  Finding the sparsest solutions to a tensor complementarity problem is
generally NP-hard due to the nonconvexity and noncontinuity of the involved
$\ell_0$ norm. In this paper, a special type of tensor complementarity problems
with $Z$-tensors has been considered. Under some mild conditions, we show that
to pursuit the sparsest solutions is equivalent to solving polynomial
programming with a linear objective function. The involved conditions guarantee
the desired exact relaxation and also allow to achieve a global optimal
solution to the relaxed nonconvex polynomial programming problem. Particularly,
in comparison to existing exact relaxation conditions, such as RIP-type ones,
our proposed conditions are easy to verify.
