2018/01/15 by Oliver Knill, Knill, Oliver · 5 citations
Computer Science · Mathematics · #05Exx #15A36 #58J50 #Advanced Algebra and Geometry #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #advanced mathematical theories #cs.DM #math.CO #math.SP #msc:05Exx #msc:15A36 #msc:58J50
paper · pdf · doi:10.48550/arxiv.1801.04639
37 pages, 12 figures
arxiv created 2018/01/15 · openalex publication_date 2018/01/15 · arxiv updated 2018/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The connection zeta function of a finite abstract simplicial complex G is defined as zetaL(s)=sumx 1/lambdaxs, where lambdax are the eigenvalues of the connection Laplacian L defined by L(x,y)=1 if x and y intersect and 0 else. (I) As a consequence of the spectral formula chi(G)=sumx (-1)dim(x) = p(G)-n(G), where p(G) is the number of positive eigenvalues and n(G) is the number of negative eigenvalues of L, both the Euler characteristic chi(G)=zeta(0)-2 i zeta'(0)/pi as well as determinant det(L)=ezeta'(0)/pi can be written in terms of zeta. (II) As a consequence of the generalized Cauchy-Binet formula for the coefficients of the characteristic polynomials of a product of matrices we show that for every one-dimensional simplicial complex G, the functional equation zeta(s)=zeta(-s) holds, where zeta(s) is the Zeta function of the positive definite squared connection operator L2 of G. Equivalently, the spectrum sigma of the integer matrix L2 for a 1-dimensional complex always satisfies the symmetry sigma = 1/sigma and the characteristic polynomial of L2 is palindromic. The functional equation extends to products of one-dimensional complexes. (III) Explicit expressions for the spectrum of circular connection Laplacian lead to an explicit entire zeta function in the Barycentric limit. The situation is simpler than in the Hodge Laplacian H=D2 case where no functional equation was available. In the connection Laplacian case, the limiting zeta function is a generalized hypergeometric function which for an integer s is given by an elliptic integral over the real elliptic curve w2=(1+z)(1-z)(z2-4z-1), which has the analytic involutive symmetry (z,w) to (1/z,w/z2).