of nonnegative-coefficient real-rooted inputs in the same degree box. The quadratic tangent theorem propagates the monomial chain through each nonpositive linear factor.

The proof first propagates the monomial chain through all PF factors, and then uses the Garloff–Wagner Krein expansion to pass from one-root deletions to every oriented interlacing pair.

References

The decomposition step uses the Krein expansion for interlacing polynomials; the operator perspective is related to P. Brändén, “Iterated sequences and the geometry of zeros,” Journal für die reine und angewandte Mathematik 658 (2011), 115–131.

Definitions

Monomial-chain condition

RealRooted.PreservesPFShiftInterlacingOnDegreesource

def PreservesPFShiftInterlacingOnDegree
    (T : ℝ[X] →ₗ[ℝ] ℝ[X]) (D : ℕ) : Prop :=
  ∀ ⦃q : ℝ[X]⦄ ⦃m : ℕ⦄,
    IsPFPolynomial q → q ≠ 0 →
      (X ^ (m + 1) * q).natDegree ≤ D →
      StrictInterl (T (X ^ m * q)) (T (X ^ (m + 1) * q))

Theorems

The monomial chain gives interlacing preservation

RealRooted.Challenges.MonomialChainOperator.preservesInterlacingsource

theorem preservesInterlacing
    {T : ℝ[X] →ₗ[ℝ] ℝ[X]} {D : ℕ} {f g : ℝ[X]}
    (hfg : StrictInterl f g)
    (hf : HasNonnegCoeffs f) (hg : HasNonnegCoeffs g)
    (hfdeg : f.natDegree ≤ D) (hgdeg : g.natDegree ≤ D)
    (hTnn : ∀ ⦃p : ℝ[X]⦄, HasNonnegCoeffs p → HasNonnegCoeffs (T p))
    (hTrr : ∀ ⦃p : ℝ[X]⦄, IsPFPolynomial p → p ≠ 0 →
      p.natDegree ≤ D → T p ≠ 0 ∧ (T p).Splits)
    (hmono : ∀ m : ℕ, m + 1 ≤ D →
      StrictInterl (T (X ^ m)) (T (X ^ (m + 1)))) :
    StrictInterl (T f) (T g)

Interlacing from a monomial chain

References

≔ Definitions

⊢ Theorems

Definitions

Theorems