Coefficient bounds for inverses of power series

Over a normed ring, the power-series inverse of a polynomial whose constant coefficient is a unit has geometrically bounded coefficients. This is what makes sums against such an inverse converge when the other factor decays faster than any geometric sequence, as for entire series.

Main results

• PowerSeries.norm_coeff_invOfUnit_coe_le: if the constant coefficient of a polynomial p is a unit u, the k-th coefficient of the inverse of p is at most ‖u⁻¹‖ * (1 + ‖u⁻¹‖ * ∑ᵢ ‖pᵢ‖) ^ k.

theorem PowerSeries.norm_coeff_invOfUnit_coe_le {R : Type u_1} [NormedRing R] (p : Polynomial R) (u : Rˣ) (k : ℕ) : ‖(coeff k) ((↑p).invOfUnit u)‖ ≤ ‖↑u⁻¹‖ * (1 + ‖↑u⁻¹‖ * ∑ i ∈ p.support, ‖p.coeff i‖) ^ k

Geometric growth of the inverse of a polynomial. If the constant coefficient of a polynomial p is a unit u, the coefficients of the power series inverse of p are bounded by ‖u⁻¹‖ * (1 + ‖u⁻¹‖ * ∑ᵢ ‖pᵢ‖) ^ k.