Countable ideal-adic approximation of bounded-degree polynomials

Over a commutative ring, if the quotients by powers of an ideal are countable, every set of polynomials of bounded degree has a countable subset that approximates each of its members coefficientwise modulo every power of that ideal. The approximants stay in the original set, so constraints such as monicity or the Eisenstein condition are preserved without separate stability arguments.

This gives countable families of defining polynomials when root counts are locally constant under ideal-adic perturbations. No topology on the coefficient ring is required.

theorem Ideal.exists_countable_polynomial_approximations {R : Type u_1} [CommRing R] (I : Ideal R) (hI : ∀ (k : ℕ), Countable (R ⧸ I ^ k)) (s : Set (Polynomial R)) {n : ℕ} (hn : ∀ f ∈ s, f.natDegree ≤ n) : ∃ t ⊆ s, t.Countable ∧ ∀ f ∈ s, ∀ (k : ℕ), ∃ g ∈ t, ∀ (i : ℕ), g.coeff i - f.coeff i ∈ I ^ k

A set of polynomials of bounded degree has a countable subset approximating all its members coefficientwise modulo every power of an ideal whose power quotients are countable. The approximants belong to the given set.