A finitely generated quadratic module in a real polynomial ring is stable when the sums of squares appearing in a representation can be kept below a degree bound depending only on the degree of the polynomial represented. Students learn how to define stability relative to a grading of the polynomial algebra by an ordered abelian group, why total stability of a quadratic module is equivalent to the highest-degree parts of its generators having trivial support, and how that turns into a geometric test: the described set must contain a tentacle, the region swept from a compact set with nonempty interior by scaling each coordinate with its own weight. Combining gradings that cover the ordinary degree grading then decides stability, and hence closedness and the failure of the Strong Moment Property, from the directions in which the set runs off to infinity.
A quadratic module in the real polynomial ring R[X1,...,Xn] is a set of polynomials containing 1 and closed under addition and under multiplication by squares; the one generated by f1,...,fs consists…