On a Conjecture Concerning Dyadic Oriented Matroids

  • Matt Scobee


A rational matrix is totally dyadic if all of its nonzero subdeterminants are in $\{\pm 2^k\ :\ k \in {\bf Z}\}$. An oriented matriod is dyadic if it has a totally dyadic representation $A$. A dyadic oriented matriod is dyadic of order $k$ if it has a totally dyadic representation $A$ with full row rank and with the property that for each pair of adjacent bases $A_1$ and $A_2$ $$2^{-k} \le \left| { {\det(A_1)} \over {\det(A_2)}}\right|\le 2^k.$$

In this note we present a counterexample to a conjecture on the relationship between the order of a dyadic oriented matroid and the ratio of agreement to disagreement in sign of its signed circuits and cocircuits (Conjecture 5.2, Lee (1990)).

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