  Mayr and Meyer found ideals $J(n,d)$ (in a polynomial ring in $10n+2$
variables over a field $k$ and generators of degree at most $d+2$) with ideal
membership property which is doubly exponential in $n$. This paper is a first
step in understanding the primary decomposition of these ideals: it is proved
here that $J(n,d)$ has $nd^2 + 20$ minimal prime ideals. Also, all the minimal
components are computed, and the intersection of the minimal components as
well.
