  As a conclusion in classical linear algebra, an underdetermined linear
equations usually have an infinite number of solutions. The sparest one among
these solutions is significant in many applications. This problem can be
modeled as the $l_0$-minimization, However, to find the sparsest solution of an
underdetermined linear equations is NP-hard. Therefore, an important approach
to solve the following $l_p$-minimization ($0<p\leq1$), The purpose of this
problem is to find a $p$-norm minimization solution $(0<p\leq1)$ instead of the
sparest one.
  In order to study the equivalence relationship between $l_0$-minimization and
$l_p$-minimization, most of related work adopt Restricted Isometry Property
(RIP) and Restricted Isometry Constant (RIC). On the premise of RIP and RIC,
those work only solve the situation when the solution $\breve{x}$ of
$l_0$-minimization satisfies that $\|\breve{x}\|_0<k$ where $k$ is a known
fixed constant with $k<\frac{spark(A)}{2}$. One of the results in this paper is
to give an analytic expression $p^*$ such that $l_p$-minimization is equivalent
to $l_0$-minimization for every $\|\breve{x}\|_0<\frac{spark(A)}{2}$.
  In this paper, we also consider the case where the matrix $A$ is a
Vandermonde matrix and we present an analytic expression $p^*$ such that the
solution of $l_p$-minimization also solve $l_0$-minimization. Compared with the
similar results based on RIP and RIC, we do not need the uniqueness assumption,
i.e., the solution $x^*$ of $l_0$-minimization do not have to be assumed to be
the unique solution which is the main breakthrough in our result. Another
superiority of our result is its computability, i.e., each part in the analytic
expression can be easily calculated.
