On the Pythagoras numbers of real analytic set germs

We show that: (i) the Pythagoras number of a real analytic set germ is the supremum of the Pythagoras numbers of the curve germs it contains, and (ii) every real analytic curve germ is contained in a real analytic surface germ with the same Pythagoras number (or Pythagoras number 2 if the curve is Pythagorean). This gives new examples and counterexamples concerning sums of squares and positive semidefinite analytic function germs. 1 Preliminaries and statement of results 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 invariant appeals specialists from many different areas, and has a very interesting behaviour in geometric cases; we refer the reader to [BCR], [CDLR], [Sch1] and [Sch2]. Here we are interested in the important case of real analytic germs, which have been extensively studied in [CaRz], [Or], [Qz], [FeQz], [Rz], [FeRz], [Fe1], [Fe2], [Fe3]. Let X ⊂ Rn be a real analytic set germ and O(X) its ring of analytic function germs. Since O(Rn) is the ring R{x} of convergent power series in x = (x1, . . . , xn), we have O(X) = R{x}/J (X), where J (X) stands for the ideal of analytic function germs vanishing on X. We will discuss the Pythagoras number p[X] = p(A) of the ring A = O(X). Clearly, if we have another real analytic set germ Y ⊂ X, then J (Y ) ⊃ J (X) and the canonical surjection O(X) → O(Y ) gives immediately the inequality p[Y ] ≤ p[X]. This easy remark can be sharpened as follows: ∗Supported by Spanish GAAR BFM2002-04797 and European RAAG HPRN-CT-2001-00271