ON THE PYTHAGORAS NUMBERS OF REAL ANALYTIC CURVES. Francesca Acquistapace, Fabrizio Broglia, José F.Fernando, Jesús M. Ruiz.

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1 ON THE PYTHAGORAS NUMBERS OF REAL ANALYTIC CURVES Francesca Acquistapace, Fabrizio Broglia, José FFernando, Jesús M Ruiz December 9, 005 Abstract We show that the Pythagoras number of a real analytic curve is the supremum of the Pythagoras numbers of its singularities, or that supremum plus This includes cases when the Pythagoras number is infinite AMS Subject Classification: Primary 4P99; Secondary E5, 3B0, 3S05 Keywords: Pythagoras number, sum of squares, analytic curve germ, global analytic curve Statements The problem of representing a positive semidefinite function (= psd) as a sum of squares (= sos), and of how many, is a very old matter appealing the specialists from widespread different areas The quantitative question is formulated for any ring of functions A in terms of the Pythagoras number p(a), which is the smallest p such that every sum of squares of A is a sum of p squares This p may well be infinity, and even deciding this can be very difficult We refer to the beautiful paper [CDLR] for a survey of most of what is known on the topic Later advances can be found in [Sch], [Sch] and [FRS] In this note we address the problem for real analytic curves To start with, we consider the local case Let X x beareal analytic curve germ (at a point x R n ), O(X x ) its ring of analytic function germs, and M(X x ) its ring of meromorphic function germs One immediately sees that every psd is the square of one All authors supported by European RAAG HPRN-CT ; first and second named authors also by Italian GNSAGA of INdAM and MIUR, third and fourth by Spanish GEOR MTM

2 meromorphic function germ, which consequently leads to concentrate on analytic function germs The question whether every psd analytic function germ is an sos of analytic function germs (psd = sos, in short) is also solved: this holds true if and only if X x is a union of independent non-singular branches (which means that the tangent lines of those branches are independent [Sch]) Now, we turn to the computation of the Pythagoras number p(o(x x )), which is always finite ([Rz]), and bounded in terms of the multiplicity of X x ([Qz], [FeQz]) Curve germs with minimum Pythagoras number are completely characterized: they are the so-called Arf curve germs ([CaRz]), of relevance in other contexts Here, for instance, the above mentioned unions of independent non-singular branches are Arf; hence, the curve germs for which psd = sos have all Pythagoras number On the other hand, the Pythagoras number of a curve germ can be arbitrarily large ([Or]) This was first proved through an explicit construction of curve germs with large embedding dimension, but later it has been discovered that curve germs with large Pythagoras number are ubiquitous ([FeRz]): Every semianalytic germ Z x of dimension d 3 contains (punctured) irreducible curve germs with arbitrarily large Pythagoras number If d, Z x contains curve germs with as large as possible Pythagoras number, the bound being the Pythagoras number of the analytic closure of Z x These local constructions are of importance to produce examples in the global analytic case All in all, there is a good systematic knowledge of p(o(x x )), even from an algorithmic viewpoint Consider now the global case Let X R n beareal analytic curve, O(X) its ring of global analytic functions and M(X) its ring of global meromorphic functions Here, every psd is an sos of global meromorphic functions; moreover, p(m(x)) = if some irreducible component of X is compact, p(m(x)) = otherwise (see [Jw] for X nonsingular; the general case can be deduced by normalization) Thus we look at analytic functions What is known so far is that for non-singular X (that is, when X is a disjoint union of circles and lines), every psd analytic function on X is an sos of analytic functions, and p(o(x)) = p(m(x)) ([Jw]) The goal of this short note is to analyse the behaviour of psd analytic fuctions and sos of analytic functions in the presence of singularities, which is far more involved The main concern is the computation of the Pythagoras number p(o(x)), but before that, and to make the discussion complete, let us say that the property that every psd analytic function on X is a sos of analytic functions (psd = sos for X, inshort) holds very rarely, as shows the following result: Proposition Let X be a real analytic curve The following conditions are equivalent: (i) The property psd = sos holds for X (ii) The property psd = sos holds for every germ X x, x X

3 (iii) Every germ X x, x X, isaunion of non-singular independent branches If that is the case, then p(o(x)) Thus we see that for real analytic curves the property psd = sos is local, and when it holds all singularities are highly special However, it is remarkable that while being psd is (trivially) a local property, being an sos needs not In fact, There are real analytic curves X R n (n 3) and psd global analytic functions f on X, such that every germ f x, x X, isansos of analytic function germs on X x, without f being an sos of global analytic functions After this preamble, our main result is the following: Theorem Let X be areal analytic curve in an open subset Ω of R n Then where ε =0or p(o(x)) = sup p(o(x x )) + ε, x X Note that since p(o(x x )) = for every regular point x X, this computes the Pythagoras number of X in terms of the Pythagoras numbers of its singularities An interesting consequence of this result is that There are analytic curves X R n (n 3), with infinite Pythagoras number: p(o(x)) = + Examples 3 Consider the following nine curves X i R ( i 9), which are singular or non-singular, compact or non compact, irreducible or reducible Note that since p(o(r )) =, all of them have Pythagoras number p We look at their Pythagoras numbers p, totheir residues ε as in Theorem, and to whether the property psd=sos holds or not for them, and find many different possibilities 3

4 curve X i R p(o(x i )) ε psd = sos an sos which is not a square X : y =0 line 0 YES X : y 3 = x cusp 0 NO X 3 : xy =0 independent lines 0 YES X 4 : y = x 4 two parabolas 0 NO (0, 0) X 5 : y 3 = x 4 singularity E 6 0 NO x + y X 6 : x + y = (, 0) circle YES (x ) + y (, 0) X 7 : y = x x 4 transversal eight YES (x ) + y (, 0) X 8 : y = x x 3 node YES (x ) + y (, 0) X 9 : y = x 4 x 6 tangent eight NO (x ) + y 4

5 Note that the curve X 5 has Pythagoras number because its singularity (at (0, 0)) has this Pythagoras number For the curves X i with 6 i 9, which have also Pythagoras number, the situation is different because their singularities have all Pythagoras number (since they are Arf) To find a function f which is a sum of two squares in O(X i ) but not a square, one proceeds as follows Consider a polynomial f of degree which is a sum of two squares of polynomials and only vanishes at a regular point x of X i which is contained in aloopofx i (see the previous pictures) Next, if f has a square root h O(X i ), then h cannot change sign because it only vanishes at x and this point does not disconnect X i Now, using a regular parametrization of X i at x, one sees that h must have order at x and consequently, must change sign at that point, a contradiction Proofs We begin by the Proof of Proposition First, we see that (i) implies (ii) We assume psd = sos for X, and will prove psd = sos for the germ X a at an arbitrary point a X Pick a psd analytic function germ f a O(X a ), and let us see that f a is an sos As every branch of X a can be parametrized, to be psd and to be an sos are properties that depend solely on finitely many terms of the Taylor expansion of f a ([Or]) In view of this, we are reduced to prove the following: Given f = ν m a ν x ν xνn n, g = f + λ(x + + x n) m (λ>0), where the germ g a is psd, then the global analytic function g is psd (hence an sos by (i)) for a suitable λ This λ is easily found Indeed, as the germ g a is psd, we can choose η>0small enough such that g is psd on X { x <η} Consequently we only care about X { x η} But there we have: f = a ν x ν xνn n a ν x ν x n νn ν m ν m ν m a ν x ν ν m a ν x m η m ν = λ x m, with λ = ν m a ν η m ν Weare done, and the first implication is proved Next, note that the fact that (ii) implies (iii) is the local result quoted at the begining of the Introduction ([Sch, 5]) 5

6 Finally, we prove that (iii) implies (i) We assume (iii) and will show that every psd global analytic function f on X is a sum of squares of global analytic functions Since every germ X x is a union of independent nonsingular branches, it has the property psd = sos and p(o(x x )) = Using this, one easily finds a locally principal sheaf of ideals J O X such that J = fo X But then J is generated by two global analytic functions g, h ([Co]) Note that for all a X there is a well defined square root f a O(X a )off a, and there exist ζ a,θ a O(X a ) such that g a = ζ a fa, h a = θ a fa Moreover, if f a 0(in O(X a )) then at least one of ζ a,θ a is a unit Hence, for all a X such that f a 0there exists a unit u a O(X a ) such that (g + h ) a = u a f a Now, let X X be the union of the irreducible components of X over which f is not identically 0 and X X the union of the others From the fact that for all a X with f a 0there exists a unit u a O(X a ) such that f a = u a (g + h ) a in O(X a ), we deduce f g +h that there exist an open subset Ω of Ω containing X so that the function α = is strictly positive and analytic in X Ω Clearly, αg, αh vanish on X Ω, hence both can be extended by zero on X, which gives a representation of f as a sum of squares on the whole X = X X Next, we prove the following: Proposition There are real analytic curves X R n (n 3) and psd global analytic functions f on X, which are not sos of global analytic functions, but every germ f x, x X, is an sos of analytic function germs on X x Proof Indeed, this is an straightforward argument based on analytic approximation and Cartan s Theorems A and B Choose a discrete set of points x k R n, and denote by m k O(R n x k ) their maximal ideals We can draw through x k a small representative of a germ X xk with Pythagoras number p k >k(these germs exist as explained in the introduction) Next, we connect smoothly each representative with the next to obtain a smooth curve Y off the x k s, and such that Y xk = X xk for all k Then we approximate Y by areal analytic curve X whose singular points are the x k s, and whose germs at them are (equivalent to) the X xk s Now for each k we choose an analytic germ f xk which is a sum of squares in X αk but not a sum of p k k squares Next, we find an analytic function g on X such that g xk f xk m m k k + J (X xk ) As was said before, for m k large enough, g xk and f xk behave the same concerning sos Whence g xk is an sos of analytic function germs, but not a sum of p k k squares 6

7 ([Or]), and consequently g is not either We conclude that g cannot be an sos of global analytic functions Now we turn to the main result Proof of Theorem First, let x k (k ) be the singular points of X, and set p(o(x xk )) = p k Fix k Wehave an sos of analytic function germs f xk = i f i,x k which is not a sum of p k squares in O(X xk ) Then there are global analytic functions g i such that g i,xk f i,xk m m k k + J (X xk ) Now we consider g = i g i, and have g xk f xk = i (g i,xk + f i,xk )(g i,xk f i,xk ) m m k k + J (X xk ) Again, for m k large, g xk is not a sum of p k squares in O(X xk ), which implies g is not either Thus, p(o(x)) p k for all k After this, we must prove the inequality p(o(x)) sup p(o(x x ))+ x X We may assume that p = sup x X p(o(x x )) < + ; otherwise, there is nothing to prove We denote by O X the sheaf of germs of analytic functions on X, which is a coherent sheaf, because analytic curves are always coherent Explicitely, O(X) =Γ(O X,X) and O(X x )=O X,x for x X Let f : X R be an analytic function on X which is a sum of squares in O(X) The O X -module M = O X /f O X is coherent, and we are to define a global cross section ξ of M The support of M consists of the zeros of f, and to look at them, we consider once again the union X (resp X )ofthe irreducible components of X on which f does not vanish (resp does vanish) identically The zeros of f split into some isolated ones y l X, and all the points in X Note that there are maybe nonisolated zeros through which some irreducible components are in X and some others not: these zeros z k are singular points of X, and important for the sequel By hypothesis, at each zero x = y l, z k there are p analytic germs α i,x O X,x such that f x = p i= α i,x in the local ring O X,x Wedefine our global cross-section ξ i Γ(M,X)by { αi,x mod f ξ i,x = O X,y for x = y l,z k, 0 mod f O X,y otherwise Notice that this cross-section is well-defined, because at a point x = z k, the function α i,x vanishes identically on the branches of X x on which f vanishes 7

8 Now, by Cartan s Theorem B, the homomorphism O X =Γ(O X,X) Γ(M,X)is surjective Hence, there is a global analytic function a i O(X) such that { a i,x α i,x f O X,x for all x = y l,z k, and a i,x f O X,x otherwise In particular, a i vanishes on the zeros of f Moreover, for each x = y l,z k there is an analytic germ λ i,x O X,x with a i,x = α i,x + λ i,x f in O X,x Hence, i a i,x = i = i (α i,x + λ i,x f ) α i,x + i α i,x λ i,x f + i (λ i,x f ) = f + α i,x λ i,x f + i i ( = f + α i,x λ i,x f + i i λ i,xf 4 λ i,xf 3) = f( + γ x )=θ x f, where the germ γ x vanishes at x, and the germ θ x =+γ x is a unit in O X,x Next, denote A = p i= a i Then, u = f is a well defined strictly positive A+f analytic function on a neighborhood W of X This is because: f and A + f are both positive off f (0), and for all x = y l,z k, the quotient f x A x + f x = θ x + f x is a positive unit in O X,x Thus, on X W we have f = p i= (ua i) +(uf) But, all ua i s and uf vanish on X ; hence, the equality holds in fact on the whole of X, ifweextend the functions ua i s and uf by zero to X \ W Therefore, f is a sum of p + squares in O(X), as wanted Finally, we deduce as an easy corollary: Proposition There are real analytic curves X R n (n 3) with infinite Pythagoras number 8

9 Proof For, as in the proof of, we can glue a discrete family of singularities X xk with Pythagoras numbers p(o(x xk )) = p k +, toobtain a real analytic curve X R n with Pythagoras number sup k p k =+ References [CaRz] A Campillo, JM Ruiz: Some remarks on pythagorean real curve germs J Algebra 8 (990) 7 75 [CDLR] MD Choi, ZD Dai, TY Lam, B Reznick: The Pythagoras number of some affine algebras and local algebras J reine angew Math 336 (98) 45 8 [Co] S Coen: Sul rango dei fasci coerenti, Boll Un Mat Ital (967) [FeQz] JF Fernando, R Quarez: A remark on the Pythagoras number of algebroid curves J Algebra 74 (004) [FeRz] JF Fernando, JM Ruiz: On the Pythagoras numbers of real analytic set germs Bull Soc Math France 33 (005) [FRS] JF Fernando, JM Ruiz, C Scheiderer: Sums of squares in real rings Trans Amer Math Soc 356 (004) [Jw] [Or] P Jaworski: Positive definite analytic functions and vector bundles Bull Ac Pol Sc 30 (98) J Ortega: On the Pythagoras number of a real irreducible algebroid curve Math Ann 89 (99) 3 [Qz] R Quarez: Pythagoras numbers of real algebroid curves and Gram matrices J Algebra 38 (00) [Rz] JM Ruiz: On Hilbert s 7th problem and real Nullstellensatz for global analytic functions Math Z 90 (985) [Sch] C Scheiderer: Sums of squares of regular functions on real algebraic varieties Trans Am Math Soc 35 (999) [Sch] C Scheiderer: On sums of squares in local rings J reine angew Math 540 (00)

10 Francesca Acquistapace Dipartimento di Matematica Universit degli Studi di Pisa Via Filippo Buonarroti, 567 Pisa, Italy José FFernando Depto Matemáticas Facultad de Ciencias Univ Autónoma de Madrid Cantoblanco 8040 Madrid, Spain Fabrizio Broglia Dipartimento di Matematica Universit degli Studi di Pisa Via Filippo Buonarroti, 567 Pisa, Italy Jesús M Ruiz Depto Geometría y Topología F Ciencias Matemáticas Univ Complutense de Madrid 8040 Madrid, Spain jesusr@matucmes 0