Mathoverflow 339137

Why do polynomials with coefficients 0,1 like to have only factors with 0,1 coefficients?

Reference: mathoverflow/339137 asked by user Sil

def Mathoverflow339137.IsZeroOne (P : Polynomial ℝ) : Prop

The predicate that all coefficients of a polynomial are either zero or one. P.coeffs is the finite set of all nonzero coefficients of the polynomial P. So IsZeroOne P means that every nonzero coefficient of P is equal to 1. Note that zero coefficients are not included in P.coeffs.

Equations

• Mathoverflow339137.IsZeroOne P = (P.coeffs ⊆ {1})

theorem Mathoverflow339137.mathoverflow_339137 (P Q R : Polynomial ℝ) (hP : P.Monic) (hQ : Q.Monic) (hp : ∀ c ∈ P.coeffs, 0 ≤ c) (hq : ∀ c ∈ Q.coeffs, 0 ≤ c) (h : R = P * Q) (hR : IsZeroOne R) : IsZeroOne P ∧ IsZeroOne Q

Let $P(x), Q(x) ∈ ℝ[x]$ be two monic polynomials with non-negative coefficients. If $R(x) = P(x)Q(x)$ is a $0,1$ polynomial (coefficients only from $\{0,1\}$), then $P(x)$ and $Q(x)$ are also $0, 1$ polynomials.