Truncated Toeplitz Unit Groups and Exact Reciprocal-Support Semigroups

Abstract

This paper develops the algebraic theory of weighted lower-triangular Toeplitz matrices as finite truncated-convolution operators and determines exactly which coefficients can occur in their reciprocals. The principal result is a reciprocal-support theorem valid in characteristic zero, positive characteristic, prime-power coefficient rings, multilevel truncations, and arbitrary finite Artinian monomial quotients. For a finitely supported polynomial f(z) = ∑ₛ∈S cₛzˢ, the coefficients of (1 + f(z))⁻¹ modulo zⁿ are given by an exact finite multinomial expansion. Over characteristic zero, the universal nonzero support of the reciprocal is precisely the truncated additive semigroup generated by S: supp((1 + f)⁻¹) = {0} ∪ (⟨S⟩ ∩ {1,…,n−1}) for a Zariski-generic specialization of the coefficients. When gcd(S) = 1, the reciprocal is therefore generically dense beyond the conductor of the numerical semigroup ⟨S⟩. In characteristic p, the support undergoes an exact arithmetic refinement. A reciprocal coefficient in degree d survives modulo p exactly when d admits a representation d = ∑ₛ smₛ for which the base-p addition of the multiplicities mₛ is carry-free. More generally, over ℤ/pᵉℤ, survival is determined by the minimum p-adic valuation of the associated multinomial coefficients. If κₚ(d;S) = min vₚ((∑ₛmₛ)!/∏ₛmₛ!), taken over all representations d = ∑ₛ smₛ, then the degree-d reciprocal coefficient vanishes identically modulo pᵉ exactly when κₚ(d;S) ≥ e. Thus reciprocal support is governed by ordinary additive-semigroup membership in characteristic zero and by an exact multinomial carry filtration in prime-power characteristic. The theorem extends without change of principle to several variables. For rectangular truncations in variables z₁,…,z_d, reciprocal support is the intersection of the affine semigroup ⟨S⟩ ⊆ ℕᵈ with the surviving truncation box. More generally, for every finite-colength monomial ideal J, the reciprocal support in R[z₁,…,z_d]/J is exactly the surviving part of ⟨S⟩ in characteristic zero, with the same carry and p-adic valuation criteria over 𝔽ₚ and ℤ/pᵉℤ. The support theorem is therefore intrinsic to affine semigroups and standard monomials rather than to one-dimensional Toeplitz matrices alone. The surrounding matrix theory is developed in parallel. Weighted triangular Toeplitz matrices are identified exactly with multiplication operators in ℂ[z]/(zⁿ), yielding the truncated convolution law, determinant and spectrum formulas, a complete minimal-polynomial and Jordan-block classification, exact image and kernel ideals, and weight-independent inversion formulas. If the first nonconstant coefficient occurs in degree r, the nilpotency index of the nonconstant part is ⌈n/r⌉, which gives the sharp termination length of the finite Neumann inverse. Kernel factorizations are shown to be equivalent to factorizations in the truncated polynomial algebra, and exact deconvolution fibers are described algebraically. Parametric geometric, binomial, and binomial-geometric kernel families form explicit additive-group representations under convolution, with finite logarithmic generators and precisely characterized parameter kernels. The scalar calculus extends to block-valued lower-triangular Toeplitz matrices. In the noncommutative block setting, invertibility is equivalent to invertibility of the constant block, the determinant is det(F₀)ⁿ, and the convolution inverse is given by an exact ordered finite Neumann expansion. The theory includes compatible truncation laws and direct structured algorithms with explicit arithmetic costs. Over characteristic-zero fields, the principal unit group 1 + 𝔪 of K[z]/(zⁿ) is polynomially isomorphic to the additive nilpotent ideal 𝔪 through the finite logarithm and exponential. Consequently, (𝔪,+) ≅ (1 + 𝔪,×), Zariski-closed principal-unit subgroups are exactly exponentials of linear subspaces, and every algebraic one-parameter subgroup has the form t ↦ exp(th). This finite-dimensional structure is related explicitly to logarithmic factor coordinates in weighted Wiener algebras. Together these results give an exact algebraic and arithmetic description of truncated Toeplitz unit groups, inversion, reciprocal support, semigroup propagation, carry obstructions, and their multivariate and block-valued extensions.