Common roots and periodic trace constraints
Abstract
Common roots at the designated defect site satisfy the periodic Chebyshev equations and vanish on the strict trace product.
Let K be a field with an algebra over the real numbers. Let r and m be natural numbers at least three, and let t and mu belong to K, with t transcendental over the real numbers. On Fin(rm), set q=(m-1)r+1 and U(i)=-1 exactly when i modulo r is zero or i=q, and U(i)=1 otherwise. The Hamiltonian H is the ordinary cycle graph Laplacian plus t diag(U). The matrix H^(q) is its actual principal submatrix deleting q; the remaining diagonal entries retain the value 2+tU(i).
Put T(d)=[[d,-1],[1,0]], a=2+t-mu, b=2-t-mu and sigma=trace(T(a)^(r-1)T(b)). The polynomials S_n satisfy S_0=1, S_1=X and S_(n+2)=X S_(n+1)-S_n. Let traceFactors be the product of sigma_r(lambda,t)-2cos(2 pi k/m) over 0<k and 2k<m. The hypotheses say that mu is a common root of the characteristic polynomials of H and H^(q).
Theorem 1.1 (The two Chebyshev equations).
Proof. Machine-checked in Lean as D5/S3/Combinatorics/Graph/Walks/AndersonDefectExactNodal.common_root_trace_constraints (✓ std3). ∎
Source. Repository-derived.
Commentary.
For a symmetric matrix and a singular principal minor, a nonzero kernel vector vanishing at the deleted coordinate is obtained by zero extension and the symmetry of the bilinear pairing. Applied to mu I-H, this gives an actual nodal eigenvector at q.
The cycle equation transports this vector through the rotated block C=T(a)T(b)T(a)^(r-2). Two consecutive zero coordinates would force the vector to vanish throughout the cycle. Hence its initial nonzero coordinate can be divided out, giving C^m(1,0)=(1,0). The trace of C is sigma.
Write s=S_(r-2)(a), v=S_(r-3)(a) and h=bs-v. The first column of C is (ah-s,h). If h=0, then (-s)^m=1. Since r is at least three, this makes a algebraic over the real numbers; s and v are then algebraic, and b=v/s is algebraic. The equality t=(a-b)/2 contradicts transcendence. Thus h is nonzero. The two coordinates of the determinant-one Chebyshev power formula now give the displayed equations.
The first Chebyshev equation places sigma/2 among the real roots of U_(m-1), even when the common root lies in an extension field. The second equation and T_m=X U_(m-1)-U_(m-2) give T_m(sigma/2)=1. The real extremum classification forces an even angular index, while the roots of U_(m-1) exclude both endpoints. Thus sigma=2cos(2 pi k/m) for an index with 0<k and 2k<m, and the strict trace product vanishes at mu.
The conclusion concerns common-root containment. It does not assert equality of the monic gcd with the periodic trace product, its exact degree, or squarefreeness.