  We show that the sparse polynomial interpolation problem reduces to a
discrete super-resolution problem on the $n$-dimensional torus. Therefore the
semidefinite programming approach initiated by Cand\`es \\& Fernandez-Granda
\cite{candes\_towards\_2014} in the univariate case can be applied. We extend
their result to the multivariate case, i.e., we show that exact recovery is
guaranteed provided that a geometric spacing condition on the supports holds
and the number of evaluations are sufficiently many (but not many). It also
turns out that the sparse recovery LP-formulation of $\ell\_1$-norm
minimization is also guaranteed to provide exact recovery {\it provided that}
theevaluations are made in a certain manner and even though the Restricted
Isometry Property for exact recovery is not satisfied. (A naive sparse recovery
LP-approach does not offer such a guarantee.) Finally we also describe the
algebraic Prony method for sparse interpolation, which also recovers the exact
decomposition but from less point evaluations and with no geometric spacing
condition. We provide two sets of numerical experiments, one in which the
super-resolution technique and Prony's method seem to cope equally well with
noise, and another in which the super-resolution technique seems to cope with
noise better than Prony's method, at the cost of an extra computational burden
(i.e. a semidefinite optimization).
