j
Hilbert's Tenth Problem over Number Fields, a Survey. Alexandra Shlapentokh AllSTRACT. \Ve survey the results concerning the analogs of Hilbert's Tcuth Problern over subrings of nun1ber fields and related issues of Diophantine dc-
finability.
Part I: What do we know ? 1. Generalizing the Problem and Taking the First Steps.
Since the tiine \vhen Yurii :rvia.tija.sevich con1pleted the proof of the undecidability of IIilbert's 1'enth Problc1n, siinilar probletns have been raised for rings other than the ring of rational integers. (For a discussion of so1nc of the in1plications of the solution of Hilbert's T'enth Problc1n and different versions of the original proof, see [15], [3], and [4].) In other words, let R be a recursive ring. Then, given an arbitrary polyno1nial equation in several variables over R, is there a. unifor1n algoritlun to detern1ine whether such an equation has solutions in R ? (For a definition of a recursive ring see [24], [8] or [23].) In this paper we survey the known answers to this question in cases \Vhere R is a subring of a nu1nber field. (For a discussion of issues of Diophantine definability and decidability over other domains see [23] .) At this point due to results of Jan Dencf, Leonard Lipshitz) Thanases Pheidas) Harold Shapiro and the author of this paper \Ve know the analog of Hilberes Tenth Proble1n to be undecidable over the rings of algebraic integers and in so111c cases bigger subrings of totally real uu1nber fields) their totally co1nplex extensions of degree 2, nurnber fields \vith one pair of nonreal conjugate e1nbcddings and sonic tot.ally complex fields of degree 4. Sect.ions 3.4 and :l.5 of Part I of this paper give inore details concerning the exact state1ncnts and the history of these results. In Pa.rt II of the paper \Ve describe the inethods used to obtain the results inentioned above. In particular 1 Sections 2.4-2.8 of Part II sketch the proofs of Diophantine undecidability over the rings of algebraic integers of the above n1c11tioned fields and Sections 3.1-3.:J of Part II sketch the proofs of the Diophantine undecidability of the larger subrings. 1.1. One Equals Finitely Many. Perhaps, we should start with the following easy observation. If the ring R under consideration is an integral do1nain and 1991 J\.fathe1natics Subject Classification. Prilnary 0.3035, 11U05. Secondary 11D57, 11072, l 1R04. 1'he author \vas supported in part hy NSA Grant MDA904-98-l-0510. ( 0 and any J{ 1 as described in Thcore111 3.5.4) there exists a .set of rational pri1nes il~~ such that the difference bet\veen l;i~u and Vl~:l i.s cont.ained in a set of prin1es of Dirichlet density less than c, and (JQ,lt';,; has a l)iophant.ine definition in its integral closure
llli
ALEXANDRA SHLAPENTOKH
in ](. (See [33], [35], [29] for proofs of the theorems.)
Part II: How do we know it ? In this part of the paper \Ve discuss in son1e detail the construction of Diophantine definitions of Z over rings of algebraic nu1nbers in so1ne nu1nber fields.
1. Imposing Bounds. \\Te start \vith a description of Diophantine 1nethods for bounding heights of algebraic nu1nbers. \\Te consider t\vo ways of bounding the height. The first 1nethod relies on quadratic for1ns and is inost effective over totally real fields. The second 1nethod relies on divisibility and thus depends on the choice of the ring. First of all 1 the influence of divisibility depends on \vhether any pri1nes are allo\ved to occur in the deno1ninator of the divisors of ele1nents of the ring. \Ve will discuss this aspect of the 1natter in 1nore detail in Section 1.3. Secondly, the expressive po\ver of divisibility in the rings of algebraic integers depends on the field at hand. Specifically, in the rings of algebraic integerD of any nu1nber field different fro1n Q or a totally co1nplex extension of degree t\vo of the rationals, divisibility together with addition can define inultiplication in Diophantine ter1ns. This result does not hold for 0, then Xi = ( y'Xi) 2 + 0 2 + ci0 2 + 02 . On the other hand if Xi < 0, then Ci < 0. Thus, x 1 /ci is a square in the con1pletion of the corresponding e1nbedding of J(. 1.1.2. Corollary. Let J( be a number field, let W be any subset of its nonarchi1nedean priines. Let a : J( ~ C be any real e1nbedding of J( into C. Then {(x, y) E K 2 IO'(x) > O'(y)) has a Diophantine definition over OK,IV· 0. Let c E J( be such that of [( not equal to er. Such a c exists by the Weak Approximation Theorem. Then D'(x) > 0 if and only if x = yf + y~ + cyg + y~ by Lemma 1.1.1 of Part II. Thus the corollary follows from Proposition 2.5 of Part I. PROOF.
It is enough to consider the case of y
O'(c) > 0 and r(c) < 0 for any real embedding
T
=
HILBEl"fl"S TENTII PROI3LEl\I OVER NUl\!13ER FIELDS, A SURVEY
117
1.2. Using Divisibility in the Rings of Algebraic Integers. 1.2.1. Lemma. Let K be a number field. Let x,y E OJ
0
0
1
.. r1-1
1 1
l~-1
n-1 {n-1 n
l"n
l :~ l ( ( l coN I 1, then jer(x)I C/2. For this u, denote F,, by M, denote G 11 by II, and denote Qu by q. 'l'hen in Af,
y
(9)
y/Jdeg(H)
=n~c--~-~J E T 0 - /,(3 - T/3
o,\/.
Since (a,(3) = 1 in OK, ((3,a -1;(3 -- T(3) = 1. Tims, for each T, (n-l,~-T/J) E OM. Therefore, INM;Ql(a - l,(3-T(3)j
(4a)
l
2
, lx,,,.;(a)i ;>
l
2
=
2, ... , n.
f' O;
(50)
y,,,(a)
(51)
Y~(a)IYm (a)
(52)
,i
mod y,,(a), boo a mod x,,,(a),
boo l
Xj(b) oo ±xk(a) mod x,,Ja),
(53) (54)
Yj(b)
"°e
mod y,(a).
By Lemma 2.3.1 of Part II and (49) we also have the following congruences.
(55)
y1 (b)
"'° j
mod (b - 1) "')- y1 (b)
(56)
j
mod y,,(a) by (52),
mod y,,(a) by (54),
Xj(b) oo x 1 (a)
(57) (58)
""e
"'° j
mod x,,,(a) hy (52),
.T1(a) oo ±x,(a) mod .T,,,(a) by (53),
(5a)
/,; oo ±j
(60)
y,,(a)lm by (51),
mod m,
"'° ±j
mod y,,(a) by (59) and (60),
±e
mod 2"f"' by (48) and (56).
(61)
/,;
(62)
k oo
It is not hard to see that k S jxk( a)j. Thus, by Lemma 1.2. l of Part II and (48), k < j.Tk(a)i S INKJQ(f)'I· Similarly, for i = 1, ... ,n,l(il S INK;Q(f)I'· Thus, by the strong version of the vertical method (see Section 2. 7 of Part II), ( = ±/,; E Z. Next we show that if ( E N, (38)·( 48) can be satisfied in all other variables in OJ ~,i = 2, ... , n. Set w = x,,(a). Then (39) and the z-part of (42) are
satisfied. By the discussion of the properties of i1'-units in Section 2.1.1 of Pa.rt II and by Lemma 2.3.1 of Part II again, we can find m EN such that yi,(a)jy,,,(a)
and jxi,ml > ~ for i = 2, ... , n. Set u = x 111 (a), v = Ym (a). Then (40), the u-part of (42) and (44) are satisfied. By Lemma 2.3.1 of Part II, there exists b E 0 [( satisfying the b-part of (42) and (45). Sets= xk(b), t = yk(b). Then (41) is satisfied. Finally, the ren1aini11g congruences are satisfied by Len1111a 2.3.l of Part II. Given Le1n1na 2.3.l of Part II) the case of the fields \Vith one pair of nonreal conjugate ernbeddings is processed in a n1anner very sin1ilar to the one used for the tot.ally real fields.
3. Horizontal Methods. 3.1. lntegrality at Finitely Many Primes in the Rings of W-integers. As \Ve have rnentioncd in Part I, the idea behind the construction of a Diophantine definition of integrality at finitely inany prin1es presented belo\v goes back to the \vork of Julia Ilobinson on aritlunctic definability of rational integers in algebraic number fields ([25] and [26]). She used quadratic forms to carry out the construction. Rurnely generalized Robinson)s construction in his paper on arithmetic definability in global fields. (Sec [27].) In this paper he used norm equations aud the Strong llassc Norn1 Principle. I
