  We study some systems of polynomials whose support lies in the convex hull of
a circuit, giving a sharp upper bound for their numbers of real solutions. This
upper bound is non-trivial in that it is smaller than either the Kouchnirenko
or the Khovanskii bounds for these systems. When the support is exactly a
circuit whose affine span is ${\Z}^n$, this bound is $2n+1$, while the
Khovanskii bound is exponential in $n^2$. The bound $2n+1$ can be attained only
for non-degenerate circuits. Our methods involve a mixture of combinatorics,
geometry, and arithmetic.
