The c=2 q-Carlitz Smith Form and q-Pascal Matrix Structures

Abstract

This paper solves the complete c = 2 case of the Kuperberg–Conlon q-Carlitz Smith-form problem over the Laurent polynomial ring ℤ[q,q⁻¹]. For every a,b ≥ 2, the q-Carlitz matrix C(a,b,2;q) is proved to be unimodularly equivalent over ℤ[q,q⁻¹] to an explicit two-factor Smith form diag(G, Δ/G), where G is the monic greatest common divisor of three adjacent Gaussian binomial coefficients and Δ is the determinant of the associated 2 × 2 reduction. Both invariant factors are proved to be squarefree q-round polynomials, with G ∣ Δ/G. Thus Smith normal form exists throughout the entire c = 2 family despite the fact that ℤ[q,q⁻¹] is not a principal ideal domain. A central ingredient is a general adjacent-Gaussian-binomial Bézout theorem. If Aⱼ(q) = [n choose j]₍q₎ and d = gcd(j+1,n−j), then the ideal generated by two adjacent Gaussian coefficients is principal: (Aⱼ,Aⱼ₊₁) = (gⱼ), with gⱼ = Aⱼ[d]₍q₎/[j+1]₍q₎. Equivalently, there exist Uⱼ,Vⱼ ∈ ℤ[q,q⁻¹] satisfying UⱼAⱼ + VⱼAⱼ₊₁ = gⱼ. This proves the Bézout identity proposed in the c = 2 q-Carlitz reduction. The Smith factors admit an explicit cyclotomic packet decomposition. Gaussian-binomial cyclotomic exponents are controlled by residue carries, and the determinant factors into disjoint cyclotomic packets. Resultant arguments then establish the pairwise Bézout relations required to perform the Smith reduction over the non-PID coefficient ring. The universal equivalence is preserved by arbitrary commutative base change. Consequently, the theorem determines the q-Carlitz cokernel, all Fitting ideals, and the exact rank and nullity at every nonzero scalar specialization. In particular, coker C(a,b,2;q) ≅ A/(G) ⊕ A/(Δ/G) after any base change ℤ[q,q⁻¹] → A. The cyclotomic support therefore gives the exact rank drop at roots of unity and at all other nonzero field specializations. Both Smith factors are nonunits for every a,b ≥ 2, proving Conlon’s conjectured count of nonzero nonunit invariant factors throughout the complete c = 2 family: 2 = min(a,b,2). The paper also develops the q-Pascal and coefficient-ring machinery surrounding the main theorem. It gives exact LDLᵀ and inverse formulas for symmetric q-Pascal matrices, geometric-node connection transforms, Gaussian q-Catalan identities, q-Pell Hankel reductions and factorization obstructions, tensor-product multivariate extensions, Gaussian row-sum and convolution identities, coefficient-ring boundaries for (p,q)-Pascal factorizations, and cyclotomic q-Lucas Kronecker self-similarity. These results provide the structural setting for the q-Carlitz Smith-form theorem and clarify which identities hold polynomially and which require localization.