Coefficient bounds for Coleman's transform

Let R be an ultrametric seminormed commutative ring with ‖1‖ = 1. Let P = ∑ aⱼ Tʲ and B = ∑ bₗ Tˡ be polynomials over R with B(0) = 0, let ρ be a nonnegative antitone sequence with ‖aⱼ‖ ≤ ∏ i < j, ρᵢ, and let β satisfy ‖bₗ‖ ρₜˡ ≤ βₜ for all t and all l ≥ 1. Then every bounded Coleman transform D_(n,m)(B, P) satisfies

‖coeff k D_(n,m)(B, P)‖ ≤ ∏ t < k, βₜ (Polynomial.norm_coeff_colemanDBounded_le).

If P = ∏ (1 - αᵢ T) with ‖αᵢ‖ ≤ ρᵢ, then D(B, P) = ∏ (1 - B(αᵢ) T) and the bound is the ultrametric estimate for the elementary symmetric functions of the B(αᵢ). The theorem asserts the same bound for every P obeying the coefficient bounds, over any coefficient ring, with no access to roots. It is uniform in the ranks n and m, so it applies to all truncations of an entire series F with Newton majorant ρ along an entire series B with B(0) = 0, with βₜ = β_B(ρₜ) the Coleman majorant (PowerSeries.norm_coeff_colemanDBounded_trunc_le). These products are the coefficient bounds of an entire series, so they control the limit of the finite transforms that defines Coleman's transform with entire functional input.

The common majorant alone does not show that the finite transforms converge. For that we compare two pairs (B, P) and (B', P') obeying the same bounds, with bₗ = b'ₗ for l ≤ L and aⱼ = a'ⱼ for j ≤ k L. If γ bounds the tails, ‖bₗ‖ ρₜˡ ≤ γₜ and ‖b'ₗ‖ ρₜˡ ≤ γₜ for l > L, then

‖coeff k D_(n,m)(B, P) - coeff k D_(n,m)(B', P')‖ ≤ ∑ r < k, γᵣ ∏ t < k, t ≠ r, βₜ

(Polynomial.norm_coeff_colemanDBounded_sub_le). For truncations of power series, γₜ is the Coleman majorant of the tail B - trunc (L + 1) B at ρₜ (PowerSeries.norm_coeff_colemanDBounded_trunc_sub_le), which tends to zero as L → ∞. Hence, for F(0) = 1, the k-th coefficients of the transforms D_(N,M)(trunc (M + 1) B, trunc (N + 1) F) form a Cauchy family as N and M tend to infinity independently (PowerSeries.cauchySeq_coeff_colemanDBounded_trunc).

The estimates are read off the Leibniz expansion of the Sylvester matrix defining the resultant, term by term. In a nonzero term, the permutation σ of the n + m columns moves each of the first n columns x (carrying coefficients of 1 - T B(X)) down to a row σ x ≥ x, picking b_(σ x - x) T when σ x > x, and moves each of the last m columns up, picking a_(x - σ x). Count both kinds of moves across each gap between consecutive rows: a column moved across j gaps is charged ρ₀, …, ρⱼ₋₁ in order on the a side, while on the b side every gap it crosses is charged ρ_r, where r is its rank by decreasing target row. At each gap the two counts agree, and the a charges are never larger than the b charges after sorting, so ∏ ‖a‖ ≤ ∏ ρ_r ^ (σ x - x) (Equiv.Perm.prod_prod_range_le_prod_pow_upRank). The ranks enumerate the moved columns, so each term is bounded by ∏ t < k, βₜ. A term contributing to the k-th coefficient has exactly k forward moves, and their total length is the total length of the backward moves. So if all forward moves have length at most L, the term only involves bₗ with l ≤ L and aⱼ with j ≤ k L, and it is the same for both pairs; otherwise one of its factors is bounded by a tail bound γᵣ instead of βᵣ.

Over a normed ring the same bound controls differences. If P₂ - P₁ and B₂ - B₁ satisfy the hypotheses after multiplication by r ≥ 1, then the k-th coefficients of D_(n,m)(B₂, P₂) and D_(n,m)(B₁, P₁) differ by at most r⁻¹ ∏ t < k, βₜ (Polynomial.dist_coeff_colemanDBounded_mul_le). To see this, join the inputs by the segments Bₛ = B₁ + s (B₂ - B₁) and Pₛ = P₁ + s (P₂ - P₁) over the polynomial ring R[s], and give R[s] the Gauss norm at radius r, in which ‖s‖ = r. The coefficient bound over this ring says that the i-th coefficient in s of the k-th coefficient of D_(n,m)(Bₛ, Pₛ) has norm at most r⁻ⁱ ∏ t < k, βₜ, and the difference of the two transforms is the sum of these coefficients for i ≥ 1. No polynomial identity for the transform is needed beyond its compatibility with ring homomorphisms.

For a power series F with Newton majorant ρ and B restricted at the radii c² ρᵢ for some c ≥ 1, the tails of the truncations from order a of B and b of F are small against the majorants at the radii c ρᵢ. So lengthening the truncations moves the k-th coefficients by at most c ^ (-min a b) ∏ i < k, β_B(c² ρᵢ) (PowerSeries.dist_coeff_colemanDBounded_trunc_mul_le). Unlike PowerSeries.norm_coeff_colemanDBounded_trunc_sub_le, whose truncation orders must grow with k, this rate is the same for every coefficient relative to its majorant.

Main results

• Polynomial.norm_coeff_colemanDBounded_le: the uniform coefficient bound for the bounded Coleman transform of polynomials.
• Polynomial.norm_coeff_colemanDBounded_sub_le: the continuity estimate under agreement of the coefficients of low degree.
• PowerSeries.norm_coeff_colemanDBounded_trunc_le: the bound ∏ i < k, β_B(ρᵢ) for the transforms of truncations of power series.
• Polynomial.dist_coeff_colemanDBounded_mul_le: the perturbation bound for the bounded Coleman transform of polynomials.
• PowerSeries.norm_coeff_colemanDBounded_trunc_sub_le: the continuity estimate for truncations, in terms of the Coleman majorants of the tails of B.
• PowerSeries.dist_coeff_colemanDBounded_trunc_mul_le: the continuity estimate in the truncation orders of B and F, with a geometric rate uniform in the ranks.
• PowerSeries.dist_coeff_colemanDBounded_trunc_mul_trunc_le: the comparison of the transforms of the truncation of F * G and of the product of the truncations of F and G, with a geometric rate in the truncation order.
• PowerSeries.cauchySeq_coeff_colemanDBounded_trunc: the coefficients of the finite transforms of truncations are Cauchy in both truncation orders.

References

• R. Coleman, p-adic Banach spaces and families of modular forms, Invent. Math. 127 (1997), 417–479, Appendix A3.

theorem Polynomial.norm_coeff_colemanDBounded_le {R : Type u_1} [SeminormedCommRing R] {B P : Polynomial R} {n m : ℕ} [NormOneClass R] [IsUltrametricDist R] {ρ β : ℕ → ℝ} (hB₀ : B.coeff 0 = 0) (hρ₀ : ∀ (i : ℕ), 0 ≤ ρ i) (hρ : Antitone ρ) (hP : ∀ j ≤ n, ‖P.coeff j‖ ≤ ∏ i ∈ Finset.range j, ρ i) (hβ : ∀ (t j : ℕ), ‖B.coeff (j + 1)‖ * ρ t ^ (j + 1) ≤ β t) (k : ℕ) : ‖(B.colemanDBounded P n m).coeff k‖ ≤ ∏ t ∈ Finset.range k, β t

Coleman's coefficient bound. Let P and B be polynomials over an ultrametric seminormed ring with ‖1‖ = 1 and B(0) = 0. If ρ is a nonnegative antitone sequence with ‖coeff j P‖ ≤ ∏ i < j, ρ i for j ≤ n, and ‖coeff l B‖ * ρ t ^ l ≤ β t for every t and every l ≥ 1, then the k-th coefficient of Coleman's transform D_(n,m)(B, P) has norm at most ∏ t < k, β t, for all ranks n and degree bounds m.

theorem Polynomial.norm_coeff_colemanDBounded_sub_le {R : Type u_1} [SeminormedCommRing R] {B P : Polynomial R} {n m : ℕ} [NormOneClass R] [IsUltrametricDist R] {B' P' : Polynomial R} {L k : ℕ} {ρ β γ : ℕ → ℝ} (hB₀ : B.coeff 0 = 0) (hρ₀ : ∀ (i : ℕ), 0 ≤ ρ i) (hρ : Antitone ρ) (hP : ∀ j ≤ n, ‖P.coeff j‖ ≤ ∏ i ∈ Finset.range j, ρ i) (hP' : ∀ j ≤ n, ‖P'.coeff j‖ ≤ ∏ i ∈ Finset.range j, ρ i) (hβ : ∀ (t j : ℕ), ‖B.coeff (j + 1)‖ * ρ t ^ (j + 1) ≤ β t) (hβ' : ∀ (t j : ℕ), ‖B'.coeff (j + 1)‖ * ρ t ^ (j + 1) ≤ β t) (hγ : ∀ (t l : ℕ), L < l → ‖B.coeff l‖ * ρ t ^ l ≤ γ t) (hγ' : ∀ (t l : ℕ), L < l → ‖B'.coeff l‖ * ρ t ^ l ≤ γ t) (hBB' : ∀ l ≤ L, B.coeff l = B'.coeff l) (hPP' : ∀ j ≤ k * L, P.coeff j = P'.coeff j) : ‖(B.colemanDBounded P n m).coeff k - (B'.colemanDBounded P' n m).coeff k‖ ≤ ∑ r ∈ Finset.range k, γ r * ∏ t ∈ (Finset.range k).erase r, β t

Continuity of Coleman's transform. Let (B, P) and (B', P') be two pairs of polynomials over an ultrametric seminormed ring with ‖1‖ = 1, both obeying the hypotheses of Polynomial.norm_coeff_colemanDBounded_le for the same ρ and β, with B(0) = 0. Suppose that coeff l B = coeff l B' for l ≤ L and coeff j P = coeff j P' for j ≤ k L, and that γ bounds the tails: ‖coeff l B‖ * ρ t ^ l ≤ γ t and ‖coeff l B'‖ * ρ t ^ l ≤ γ t for l > L. Then the k-th coefficients of the two Coleman transforms of rank n and degree bound m differ by at most ∑ r < k, γ r * ∏ t < k, t ≠ r, β t.

theorem Polynomial.dist_coeff_colemanDBounded_mul_le {R : Type u_1} [NormedCommRing R] [NormOneClass R] [IsUltrametricDist R] {B₁ B₂ P₁ P₂ : Polynomial R} {n m : ℕ} {ρ β : ℕ → ℝ} {r : ℝ} (hB₁ : B₁.coeff 0 = 0) (hB₂ : B₂.coeff 0 = 0) (hρ₀ : ∀ (i : ℕ), 0 ≤ ρ i) (hρ : Antitone ρ) (hr : 1 ≤ r) (hP₁ : ∀ j ≤ n, ‖P₁.coeff j‖ ≤ ∏ i ∈ Finset.range j, ρ i) (hP : ∀ j ≤ n, ‖(P₂ - P₁).coeff j‖ * r ≤ ∏ i ∈ Finset.range j, ρ i) (hβ₁ : ∀ (t j : ℕ), ‖B₁.coeff (j + 1)‖ * ρ t ^ (j + 1) ≤ β t) (hβ : ∀ (t j : ℕ), ‖(B₂ - B₁).coeff (j + 1)‖ * r * ρ t ^ (j + 1) ≤ β t) (k : ℕ) : dist ((B₂.colemanDBounded P₂ n m).coeff k) ((B₁.colemanDBounded P₁ n m).coeff k) * r ≤ ∏ t ∈ Finset.range k, β t

Coleman's perturbation bound. Let ρ be a nonnegative antitone sequence and r ≥ 1. If P₁ has Newton majorant ρ up to degree n, the perturbation P₂ - P₁ satisfies the same bound after multiplication by r, and the coefficients of positive degree of B₁ and of r (B₂ - B₁) are bounded by β at every radius ρ t, where B₁(0) = B₂(0) = 0, then the coefficients of the bounded Coleman transforms satisfy dist (coeff k D_(n,m)(B₂, P₂)) (coeff k D_(n,m)(B₁, P₁)) * r ≤ ∏ t < k, β t.

Thus perturbing the inputs by a factor r⁻¹ relative to their majorants moves every coefficient of the transform by at most r⁻¹ times its coefficient bound Polynomial.norm_coeff_colemanDBounded_le, uniformly in the ranks n and degree bounds m.

theorem PowerSeries.norm_coeff_colemanDBounded_trunc_le {R : Type u_1} [NormedCommRing R] [NormOneClass R] [IsUltrametricDist R] {F B : PowerSeries R} {ρ : ℕ → ℝ} (hρ₀ : ∀ (i : ℕ), 0 ≤ ρ i) (hρ : Antitone ρ) (hF : ∀ (j : ℕ), ‖(coeff j) F‖ ≤ ∏ i ∈ Finset.range j, ρ i) (hB : ∀ (i : ℕ), IsRestricted (ρ i) B) (hB₀ : (coeff 0) B = 0) (n m a b k : ℕ) : ‖(((trunc a) B).colemanDBounded ((trunc b) F) n m).coeff k‖ ≤ ∏ i ∈ Finset.range k, B.colemanMajorant (ρ i)

Coleman's bound for truncations. Let ρ be a nonnegative antitone Newton majorant of a power series F, ‖coeff j F‖ ≤ ∏ i < j, ρ i, and let B be restricted at every radius ρ i with B(0) = 0. Then the Coleman transforms of the truncations of F along the truncations of B satisfy ‖coeff k D_(n,m)(trunc a B, trunc b F)‖ ≤ ∏ i < k, β_B(ρ i), uniformly in the ranks, degree bounds and truncation orders. In particular this bounds D_(N,M)(trunc (M + 1) B, trunc (N + 1) F).

theorem PowerSeries.norm_coeff_colemanDBounded_trunc_sub_le {R : Type u_1} [NormedCommRing R] [NormOneClass R] [IsUltrametricDist R] {F B : PowerSeries R} {ρ : ℕ → ℝ} (hρ₀ : ∀ (i : ℕ), 0 ≤ ρ i) (hρ : Antitone ρ) (hF : ∀ (j : ℕ), ‖(coeff j) F‖ ≤ ∏ i ∈ Finset.range j, ρ i) (hB : ∀ (i : ℕ), IsRestricted (ρ i) B) (hB₀ : (coeff 0) B = 0) (n m : ℕ) {L k a a' b b' : ℕ} (ha : L < a) (ha' : L < a') (hb : k * L < b) (hb' : k * L < b') : ‖(((trunc a) B).colemanDBounded ((trunc b) F) n m).coeff k - (((trunc a') B).colemanDBounded ((trunc b') F) n m).coeff k‖ ≤ ∑ r ∈ Finset.range k, (B - ↑((trunc (L + 1)) B)).colemanMajorant (ρ r) * ∏ t ∈ (Finset.range k).erase r, B.colemanMajorant (ρ t)

Continuity of Coleman's transform in the truncations. Let ρ be a nonnegative antitone Newton majorant of a power series F, and let B be restricted at every radius ρ i with B(0) = 0. If the truncation orders satisfy L < a, a' and k L < b, b', then the k-th coefficients of the Coleman transforms D_(n,m)(trunc a B, trunc b F) and D_(n,m)(trunc a' B, trunc b' F) differ by at most ∑ r < k, β_(B - trunc (L + 1) B)(ρ r) * ∏ t < k, t ≠ r, β_B(ρ t), a bound involving the Coleman majorants of the tail of B beyond degree L.

theorem PowerSeries.cauchySeq_coeff_colemanDBounded_trunc {R : Type u_1} [NormedCommRing R] [NormOneClass R] [IsUltrametricDist R] {F B : PowerSeries R} {ρ : ℕ → ℝ} (hρ₀ : ∀ (i : ℕ), 0 ≤ ρ i) (hρ : Antitone ρ) (hF : ∀ (j : ℕ), ‖(coeff j) F‖ ≤ ∏ i ∈ Finset.range j, ρ i) (hF₀ : (coeff 0) F = 1) (hB : ∀ (i : ℕ), IsRestricted (ρ i) B) (hB₀ : (coeff 0) B = 0) (k : ℕ) : CauchySeq fun (p : ℕ × ℕ) => (((trunc (p.2 + 1)) B).colemanDBounded ((trunc (p.1 + 1)) F) p.1 p.2).coeff k

The finite Coleman transforms are Cauchy. Let F be a power series with F(0) = 1 and a nonnegative antitone Newton majorant ρ, and let B be restricted at every radius ρ i with B(0) = 0. Then, for each k, the k-th coefficients of the Coleman transforms D_(N,M)(trunc (M + 1) B, trunc (N + 1) F) form a Cauchy family as the rank N and the degree bound M tend to infinity independently.

theorem PowerSeries.dist_coeff_colemanDBounded_trunc_mul_le {R : Type u_1} [NormedCommRing R] [NormOneClass R] [IsUltrametricDist R] {F B : PowerSeries R} {ρ : ℕ → ℝ} (hρ₀ : ∀ (i : ℕ), 0 ≤ ρ i) (hρ : Antitone ρ) (hF : ∀ (j : ℕ), ‖(coeff j) F‖ ≤ ∏ i ∈ Finset.range j, ρ i) {c : ℝ} (hc : 1 ≤ c) (hB : ∀ (i : ℕ), IsRestricted (c ^ 2 * ρ i) B) (hB₀ : (coeff 0) B = 0) {a a' b b' : ℕ} (ha : a ≤ a') (hb : b ≤ b') (n m k : ℕ) : dist ((((trunc a') B).colemanDBounded ((trunc b') F) n m).coeff k) ((((trunc a) B).colemanDBounded ((trunc b) F) n m).coeff k) * c ^ min a b ≤ ∏ i ∈ Finset.range k, B.colemanMajorant (c ^ 2 * ρ i)

Continuity of Coleman's transform in the truncation orders. Let ρ be a nonnegative antitone Newton majorant of F, let c ≥ 1, and let B be restricted at every radius c² ρ i with B(0) = 0. Lengthening the truncations of F and B from orders b and a to b' ≥ b and a' ≥ a moves the k-th coefficient of every bounded Coleman transform by at most c ^ (-min a b) ∏ i < k, β_B(c² ρ i): the tails of F and B from these orders on are smaller than their majorants at the radii scaled by c.

theorem PowerSeries.dist_coeff_colemanDBounded_trunc_mul_trunc_le {R : Type u_1} [NormedCommRing R] [NormOneClass R] [IsUltrametricDist R] {F G B : PowerSeries R} {ρ : ℕ → ℝ} (hρ₀ : ∀ (i : ℕ), 0 ≤ ρ i) (hρ : Antitone ρ) (hFG : ∀ (j : ℕ), ∑ x ∈ Finset.antidiagonal j, ‖(coeff x.1) F‖ * ‖(coeff x.2) G‖ ≤ ∏ i ∈ Finset.range j, ρ i) {c : ℝ} (hc : 1 ≤ c) (hB : ∀ (i : ℕ), IsRestricted (c * ρ i) B) (hB₀ : (coeff 0) B = 0) (a b n m k : ℕ) : dist ((((trunc a) B).colemanDBounded ((trunc b) (F * G)) n m).coeff k) ((((trunc a) B).colemanDBounded ((trunc b) F * (trunc b) G) n m).coeff k) * c ^ b ≤ ∏ i ∈ Finset.range k, B.colemanMajorant (c * ρ i)

Let ρ be a nonnegative antitone sequence dominating the convolution of the coefficient norms of F and G, let c ≥ 1, and let B be restricted at every radius c ρ i with B(0) = 0. Replacing the product of the truncations of F and G at order b by the truncation of F * G moves the k-th coefficient of every bounded Coleman transform by at most c ^ (-b) ∏ i < k, β_B(c ρ i): the two polynomials agree below degree b.