Maximum modulus for Tate algebras at roots of unity

Let f = ∑ₜ fₜ Xᵗ be a restricted power series in k variables at unit polyradius over an ultrametric normed ring R, and let φ : R → S be an isometric ring homomorphism into a complete ultrametric normed field. Suppose that S contains a primitive m-th root of unity ζ, where ‖m‖ = 1 in S and m exceeds every exponent of every coefficient fₜ with ‖fₜ‖ = ‖f‖. Then f attains its Gauss norm at a point of S ^ k whose coordinates are powers of ζ.

The proof is discrete Fourier inversion. Let t₀ be an exponent with ‖fₜ₀‖ = ‖f‖. Averaging the values f(ζʲ) against the characters ζ^(-⟨j, t₀⟩) over j ∈ (ℤ ⧸ m) ^ k gives m ^ k ∑_{t ≡ t₀} φ(fₜ), the sum over the exponents congruent to t₀ modulo m. By the choice of m the term at t₀ is the only one of norm ‖f‖, so this sum has norm ‖f‖, and since ‖m‖ = 1 the ultrametric inequality forces ‖f(ζʲ)‖ ≥ ‖f‖ for some j. The reverse inequality holds at every point of the closed unit polydisc.

The condition ‖m‖ = 1 excludes multiples of the residue characteristic, so that the reductions of the m-th roots of unity are distinct. Points with coordinates in R do not suffice in general: over ℚ_p the polynomial X ^ p - X has Gauss norm one but takes values of norm at most p⁻¹ on ℤ_p.

Main results

• MvPowerSeries.exists_norm_restrictedEval₂_eq_norm: a restricted series attains its Gauss norm at a point whose coordinates are m-th roots of unity.

References

• [Bosch, Güntzer, Remmert, Non-Archimedean Analysis][bosch-guntzer-remmert], §5.1.4.

theorem MvPowerSeries.exists_norm_restrictedEval₂_eq_norm {k : ℕ} {R : Type u_1} {S : Type u_2} [NormedCommRing R] [IsUltrametricDist R] [NormedField S] [IsUltrametricDist S] [CompleteSpace S] (φ : R →+* S) (hφ : ∀ (r : R), ‖φ r‖ = ‖r‖) {m : ℕ} {ζ : S} (hζ : IsPrimitiveRoot ζ m) (hm : ‖↑m‖ = 1) (f : ↥(IsRestricted.subring fun (x : Fin k) => 1)) (hfm : ∀ (t : Fin k →₀ ℕ), ‖(coeff t) ↑f‖ = ‖f‖ → ∀ (i : Fin k), t i < m) : ∃ (x : Fin k → S) (hx : ∀ (i : Fin k), ‖x i‖ ≤ 1), (∀ (i : Fin k), ∃ a < m, x i = ζ ^ a) ∧ ‖(restrictedEval₂ φ x ⋯ hx) f‖ = ‖f‖

Maximum modulus at roots of unity. Let φ : R → S be an isometric ring homomorphism into a complete ultrametric normed field containing a primitive m-th root of unity ζ, where ‖m‖ = 1 in S and m exceeds every exponent of every coefficient of f of norm ‖f‖. Then there is a point x of the closed unit polydisc of S ^ k, with coordinates powers of ζ, at which ‖f(x)‖ = ‖f‖.