2010/01/01 by David G. Glynn · 1 citation
Engineering · Mathematics · Computer Science · #graph theory and CDMA systems #Mathematics and Applications #Polynomial and algebraic computation #Mathematics #Combinatorics #Conjecture #Dimension (graph theory) #Vector space #Product (mathematics) #Matrix (chemical analysis) #Order (exchange) #Prime (order theory) #Basis (linear algebra) #Latin square #Finite field #Prime number #Permutation (music) #Discrete mathematics #Pure mathematics #Geometry
paper · doi:10.1137/090773751
openalex publication_date 2010/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
A formula for Glynn's hyperdeterminant detp (p prime) of a square matrix shows that the number of ways to decompose any integral doubly stochastic matrix with row and column sums p-1 into p-1 permutation matrices with even product, minus the number of ways with odd product, is 1 (mod p). It follows that the number of even Latin squares of order p-1 is not equal to the number of odd Latin squares of that order. Thus Rota's basis conjecture is true for a vector space of dimension p-1 over any field of characteristic zero or p, and all other characteristics except possibly a finite number. It is also shown where there is a mistake in a published proof that claimed to multiply the known dimensions by powers of two, and that also claimed that the number of even Latin squares is greater than the number of odd Latin squares. Now, 26 is the smallest unknown case where Rota's basis conjecture for vector spaces of even dimension over a field is unsolved.