Integer polynomials as ℕ-coefficient polynomials: shift to a large Taylor point

Exponent arguments (T^{p(n)}) need ℕ-valued polynomials, and an induction on ℕ-coefficient polynomials does not cover p ∈ ℤ[X] such as n² − n. Trick: for large r, every Taylor coefficient (taylor r p).coeff i = (hasseDeriv i p).eval r with i ≤ natDegree is positive. Its leading coefficient is choose(d,i)·lead > 0 by natDegree_hasseDeriv and hasseDeriv_coeff. Eventual positivity over ℤ: map to ℝ[X] and use Polynomial.tendsto_atTop_of_leadingCoeff_nonneg (import Mathlib.Analysis.Polynomial.Basic) composed with tendsto_intCast_atTop_atTop. Treat degree 0 separately via eq_C_of_degree_le_zero. Combine the finitely many i with Filter.eventually_all_finset. Then p(Qm + r) = (taylor r p).eval (Qm) = ∑ tᵢ Qⁱ mⁱ (taylor_eval, eval_eq_sum_range, natDegree_taylor) has natural coefficients (tᵢQⁱ).toNat. Take r = Q(R+1) so that Q ∣ p(Q(m+R+1)) when p(0) = 0 (X_dvd_iff).