ARTIN''S THEOREM FOR $f$-RINGS Текст научной статьи по специальности «Математика»

Владикавказский математический журнал 2015, Том 17, Выпуск 2, С. 32-36

УДК 510.67+512.55

ARTIN'S THEOREM FOR /-RINGS1

A. G. Kusraev

To Vladimir Kojbaev on occasion of his 60th birthday

The main result states that each positive polynomial p in N variables with coefficients in a unital

Archimedean /-ring K is representable as a sum of squares of rational functions over the complete ring of

quotients of K provided that p is positive on the real closure of K. This is proved by means of Boolean

valued interpretation of Artin's famous theorem which answers Hilbert's 17th problem affirmatively.

Mathematics Subject Classification (2010): 03C25, 12D15, 13B25. /

theorem, Hilbert 17th problem, Boolean valued representation.

The aim of this note is to prove that each positive polynomial p in N variables with coefficients in a unital Archimedean f-ring K is representable as a sum of squares of rational functions over the complete ring of quotients of K provided that p is positive on the real closure of K. For an ordered field K this is Artin's famous theorem which answers Hilbert's 17th problem affirmatively.

Recall some basic notions of the theory of rings; see J.Lambek [11]. Everywhere below K is a commutative unital ring. The complete ring of quotients of a commutative ring K is denoted by Q(K). We call K rationally complete if Q(K) ~ K canonically or, equivalently, every irreducible fraction has domain K. Given a subset A of a commutative ring K, define the annihilator of A as A* := {k e K : kA = {0}}. The ideals of the form A* are called annihilator ideals. Thus J is an annihilator ideal if and only if J = A* for some subset A of K, and this is equivalent to saying that J** := (J* )* = J .A commutative ring K is called semiprime if its prime radical is 0, that is if it has no nonzero nilpotent elements. The annihilator ideals in a commutative semiprime ring K form a complete Boolean algebra A(K), with intersection as infimum and annihilator as complementation. If K is commutative, semiprime, and rationally complete, then every annihilator of K is a direct summand and P(K) ~ A(K) with P(K) being the Boolean algebra of idempotents of K. The lateral (or orthogonal) completion of a commutative semiprime ring K is the least subring K' c Q(K) such that for all families (x)^es in K and (e^in P(Q(K)) with e^en = 0 (£ = n) there exists x e K' such that e^x = e^x^ for all £ e S.

The ring K is called formally real if af +-----+ a^, e J implies ai,..., an e J for every finite

collection af,... ,an e K and every J e A(K) or, in terminology of J. Bochnak, M. Coste,

^The studv was supported by a grant from the Russian Foundation for Basic Research, project 14-01-91339.

© 2015 Kusraev A. G.

and M.-F. Roy [3, Definition 4.1.3], every annihilator ideal in K is real. A semiprime regular

ring K is real if and only if a2 +-----+ a^ = 0 implies a1 = ■ ■ ■ = an = 0 for all ai,..., an G K

and n G N, since every principal annihilator ideal is a direct summand.

Consider commutative unital rings K and L. Say that L extends K if K is a subring of L and the mapping J ^ J n K is one-to-one from A(L) onto A(K). Say also that L is locally algebraic over K whenever L extends K and, given x G L and a nonzero I G A(K), there exist a nonzero J G A(K), natural n G N, and a0,...,an G K such that J C I and a0 + a1x + ■ ■ ■ + anxn G J *. In the case of of semiprime regular rings L is locally algebraic over K if and only if P(K) = P(L) and, given x G L, for every nonzero d G P(K) there exist a nonzero e G P(K), natural n G N, and a0,... ,an G K such that e ^ d and e(a0 + a1x + ■ ■ ■ + anxn) = 0.

An f-ring is a lattice-ordered ring K such that y A z = 0 implies xy A z = yx A z = 0 for all x,y,z G K+. A band (or 'polar) in K is each set of the form A± := {k G K : (Va G A) |k| A |a| = 0} with 0 = A C K. The set of all bands B(K) in a semiprime Archimedean f-ring K coincides with A(K) and hence is a complete Boolean algebra, since A* = A^ for every A C K. In this note we consider only Archimedean f-rings. See more details in [2].

For a unital f-ring K the complete ring of quotients Q(A) can be uniquely made an f-ring with K a sublattice of Q(A). This result is due to F. W. Anderson [1]; see also [8, § 10]. Moreover, the Boolean algebras P(Q(K)) and A(K) are isomorphic.

A real closure of a unital /-ring K is a rationally complete /-ring K satisfying the following conditions: 1) Q(K) is a subring and sublattice of K with K extending Q(K), 2) K is locally algebraic over Q(A), and 3) if K' is rationally complete /-ring containing K as a subring and sublattice and locally algebraic over Q(K) then K' = K. Say that K is real closed whenever K = K. See the general concept of real closed rings in [15]. The main result is stated next.

Theorem. Let K be an Archimedean unital f-ring and let K be its real closure, so that the embeddings K c Q(K) C K hold. If a polynomial p G K[x\,..., xn] is positive, that is p(ai,..., a«) ^ 0 for all (ai,...,ajv) G KN, then the representation q2P = Sm=1 kjPf holds for some non-zero-divisors 0 < k1,..., km G Q(K) and some polynomials p1,... ,pm, q G Q(K)[x1,..., xN] with eq(a1,..., aN) = 0 equivalent to ep(a1,..., aN) = 0 for all e G p(q(K )) and a1,...,aN G K.

Remark 1. Our proof uses Boolean valued analysis which signifies the technique of studying properties of an arbitrary mathematical object by means of comparison between its representations in two different set-theoretic models, the von Neumann universe V and a specially-trimmed Boolean-valued universe V(B). Comparative analysis is carried out by means of some interplay between V and V(B) which rests on the functors of canonical embedding (or standard name) X ^ XA G V(B) (X G V), descent X ^ X| G V (X G V(B)), and ascent Y ^ Yt G V(B) (Y C V(B)). Accordingly, our proof is merely an interpretation of Artin's theorem within V(B), thus demonstrating how does a Boolean valued transfer principle work in real algebra (as presented in [3] and [14]).

Remark 2. In particular, each Archimedean unital f-ring K has a real closure unique up to K-isomorphism. This is a Boolean valued interpretation of Artin-Schreier Theorem for ordered fields; see [14, Theorem 1.3.14] and [13, Theorem 28.7].

Remark 3. For every 0 = a G Q(K) there exists a least element ea G P(Q(K)) with eaa = a. Moreover, a is a non-zero-divisor of Q(eaK) and a-1 exists in Q(eaK). Now, given p, q G Q(K)[x1,..., xN], we can define (p/q)(a1,..., aN) := p(a1,..., aN)q(a1,..., aN)-1 if q(a1,..., aN) = 0, while (p/q)(a1,..., aN) := 0, whenever q(a1,..., aN) = 0. Say that p/q is a rational function over Q(K) and denote by Q(K)(x1,..., xN) the set of all rational functions over Q(K). Thus the above representation can be written as p = YjjL1 kj(pf/q2).

Throughout the sequel B is a complete Boolean algebra with unit 1 and zero O, while is the corresponding Boolean valued model of set theory and e B is the Boolean truth value of a set theoretic formula All necessary information concerning Boolean values analysis can be found in [9] and [10].

We need the following important result due to E. I. Gordon [6]. Let K be a commutative semiprime ring and B the Boolean algebra of its annihilator ideals. Then there exist K, F e V(B) such that [ K is an integral domain and F is the quotient field of K ] = 1, Kj is the lateral completion of K and Fj is the complete ring of quotients of K. In this event K is called the Boolean valued representation of K. Details can be found in [9, Theorem 8.3.5].

Lemma 1. Let K be a commutative semiprime ring and K be its Boolean valued representation in V(B) with B = A(K). If L is a subring of K and L:= L| e V(B) then

(1) [L is a subring of K ] = 1 ^^ K extends L.

(2) [ K is algebraic over L ] = 1 ^^ K is locally algebraic over L.

(3) [ K is algebraic closure of L ] = 1 ^^ K is algebraic closure of L.

< This fact can be derived from Gordon's result [6] by straightforward calculation of Boolean truth values; cp. [9, Section 8.3]. >

Lemma 2. Let K, K, L, and K be the same as in Lemma 1 and, moreover, K is an Archimedean f-ring extending L, while L is a sublattice of K. Then [K and L are totally ordered integral domains with K extending L]] = and K is a real algebraic closure of L if and only if [K is a field, a real algebraic closure of L ] = 1.

< The claim can be proved combining the above mentioned result by Anderson and Lemma 1 in the manner similar to that of [9, Theorem 8.5.6] taking into account the fact that an ordered field K admits a unique real closure up to K-isomorphism; see [3, Theorem 1.3.2]. >

Lemma 3. Let Jf be a totally ordered integral domain, Jf be its field of quotients and Jf is a real closure of K'. If a polynomial p e K[xf,..., xN] is positive, i. e. p(af,..., aN) ^ 0

for all (cii,..., ajv) £ Jf , then q2p = k\p\ + • • • + A:mPm f°r some 0 < k\,..., km e K', pf,... ,pm,q e K'[xf,... ,xN ] with q(xf,... ,xN) = 0 if and only if p(xf,... ,xN) = 0.

< This is an improved version of Artin's theorem; see [3, Theorem 6.1.3], [12, Theorem 1.4.4], [13, Theorem 28.11], and [14, Theorem 2.1.12]. >

Lemma 4. Let K be an Archimedean unital f-algebra, B = P(K), and let K e V(B) be its Boolean valued representation. If [p e K[xf,... , xN]] = 1 and [deg(p) ^ dA] for some d e N then pj e K'[xf,... ,xN] with K' := Kj.

< Assume that [p e K[xf,..., xN] ] = 1 and fix uf,..., uN e K. Define Nd := {1,..., d}

A)na = (NN

and identify N with {0,1,...,N - 1}. Observe that (N^)nA = (NN)a. There exist two

mappings a, k : (N^)NA ^ K such that

d)

p(ui,...,UN )= ^ a(v )k(v ), k(v )=

uj

,e(NJv) a jeNA

Let a := a\ and k\ stand for the modified descents of a and m, respectively, so that a,k :

dN

NN ^ K' and [a(v) = a(va)] = 1, [k(v) = k(va)J = 1 for all v e NN, see [9, 5.7.7] and [10,

1.5]. Define p e K'[xf,... ,xN] as p(xf,... ,xN) := ^veNv avxf(i) ■ ... ■ x^N), av := a(v), and observe that for all uf,..., uN e K' we have

p(ni,...,nN) = a^Yi nv(j) = p(Ui,...,Un)

It follows that pj = p. >

I | uj

veNv jeN

=.

< Proof of the Theorem. Let K G V(B) be the Boolean valued representation of K with B = A(K). Then K is an integral domain within V(B). By the Boolean valued transfer principle and the maximum principle, within V(B) there exist the field of quotients Jf' of Jf and the real closure Jf of Jf'. We may assume that Q(K) = by the above mentioned Gordon's result and K = by Lemma 2. Take a polynomial p G K[x\,... ,xn] and assume that p(ai,..., aw) ^ 0 for all (ai,..., a^v) G KN. Putting it := p\, one can prove by direct calculation of Boolean truth values that n G K[x1,..., xN] and n(a1,..., aN) ^ 0 for all (a\,... ,(in) G Jf within V®. By the transfer principle, Lemma 3 holds true within V(B) and by the maximum principle there exist m G NA, n1,..., p G K'[x1,..., xN] such that p2n = nf + ■ ■ ■ + n'm and p(x1,..., xN) = 0 if and only if n(x1,..., xN) = 0 for all x1,... ,xN G K'. Moreover, the number of squares m ^ 2an (see [13, Theorem (Pfister)

NA

29.3]) and the degrees of n2 for every j and p2 bounded by D = 22 , a := deg(n)4 (see [12, Theorem 1.4.4]). Observe now that m ^ (2A= (2A)A for any finite set A, since (Pfin(A))A = Pfin(AA), see [9, Proposition 5.1.9]. Thus, we have (2N)' = 2N'' and a ^ (deg(p)4 )A. (We identify 2 with 2A and 4 with 4A.) It follows the existence of l G N with D ^ lA. Denote q := p| and p := nj| (j := 1,... ,m). Then q2p = Yjni kjp2 and nj, q G Q(K )[xi,... ,xn ] by Lemma 4. >

Let R be the field of reals within V(B). Then R := R| G V (with the descended operations and order; see [9]) is a universally complete vector lattice, i.e. the externalization R of the internal Boolean valued reals R is a universally complete vector lattice. This remarkable result discovered by E. I. Gordon [5] tells us that each theorem on the reals (in the framework of Zermelo-Fraenkel set theory) has its counterpart for the corresponding universally complete vector lattices. In particular, R admits a unique f-ring multiplication for which a given order unit, a positive element 1 G R with {1}^ = {0}, is a ring unit.

Corollary 1. The vector lattice R is a real closed f -ring and each positive polynomial in R[x1,..., xN] is a sum of squares of rational functions in R(x1,..., xN).

Two important particular cases of R were independently studied by G. Takeuti, who observed that the vector lattice of cosets of (almost everywhere equal) measurable function and a commutative algebra of (unbounded) self-adjoint operators in Hilbert space can be considered as instances of Boolean valued reals [16, 17].

Corollary 2. Let (fi, be a Maharam measure space and let L0 := L°(^, X,^) be the f -ring of all cosets of real measurable functions on fi. Then any positive polynomial in L0[x1,..., xN] is a sum of squares of rational functions in L0(x1,..., xN).

< A vector lattice L0(fi, X,^) is a universally complete f-algebra (with identically one function as a ring unit) if and only if the measure space (fi, X,^) is Maharam (= localiz-able). >

Given a complete Boolean algebra B of projections in a Hilbert space H, denote by S(B) the space of all selfadjoint operators on H whose spectral decompositions are in B; i. e., A G 6(B) if and only if A = fR AdEA with Ex G B for all A G R, see [17]. For A, B G 6(B) put A ^ B if and only if (Ax,x) ^ (Bx,x) for all x G D(A) n D(B).

Corollary 3. Let H be a complex Hilbert space and B a complete Boolean algebra of projections on H. Then any positive polynomial in 6(B)[x1,..., xN] is a sum of squares of rational functions in 6(B)(x1,..., xN).

< 6(B) is a universally complete f-algebra (and hence a unital f-ring) [17, Ch. 1, § 3]. >

Remark 4. Corollary 2 can be considered (and also proved) as a measurable version of the

'continuous' solution of Hilbert's 17th problem obtained by C. N. Delzell and re-discovered by L. Gonzalez-Vega and H. Lombardi, see [4] and [14, Theorem 4.3.4].

References

1. Anderson F. W. Lattice-ordered rings of quotients // Cañad. J. Math.—1965.—Vol. 17.—P. 434-448.

2. Bigard A., Keimel K., and Wolfenstein S. Groupes et Anneaux Réticulés.—Berlin etc.: Springer-Verlag, 1977.-xi+334 p.—(Lecture Notes in Math., Vol. 608).

3. Bochnak J., Coste M., and Roy M.-F. Real Algebraic Geometry.—Berlin a. o.: Springer, 1998.—x+430 p.

4. Delzell C. N., González-Vega L., and Lombardi H. A continuous and rational solution to Hilbert's 17th problem and several cases of the Positivstellensatz // Computational Alg. Geom. / Eds. F. Eyssette and A. Galligo.—Birkhâuser: Boston a. o., 1993.—P. 61-75.—(Progress in Math., Vol. 109).

5. Gordon E. I. Real numbers in Boolean-valued models of set theory and K-spaces // Dokl. Akad. Nauk SSSR. 1977. Vol. 237, № 4.-P. 773-775.

6. Gordon E. I. Rationally Complete Semiprime Commutative Rings in Boolean Valued Models of Set Theory.-Gor'kiï, 1983.-35 p.-(VINITI, № 3286-83).

7. Henriksen M. and Isbell J. R. Lattice-ordered rings and function rings // Pacific J. Math.—1962.— Vol. 12.-P. 533-565.

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9. Kusraev A. G. and Kutateladze S. S. Introduction to Boolean Valued Analysis [in Russian].—M.: Nauka, 2005.-526 p.

10. Kusraev A. G. and Kutateladze S. S. Boolean Valued Analysis: Selected Topics.—Vladikavkaz: SMI VSC RAS, 2014.—iv+400 p.

11. Lambek J. Lectures on rings and modules.—Toronto: Blaisdell Publ. Comp., 1966.-183 p.—(AMS Chelesea publishing. Providence, R. I.).

12. Lombardi H., Perrucci D., and Roy M.-F. An elementary recursive bound for effective Positivstellensatz and Hilbert 17th problem.-2014.-arXiv:1404.2338v2 [math.AG].

13. Prasolov V. V. Polynomials.—Berlin-Heidelberg: Springer-Verlag, 2010.-301 p.

14. Prestel A. and Delzell Ch. N. Positive Polynomials: From Hilbert's 17th Problem to Real Algebra.— Berlin a. o.: Springer, 2001.—viii+267 p.

15. Schwartz N. The basic theory of real closed spaces.—Providence (R.I.): Amer. Math. Soc., 1989.— 122 p.-(Mem. Amer. Math. "Soc. Vol. 77(397)).

16. Takeuti G. Two Applications of Logic to Mathematics.—Princeton: Princeton Univ. Press, 1978.

17. Takeuti G. A transfer principle in harmonic analysis // Symbolic Logic.—1979.—Vol. 44, № 3.—P. 417440.

Received February 16, 2015.

Kusraev Anatoly Georgievich Southern Mathematical Institute Vladikavkaz Science Center of the RAS, Director 22 Markus street, Vladikavkaz, 362027, Russia E-mail: kusraev®smath,ru

ТЕОРЕМА АРТИНА ДЛЯ /-КОЛЕЦ Кусраев А. Г.

Основной результат заметки утверждает, что полином p от N переменных с коэфиициентами из унитарного архимедова /-кольца K представляется в виде суммы квадратов рациональных фукнций над полным кольцом частных кольца K, если только p положителен на вещественном замыкании K. Доказательство состоит в булевозначной интерпретации классической теоремы Артина, содержащей

положительное решение 17-й проблемы Гильберта. /

нальная функция, теорема Артина, 17-я проблема Гильберта, булевозначное представление.