Noncoercive sums of squares in \(\mathbb R[x_1,\ldots ,x_n]\) (Q1040597)

Motivated by questions about partial differential equations the author introduces the notion of a coercive form. This is a homogeneous polynomial \(f\) that is a sum of squares in the polynomial ring \(\mathbb R[X_1,\dots, X_n]\) and has at least one representation \(f= \sum^k_{i=1} p^2_i\) such that \(\{z\in\mathbb C^n\mid p_1(z)=\cdots= p_k(z)= 0\}= \{0\}\). The author asks whether there exist positive non-coercive forms with \(n\) variables and degree \(2\cdot d\). The answer is related to the existence of positive definite forms that are not sums of squares. It is shown that there are non-coercive forms of degree 4 if \(n\geq 6\) and of degree 6 if \(n\geq 4\). The constructions use the method of Gram matrices, which is familiar from representations of sums of squares of polynomials.