Positive semidefinite diagonal minus tail forms are sums of squares

By a diagonal minus tail form (of even degree) we understand a real homogeneous polynomial F(x1, . . . , xn) = F(x) = D(x) − T(x), where the diagonal part D(x) is a sum of terms of the form $${b_i x_i^{2d}}$$ with all bi ≥ 0 and the tail T(x) a sum of terms $${a_{i_1i_2\cdots i_n}x_1^{i_1}\cdots x_n^{i_n}}$$ with $${a_{i_1i_2\cdots i_n} > 0}$$ and at least two iν ≥ 1. We show that an arbitrary change of the signs of the tail terms of a positive semidefinite diagonal minus tail form will result in a sum of squares of polynomials. We also give an easily tested sufficient condition for a polynomial to be a sum of squares of polynomials (sos) and show that the class of polynomials passing this test is wider than the class passing Lasserre’s recent conditions. Another sufficient condition for a polynomial to be sos, like Lasserre’s piecewise linear in its coefficients, is also given.