The conjectures of Alon-Tarsi and Rota in dimension prime minus one

David Glynn

    Research output: Contribution to journalArticle

    23 Citations (Scopus)

    Abstract

    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 - 1is 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.

    Original languageEnglish
    Pages (from-to)394-399
    Number of pages6
    JournalSiam Journal on Discrete Mathematics
    Volume24
    Issue number2
    DOIs
    Publication statusPublished - 2010

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