The pp conjecture for the space of orderings of the field R(x,y)

Transcription

1 The pp cojecture for the space of orderigs of the field R(x,y) Pawe l G ladki Departmet of Mathematics ad Statistics, Uiversity of Saskatchewa, 106 Wiggis Road, Saskatoo, SK, Caada, S7N 5E6 gladki@math.usask.ca Murray Marshall Departmet of Mathematics ad Statistics, Uiversity of Saskatchewa, 106 Wiggis Road, Saskatoo, SK, Caada, S7N 5E6 marshall@math.usask.ca Abstract The paper cosiders the space of orderigs (X R(x,y),G R(x,y) ) of the field of ratioal fuctios over R i two variables. It is show that the pp cojecture fails to hold for such a space; a example of a positive primitive formula which is ot product-free ad oe-related is ivestigated ad it is prove, that although the formula holds true for every fiite subspace of (X R(x,y),G R(x,y) ), it is false i geeral. This provides a egative aswer to oe of the questios raised i: M. Marshall, Ope questios i the theory of spaces of orderigs, J. Symbolic Logic 67 (2002), This work is a sequel of previous results preseted i: P. G ladki, M. Marshall, The pp cojecture for spaces Correspodig author: phoe , fax

2 of orderigs of ratioal coics, to appear i J. Algebra Appl.; both spaces of orderigs of coic sectios ad the space (X R(x,y),G R(x,y) ) are importat examples of spaces of stability idex 2 that are i the scope of our research. Keywords: quadratic forms, spaces of orderigs. Throughout this paper (X, G) deotes a space of orderigs i the sese of [6, pp ]. We will be mostly dealig with spaces of orderigs of the form (X K, G K ), where K is a formally real field, X K deotes the set of all orderigs of K ad G K = K /(ΣK 2 \ {0}), ΣK 2 beig the set of sums of squares of K [6, Theorem 2.1.4]. I such a case G K is idetified with a subgroup of the group {1, 1} X K [6, Lemma 2.1.1]. With a slight abuse of the otatio we shall use the same symbol to deote a elemet of K, a coset i G K ad a fuctio i { 1, 1} X K. For a fixed space of orderigs (X, G) ad a G let U(a) = {x X : a(x) = 1}. As a subspace of (X, G) we uderstad a pair (Y, G Y ), where Y is some itersectio of sets of the form U(a) ad G Y is the group of all restrictios a Y, a G [6, pp ]. A subspace of a space of orderigs is a space of orderigs itself [6, Theorem 2.4.3]. While cosiderig subspaces, we will usually use the same otatio for elemets a G ad their restrictios a Y. If (Y, H) is a subspace of (X, G) ad a, b H, we defie the value set D Y (a, b) = {c H : x Y (c(x) = a(x) or c(x) = b(x))}. I the case whe Y = X or whe it is clear i which subspace we work, we shall write D(a, b) istead of D Y (a, b). With the otio of value sets we defie positive primitive (pp for short) formulae as the oes of the form m P(a) = t p j (t, a) D(1, q j (t, a)), j=1 where t = (t 1,...,t ), a = (a 1,...,a k ), for t i, a l G, i {1,..., }, l {1,..., k}, ad p j (t, a), q j (t, a) are ± products of some of the t i s ad a l s, i {1,...,}, l {1,..., k}. Clearly, whe we speak of a pp formula P(a) i a subspace (Y, H), we thik of all parameters a l as their restrictios a l H ad of all value sets D(1, q j (t, a)) as value sets D Y (1, q j (t, a)). 2

3 The followig problem, kow as the pp cojecture, has bee posed i [7]: Is it true that every pp formula P(a) with parameters a i G which holds i every fiite subspace of (X, G) ecessarily holds i (X, G)? The aswer to the problem is affirmative for umerous pp formulae describig importat properties of quadratic forms over spaces of orderigs (see [7] for details) ad for - itroduced i [8] - product-free ad oe-related formulae i spaces of fiite stability idex. The class of spaces for which the cojecture is true cotais spaces of fiite chai legth, spaces of stability idex 1 ad is closed uder direct sum ad group extesio [7]. As to spaces of stability idex 2, the followig examples are of our iterest: spaces of orderigs of formally real fiitely geerated extesios of Q of trascedece degree 1 (i particular Q(x) ad fuctio fields of coic sectios) ([1, Propositio VI.3.5]), spaces of orderigs of formally real fiitely geerated extesios of real closed fields of trascedece degree 2 (i particular R(x, y) ad its fiitely geerated algebraic extesios) ([1, Propositio VI.3.2]), ad spaces of orderigs of a field of formal power series R((x, y)) i two variables, or a field of algebraic power series R((x, y)) alg, or a field of aalytic power series R{{x, y}} over a real closed field R (i particular R((x, y)), R((x, y)) alg, ad R{{x, y}}) ([1, Example VII.2.3 b), c), Remark VII.5.6]). The pp cojecture holds true for the space of orderigs of the field Q(x) [4]. For spaces of orderigs of coic sectios the complete classificatio with respect to the cojecture is give i [5]. Due to rather complicated real valuatios of the field R(x, y), methods used i [4] ad [5] could ot be applied to the space (X R(x,y), G R(x,y) ). This paper circumvets this obstacle ad here ew, valuatio theory free methods are developed ad used. Our mai result is the followig theorem: Theorem 1. The pp cojecture fails for the space of orderigs (X R(x,y), G R(x,y) ). Proof. For N \ {0} cosider the subspaces (X, G ), where X = U(x 2 + y 2 1) U(1 + 1 x2 y 2 ) ad G = G R(x,y) X. Defie the subspace (X, G), where X = N\{0} ad G = G R(x,y) X. It is sufficiet to show that the cojecture fails i the space (X, G) [2, Propositio 6]. For N \ {0} deote X A = {(a, b) R 2 : 1 < a 2 + b 2 < } 3

4 ad let π 1,...,π 6 R(x, y) be liear irreducibles which, for large eough itersect with rigs A as follows: π6 p26 p11 p22 π2 p12 p13 π5 p25 p21 π1 A p16 p14 p24 π4 p15 p23 Fig. 1 π3 Here p 1i, p 2i deote the two coected compoets of Z(π i ) A, i {1,...,6}, N \ {0}, ad are arraged i the above order, where Z(π i ) is the set of real zeros of π i. Replacig π i by π i we may assume that every π i is positive at the origi. For two sets p i 1 j 1 ad p i 2 j 2, i 1, i 2 {1, 2}, j 1, j 2 {1,..., 6}, deote also by A i 1j 1,i 2 j 2 the rig sector startig at p i 1 j 1 ad, whe movig clockwise alog A, edig at p i 2 j 2. Let a 1 = π 1 π 6, a 2 = π 1 π 4 ad d = π 1 π 2 π 3 π 5. Cosider the followig pp formula: P(a 1, a 2, d) = t 1 t 2 (t 1 D(1, a 1 ) t 2 D(1, a 2 ) dt 1 t 2 D(1, a 1 a 2 )). We shall show that P(a 1, a 2, d) fails to hold i the space (X, G). Suppose, a cotrario, that the formula holds true i (X, G) with certai t 1, t 2 G verifyig it. Without loss of geerality we may assume that t 1, t 2 are square-free polyomials. Let S = {σ : σ is irreducible ad σ t 1 or t 2, or σ = π i for some i {1,...,6}}. 4

5 Observe, that there exists N 1 N \ {0} such that for N 1 : ad for each σ S the set Z(σ) A is a fiite disjoit uio of smooth arcs γ : (0, 1) R 2 homeomorphic to a ope lie segmet ad such that lim t 0 γ(t) is a poit o the circle x 2 + y 2 = 1, whilst lim t 1 γ(t) is a poit o x 2 + y 2 = 1 + 1, for σ, τ S, σ τ: 1 Z(σ) Z(τ) A =. This is ituitively clear, however if oe wats to prove it formally, oe should use the half-braches theorem [3, Propositio 9.5.1] ad the fact that we may restrict ourselves to those σ S for which ideals (σ) are real (see [3, Theorem 4.5.1]). Observe also that for sufficietly large (say, N 2 for some N 2 N\{0}) P(a 1, a 2, d) already holds i the subspace (X, G ). Ideed, cosider the ope set U = (U( a 1 ) U(t 1 )) (U( a 2 ) U(t 2 )) (U( a 1 a 2 ) U(dt 1 t 2 )), viewed as a subset i (X R(x,y), G R(x,y) ). Sice t 1 D(1, a 1 ) t 2 D(1, a 2 ) dt 1 t 2 D(1, a 1 a 2 ) holds true i (X, G), X U. But X = N\{0} X, where X 1 X 2... is a chai of closed subsets, ad (X R(x,y), G R(x,y) ) is compact [6, Theorem 2.1.5], so for large eough X U. That meas that P(a 1, a 2, d) holds true i (X, G ). Fix N \ {0} satisfyig all of the above coditios (that is max{n 1, N 2 }) ad cosider the space (X, G ). By lookig at umber of sig chages of each irreducible factor σ of t 1 or t 2 whe we travel alog the circle x 2 + y 2 = we see, that each such Z(σ) itersects with A i a eve umber of coected compoets [3, Theorem 4.5.1]. Furthermore, the sigs of a 1, a 2 ad d o the rig sectors betwee the successive p ij, i {1, 2}, j {1,..., 6}, are the followig: 1 Note that some of π 1,...,π 6 might be also divisors of t 1 or t 2 5

6 A 11,22 A 22,13 A 13,21 A 21,14 A 14,23 A 23,15 A 15,24 A 24,16 A 16,25 A 25,12 A 12,26 A 26,11 a a d Tab. 1 We yield a cotradictio by ivestigatig the behaviour of t 1 ad t 2 o A. The followig criterio for represetativity of biary forms shall be of costat use: f D X (1, g) (a, b) A (f(a, b) 0 or f(a, b) g(a, b) 0) (see [4, Corollary 3.2]). O A 21,14, A 24,16 ad A 26,11 both a 1 ad a 2 are positive, so t 1 ad t 2 are oegative. Moreover, sice t 1 ad t 2 are square-free ad sice there are o sigular poits of irreducible factors of t 1, t 2 iside of A, by the Sig Chagig Criterio [3, Theorem 4.5.1], t 1 ad t 2 are, i fact, positive. Near p 23 a 1 is positive, so t 1 is positive. It follows that Z(t 1 ) (from ow o we shall simply write t 1 ) does ot itersect with A alog p 13 : if it did, the π 3 would divide t 1 (sice they would have ifiitely may poits i commo), so t 1 = 0 o p 23. Furthermore, a 1 a 2 > 0 ear p 13, so dt 1 t 2 is oegative. Sice d chages sig betwee A 22,13 ad A 13,21 ad t 1 does ot itersect with A alog p 13, t 2 has to pass A at p 13. Thus π 3 t 2 ad t 2 also cuts across A at p 23. Similarly, a 2 > 0 ear p 12, so t 2 > 0 ad, as before, t 2 does ot itersect with A alog p 22. Close to p 22 a 1a 2 > 0, so dt 1 t 2 0 ad thus t 1 passes A at p 22 ad also at p 12. Next, ear p 11 a 1a 2 > 0, so dt 1 t 2 0, whilst d chages sig betwee A 26,11 ad A 11,22. Thus t 1 t 2 chages sig, so either t 1 itersects with A alog p 11 ad t 2 does ot, or t 2 does ad t 1 does ot. Similarly, ear p 21 a 1a 2 > 0, so dt 1 t 2 0. d chages sig at p 21 ad so does t 1 t 2, which implies that either t 1 crosses A at p 21 ad t 2 does ot, or t 1 does ot cross ad t 2 does. Of course if t 1 passes A at p 11, the π 1 t 1, so t 1 also passes A at p 21. Therefore t 1 cuts across A at p 11 if ad oly if it cuts across A at p 21 ad, similarly, t 2 traverses A at p 11 if ad oly if it traverses A at p 21. 6

7 O A 11,22 a 1 a 2 > 0, so dt 1 t 2 0. Sice d < 0, t 1 t 2 0, so t 1 itersects with A if ad oly if t 2 does - say, there are m 1 such itersectios withi A 11,22. Similarly, o A 13,21 a 1 a 2 > 0, so dt 1 t 2 0. At the same time d < 0, so t 1 t 2 0. Thus t 1 itersects with A if ad oly if t 2 does; there are m 2 such itersectios withi A 13,21. Fially, o A 22,13 a 1 a 2 > 0 ad d > 0, so dt 1 t 2 0 ad t 1 t 2 0. Therefore t 1 itersects with A if ad oly if t 2 does ad we have m 3 such simultaeous itersectios withi A 22,13. To sum up, there are m 1 +m 2 +m 3 simultaeous itersectios of t 1 ad t 2 with A i A 11,21. Furthermore, t 1 crosses through both p 22 ad t 2 through p 13. Ad fially, exactly oe of t 1, t 2 crosses through both p 11 ad p 21 : say t i does ad t j does ot. The t j chages sig m 1 + m 2 + m from A 26,11 to A 21,14, to go from positive to positive, hece m 1 + m 2 + m is eve ad m 1 + m 2 + m 3 is odd. Note ow that the oly simultaeous itersectios of t 1 ad t 2 with A are the m 1 + m 2 + m 3 listed above; o all other sectors of A at least oe of a 1, a 2 is positive, forcig either t 1 or t 2 to be positive as well. Simultaeous itersectios may occur oly at the commo irreducible factors of t 1, t 2. Accordig to our assumptios, each such factor has a eve umber of crossigs with A - so m 1 + m 2 + m 3 is eve, which is a cotradictio. This fiishes the first half of the proof. It remais to show that P(a 1, a 2, d) holds true o every fiite subspace of (X, G). Suppose the that there is a fiite subspace (Y, H) of (X, G) o which P(a 1, a 2, d) fails to hold. Without loss of geerality we may assume that (Y, H) is miimal with such property. We eed to cosider two cases. Firstly, suppose that d / D((1, a 1 ) (1, a 2 )) holds o (Y, H). We shall use the followig desriptio of value sets of Pfister forms: for ay f 1,...,f k H, g D((1, f 1 )... (1, f k )) if ad oly if: ρ Y [(f 1 ρ = 1... f k ρ = 1) gσ = 1] ([6, Theorem 2.4.1]). Thus, for some σ Y, a 1 σ = 1, a 2 σ = 1 ad dσ = 1. Clearly σ X for ay fixed N\{0}, so - by the Tarski Trasfer Priciple [3, Corollary 5.2.4] - there is a poit (a, b) A such that a 1 (a, b) > 0, a 2 (a, b) > 0 ad d(a, b) < 0. But there is o such poit i A (see Tab. 1) - a cotradictio. Now assume that d D((1, a 1 ) (1, a 2 )) holds i Y. Sice (Y, H) is fiite, it is a direct sum of fiitely may coected compoets, that is 7

8 subspaces which correspod to equivalece classes of the followig relatio: if ρ 1, ρ 2 Y, the ρ 1 ρ 2 if ad oly if either ρ 1 = ρ 2 or there exist ρ 3, ρ 4 Y such that {ρ 1,...,ρ 4 } is a 4-elemet fa i Y ([6, Theorem 4.2.1]). By [8, Corollary 3.6] there exists a coected compoet (Y 0, H 0 ) of (Y, H), which is ot a fa, such that, if (Y, H) deotes the residue space of (Y 0, H 0 ) (that is a miimal space i the sese that if (Y 0, H 0 ) is a group extesio of some space of orderigs (Ŷ, Ĥ), the H Ĥ), a 1, a 2 H, either a 1, a 2 or a 1 a 2 is equal to 1, (1, a 1 ) (1, a 2 ) is isotropic over (Y 0, H 0 ) ad d / H. Clearly P(a 1, a 2, d) already fails to hold i (Y 0, H 0 ), so - due to miimality of (Y, H) - (Y, H) = (Y 0, H 0 ). Sice a 1, a 2, a 1 a 2 1, there are elemets of Y makig a 1, a 2 ad a 1 a 2 positive. At the same time, sice (1, a 1 ) (1, a 2 ) is isotropic, there is o elemet of Y makig both a 1 ad a 2 positive. Fix σ 1, σ 2, σ 3 Y such that a 1, a 2 ad a 1 a 2 have the followig sigs: σ 1 σ 2 σ 3 a a a 1 a Tab. 2 Cosider the subspace (Ỹ, H) which is ot a fa ad for which {σ 1, σ 2, σ 3 } is a miimal geeratig set. Thus elemets of Ỹ, viewed as characters, are products 3 i=1 σe i i such that 3 i=1 e i 1 mod 2 ad do ot cotai the elemet σ 1 σ 2 σ 3 ([6, Theorem 3.1.3]) cosequetly, Ỹ = {σ 1, σ 2, σ 3 }. Let (Y 1, H 1 ) be the group extesio of (Ỹ, H) by d. It cosists of 6 orderigs σ 1 +, σ 2 +, σ 3 +, σ1, σ2, σ3, with respect to which the sigs of a 1, a 2, a 1 a 2, d are as follows: σ 1 + σ 2 + σ 3 + σ1 σ2 σ3 a a a 1 a d Tab. 3 P(a 1, a 2, d) fails to hold o (Y 1, H 1 ), so (Y, H) = (Y 1, H 1 ). 8

9 Defie the followig subspaces of (X, G): V 11,22 = U( π 1 ) U( π 2 ) U(π 3 ) U(π 4 ) U(π 5 ) U(π 6 ) V 22,13 = U( π 1 ) U(π 2 ) U(π 3 ) U(π 4 ) U(π 5 ) U(π 6 ) V 13,21 = U( π 1 ) U(π 2 ) U( π 3 ) U(π 4 ) U(π 5 ) U(π 6 ) V 21,14 = U(π 1 ) U(π 2 ) U( π 3 ) U(π 4 ) U(π 5 ) U(π 6 ) V 14,23 = U(π 1 ) U(π 2 ) U( π 3 ) U( π 4 ) U(π 5 ) U(π 6 ) V 23,15 = U(π 1 ) U(π 2 ) U(π 3 ) U( π 4 ) U(π 5 ) U(π 6 ) V 15,24 = U(π 1 ) U(π 2 ) U(π 3 ) U( π 4 ) U( π 5 ) U(π 6 ) V 24,16 = U(π 1 ) U(π 2 ) U(π 3 ) U(π 4 ) U( π 5 ) U(π 6 ) V 16,25 = U(π 1 ) U(π 2 ) U(π 3 ) U(π 4 ) U( π 5 ) U( π 6 ) V 25,12 = U(π 1 ) U(π 2 ) U(π 3 ) U(π 4 ) U(π 5 ) U( π 6 ) V 12,26 = U(π 1 ) U( π 2 ) U(π 3 ) U(π 4 ) U(π 5 ) U( π 6 ) V 26,11 = U(π 1 ) U( π 2 ) U(π 3 ) U(π 4 ) U(π 5 ) U(π 6 ). By the Tarski Trasfer Priciple subspaces V i 1j 1,i 2 j 2 form a partitio of (X, G) ad, clearly, sigs of a 1, a 2 ad d o the V i 1j 1,i 2 j 2 are exactly the same as o the sector A i 1j 1,i 2 j 2, for respective i 1, i 2, j 1, j 2. Comparig those sigs we see that σ1 V 23,15, σ 1 + V 14,23 or σ 1 + V 15,24, σ2 V 25,12, σ 2 + V 16,25 or σ 2 + V 12,26 ad σ 3 + V 22,13, σ3 V 11,22 or σ3 V 13,21. Cosider the followig two 4-elemet fas: {σ + 1, σ 1, σ + 2, σ 2 } ad {σ + 1, σ 1, σ + 3, σ 3 }. If σ 1 + V 14,23 ad σ 2 + V 12,26, the, i particular, π 3 (σ 1 + σ1 σ 2 + σ2 ) = 1 - a cotradictio, sice for every 4-elemet fa {ρ 1,...,ρ 4 } 4 i=1 ρ i = 1 (ote that we ca also use π 2 istead of π 3 ). O the other had, if σ 1 + V 14,23 ad σ 2 + V 16,25, the π 5 (σ 1 + σ1 σ 2 + σ2 ) = 1 - a cotradictio. Thus σ 1 + V 15,24. If σ 1 + V 15,24 ad σ3 V 13,21, the π 3 (σ 1 + σ 1 σ+ 3 σ 3 ) = 1 - a cotradictio. But if σ 1 + V 15,24 ad σ3 V 11,22, the π 2 (σ 1 + σ1 σ 3 + σ3 ) = 1, which elimiates the last case ad yields a fial cotradictio. To obtai a cocrete couterexample i the space (X R(x,y), G R(x,y) ) we use a stadard trick. The formula P(a 1, a 2, d) costructed i the proof ca be writte i the followig form: t 1, t 2 [(t 1, a 1 t 1 ) = (1, a 1 )] [(t 2, a 2 t 2 ) = (1, a 2 )] [(dt 1 t 2, a 1 a 2 dt 1 t 2 ) = (1, a 1 a 2 )] 9

10 ad we kow that, for suitably chose, it fails i the space (X, G ), although it holds true i each of its fiite subspaces [2, Propositio 6]. Let p 1 = x 2 + y 2 1 ad p 2 = x2 y 2, so that X = U(p 1 ) U(p 2 ). Clearly the formula t 1 t 2 [(t 1, a 1 t 1 ) (1, p 1 ) (1, p 2 ) = (1, a 1 ) (1, p 1 ) (1, p 2 )] [(t 2, a 2 t 2 ) (1, p 1 ) (1, p 2 ) = (1, a 2 ) (1, p 1 ) (1, p 2 )] [(dt 1 t 2, a 1 a 2 dt 1 t 2 ) (1, p 1 ) (1, p 2 ) = (1, a 1 a 2 ) (1, p 1 ) (1, p 2 )] holds true i every fiite subspace of (X R(x,y), G R(x,y) ), but fails i geeral. Remarks: (1) The case of the field Q(x, y) is already well-uderstood. Let f(x, y) = 0 be a equatio of a irreducible coic sectio without ratioal poits, for example let f(x, y) = x 2 + y 2 3. The the space (X f, G f ) of orderigs compatible with the valuatio v iduced by f is a subspace of the space (X Q(x,y), G Q(x,y) ). Moreover, this space is also a group extesio of the space of orderigs of the residue field Q(x, y) v, that is the fuctio field of the curve f(x, y) = 0. If the pp cojecture was true for the space (X Q(x,y), G Q(x,y) ), the it would be also true for the space (X f, G f ) ([2, Propositio 6]) ad, cosequetly, for the space (X Q(x,y)v, G Q(x,y)v ) ([7, Propositio 2.3]), which is a cotradictio ([5, Theorem 6]). (2) Oe would expect the pp cojecture to fail for spaces of orderigs of fiitely geerated algebraic extesios of the field R(x, y) or, more geerally, R(x, y), for a real closed field R. (3) Up to date, othig is kow about the pp cojecture for spaces of orderigs of fields R((x, y)), R((x, y)) alg, or R{{x, y}}, as well as R((x, y)), R((x, y)) alg, or R{{x, y}}, for R beig a real closed field. Refereces [1] C. Adradas, L. Bröcker, J.M. Ruiz, Costructible sets i real geometry, Ergebisse der Mathematik ud ihrer Grezgebiete, Folge 3, Vol. 33, Spriger, [2] V. Astier, M. Tressl, Axiomatizatio of local-global priciples for ppformulas i spaces of orderigs, Arch. Math. Logic 44, No. 1 (2005), [3] J. Bochak, M. Coste, M.-F. Roy, Real algebraic geometry, Ergebisse der Mathematik ud ihrer Grezgebiete, Folge 3, Vol. 36, Spriger,

11 [4] M. Dickma, M. Marshall, F. Miraglia, Lattice-ordered reduced special groups, A. Pure Appl. Logic 132 (2005), [5] P. G ladki, M. Marshall, The pp cojecture for spaces of orderigs of ratioal coics, to appear i J. Algebra Appl. [6] M. Marshall, Spaces of orderigs ad abstract real spectra, Lecture Notes i Mathematics 1636, Spriger, [7] M. Marshall, Ope questios i the theory of spaces of orderigs, J. Symbolic Logic 67 (2002), [8] M. Marshall, Local-global properties of positive primitive formulas i the theory of spaces of orderigs, to appear i J. Symb. Log. 11