Positive Semidefinite Germs on the Cone

The problem of representing a positive semidefinite function (=psd) as a sum of squares (=sos) is a very old matter in real algebra and real geometry. Still, it is a difficult question always appealing the specialists. Concerning real analytic germs we can summarize what is known in a few statements. Let X be a irreducible real analytic set germ of dimension d. Any psd f of X is an sos of meromorphic germs (see [AnBrRz]) and the qualitative question is whether every psd is an sos of analytic germs. The quantitative question: How many squares, can refer either to meromorphic or analytic germs. We know the answer to the qualitative question in the regular case: yes for the line and the plane, no otherwise (see [Rz1]). For singular curve germs we immediately realize that the answer is no. For singular surfaces we have many examples where the answer is no, and think there are very few where the answer is yes ([Rz2]). For higher dimensional germs we guess the answer is no. Concerning the quantitative matter, in the regular case the sharp bound for the number of squares of analytic germs is d for d = 1, 2 (see [Rz1]) and there are sos’s of arbitrary length for d ≥ 3 ([Ch et al ]); bounds for the number of squares of meromorphic germs are 1, 2 for d = 1, 2 (see [Rz1]) and 8 for d = 3 ([Jw]), (the general conjecture appears to be 2d−1, but for d > 3 no bound is available yet). In the singular case the only systematic works concern curves ([Or], [Qz]): For sos’s of meromorphic germs the bound is trivially 1, and for analytic germs the bound is the multiplicity (a conjecture suggested by Becker and proved by Quarez, and that seems very good as there can be no uniform bound for curve germs in R3). The next case are surface germs. Concerning this we know that twice the multiplicity is a bound for the number of squares of meromorphic germs, and that is all. We can add that for d ≥ 4 there are always sos’s of analytic germs of arbitrary length ([Rz2]), and notice the gap of knowledge for singular X of dimension 3. With this state of affairs, the simplest possible question, putting together the quantitative and qualitative aspects is: For which surface germs any psd