Zeros of polynomials

Location, counting and multiplicity of the zeros of polynomials: real-rootedness, stability, interlacing, root bounds and Sturm sequences.

• L1 norm and root bounds for polynomials
• Parity-change bound on real zeros
• Real-rooted polynomials
• Interlacing families of polynomials
• Stable polynomials
• Sturm's theorem
• Taylor expansion and double roots

TheoremAuerbach's problem on zero sets of real polynomials#●ready to prove

Let P be a nonzero real polynomial in n variables. Its zero set does not cut (separate) Euclidean space, i.e. the complement of the zero set is connected, if and only if every irreducible factor of P over the reals is always nonnegative or always nonpositive (as a function). The complement of a real algebraic hypersurface is semialgebraic, so connectedness of the complement is equivalent to the path-connectedness used by Rojas.

theorem_wanted auerbachPolynomialZeroSetSeparatesIffFactorsSignDefinite :
    ∀ (n : ℕ) (P : MvPolynomial (Fin n) ℝ), P ≠ 0 →
      IsConnected {x : Fin n → ℝ | (MvPolynomial.eval x) P ≠ 0} ↔
        ∀ Q : MvPolynomial (Fin n) ℝ, Irreducible Q → Q ∣ P →
          (∀ x : Fin n → ℝ, 0 ≤ (MvPolynomial.eval x) Q) ∨
            ∀ x : Fin n → ℝ, (MvPolynomial.eval x) Q ≤ 0

Sources

• H. Auerbach, Problem 148 of the Scottish Book; both necessity and sufficiency proved by J. M. Rojas, "About Herman Auerbach's problem on factors of real polynomials with connected zero set complement", 2015 (see the commentary in Mauldin (ed.), The Scottish Book, 2nd ed., 2015).

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