  If $f$ is a positive definite form, Reznick's Positivstellensatz
[Mathematische Zeitschrift. 220 (1995), pp. 75--97] states that there exists
$k\in\mathbf{N}$ such that ${\| x \|^{2k}_2}f$ is a sum of squares of
polynomials. Assuming that $f$ can be written as a sum of forms $\sum_{l=1}^p
f_l$, where each $f_l$ depends on a subset of the initial variables, and
assuming that these subsets satisfy the so-called running intersection
property, we provide a sparse version of Reznick's Positivstellensatz. Namely,
there exists $k \in \mathbf{N}$ such that $f=\sum_{l = 1}^p
{{\sigma_l}/{H_l^{k}}}$, where $\sigma_l$ is a sum of squares of polynomials,
$H_l$ is a uniform polynomial denominator, and both polynomials $\sigma_l,H_l$
involve the same variables as $f_l$, for each $l=1,\dots,p$. In other words,
the sparsity pattern of $f$ is also reflected in this sparse version of
Reznick's certificate of positivity. We next use this result to also obtain
positivity certificates for (i) polynomials nonnegative on the whole space and
(ii) polynomials nonnegative on a (possibly non-compact) basic semialgebraic
set, assuming that the input data satisfy the running intersection property.
Both are sparse versions of a positivity certificate due to Putinar and
Vasilescu.
