  We study fundamental solutions of elliptic operators of order $2m\geq4$ with
constant coefficients in large dimensions $n\ge 2m$, where their singularities
become unbounded. For compositions of second order operators these can be
chosen as convolution products of positive singular functions, which are
positive themselves. As soon as $n\geq3$, the polyharmonic operator
$(-\Delta)^m$ may no longer serve as a prototype for the general elliptic
operator. It is known from examples of [V.G. Maz'ya, S. A. Nazarov, Math. Notes
39 (1986); Transl. of Mat. Zametki 39 (1986)] and [E.B. Davies, Journal Differ.
Equations 135 (1997)] that in dimensions $n\ge 2m+3$ fundamental solutions of
specific operators of order $2m\geq4$ may change sign near their singularities:
there are ``positive'' as well as ``negative'' directions along which the
fundamental solution tends to $+\infty$ and $-\infty$ respectively, when
approaching its pole. In order to understand this phenomenon systematically we
first show that existence of a ``positive'' direction directly follows from the
ellipticity of the operator. We establish an inductive argument by space
dimension which shows that sign change in some dimension implies sign change in
any larger dimension for suitably constructed operators. Moreover, we deduce
for $n=2m$, $n=2m+2$ and for all odd dimensions an explicit closed expression
for the fundamental solution in terms of its symbol. From such formulae it
becomes clear that the sign of the fundamental solution for such operators
depends on the dimension. Indeed, we show that we have even sign change for a
suitable operator of order $2m$ in dimension $n=2m+2$. On the other hand we
show that in the dimensions $n=2m$ and $n=2m+1$ the fundamental solution of any
such elliptic operator is always positive around its singularity.
