Before revision:  that are sums of sparse SOS matrices , each of which is zero outside a small principal submatrix is not compact or does not satisfy the Archimedean condition, we obtain a similar decomposition for (x_1^2 + \ldots + x_n^2)^\nu P(x) with some integer \nu\geq 0 when P and g_1,\ldots,g_m are homogeneous of even degree} . Using these decomposition results, we obtain sparse SOS representation theorems for polynomials that are quadratic and correlatively sparse in a subset of variables .
After revision:  that are sums of sparse SOS matrices . Finally, if \mathcal{K is not compact or does not satisfy the Archimedean condition, we obtain a similar decomposition for (x_1^2 + \ldots + x_n^2)^\nu P(x) with some integer \nu\geq 0 when P and g_1,\ldots,g_m are homogeneous of even degree} . Using these decomposition results, we obtain sparse SOS representation theorems for polynomials that are quadratic and correlatively sparse in a subset of variables .