On the evaluation at $(j,j^2)$ of the Tutte polynomial of a ternary matroid

International audienceF. Jaeger has shown that up to a $\pm$ sign the evaluation at $(j,j^2)$ of the Tutte polynomial of a ternary matroid can be expressed in terms of the dimension of the bicycle space of a representation over $GF(3)$. We give a short algebraic proof of this result, which moreover yields the exact value of $\pm$, a problem left open in Jaeger's paper. It follows that the computation of $t(j,j^2)$ is of polynomial complexity for a ternary matroid