By a "diagonal minus tail" form (of even degree) we understand a real homogeneous polynomial
F(x_1,...,x_n)=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 b_i\geq 0 and the tail T(X) a sum of terms a_{i_1i_2...i_n}x_1^{i_1}... x_n^{i_n} with a_{i_1i_2...i_n}> 0
and at least two i_\nu\geq 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 (sos) of polynomials. The work uses Reznick's theory of agiforms [Re] and gives easily tested sufficient conditions for a form to be sos; one of these is piecewise linear in the coefficients of a polynomial and reminiscent of Lassere's recent conditions [La] but proved in completely a different manner.[La] J. B. Lasserre, Sufficient conditions for a polynomial to be a sum of squares, Arch. Math. 89, 390-398 (2007).
[Re] B. Reznick, Forms derived from the arithmetic geometric inequality, Math. Ann. 283, 431-464, (1989
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