On the Pythagoras Numbers of Real Analytic Rings

The Pythagoras number of a ring A is the smallest integer p A = p ≥ 1 such that any sum of squares of A is a sum of p squares, and p A = +∞ if such an integer does not exist. This is a very delicate invariant whose study has deserved a lot of attention from specialists in number theory, quadratic forms, real algebra, and real geometry. Wellknown examples are the following: p = 4 (Lagrange’s famous theorem), n + 2 ≤ p x1 xn ≤ 2 [Pf, CEP], p x1 xn = +∞ for n ≥ 2 [ChDLR]. We refer the reader to [ChDLR, BCR] for further details. A special important case is that of local rings. The most general result here is that p A = +∞ for any local regular ring A of dimension ≥ 3 [ChDLR]; there are also several finiteness results for local regular rings of dimension 2 in the so-called geometric cases [Sch]. However, there is a serious lack of information without the regularity assumption. In this paper we deal with this matter for real local analytic rings, that is, for the local rings of real analytic spaces.