Positive definite forms
f 2 R[x1, . . . , xn] which are sums of squares of forms of R[x1, . . . , xn] are constructed to have the additional property that the members of any collection of forms whose squares sum to f must share a nontrivial complex root in Cn.
Verchota, Gregory C., "Existence of Positive Definite Noncoercive Sums of Squares" (2007). Mathematics Faculty Scholarship. Paper 136.
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