  The concept of sums of nonnegative circuit polynomials (SONC) was recently
introduced as a new certificate of nonnegativity especially for sparse
polynomials. In this paper, we explore the relationship between nonnegative
polynomials and SONC polynomials. As a first result, we provide sufficient
conditions for nonnegative polynomials with general Newton polytopes to be SONC
polynomials, which generalizes the previous result on nonnegative polynomials
with simplex Newton polytopes. Secondly, we prove that every SONC polynomial
admits a SONC decomposition without cancellation. In other words, SONC
decompositions can exactly preserve the sparsity of nonnegative polynomials,
which is dramatically different from the classical sum of squares (SOS)
decompositions and is a key property to design efficient algorithms for sparse
polynomial optimization based on SONC decompositions.
