Model Theory of Algebra and Arithmetic: Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic Held at Karpacz, Poland, September 1 – 7, 1979

412 Pages·1980·2.51 MB·English

by  J. Becker

Lecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
834
IIIIIIIII
Model Theory fo Algebra
dna citemhtirA
Proceedings of the Conference on Applications
of Logic to Algebra and Arithmetic Held at
Karpacz, Poland, September - 1 ,7 1979
Edited by
.L Pacholski, .J Wierzejewski, and .A .J Wilkie
galreV-regnirpS
Berlin Heidelberg New York 1980
Editors
Leszek Pacholski
Instytut Matematyczny PAN
Kopernika 18
51-617 wa/J.corW
Poland
Jedrzej Wierzejewski
Instytut Matematyki
Politechniki Wroc, t'aws kiej
Wybrze~e Wyspiar~skiego 27
50-370 WrocYaw
Poland
Alec .J Wilkie
Mathematical Institute
University of Oxford
24-29 St. Giles
Oxford OX1 3LB
England
AMS Subject Classifications (1980): 03 Cxx
ISBN 3-540-10269-8 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-387-10269-8 Springer-Verlag NewYork Heidelberg Berlin
This work is subject to copyright. All rights are reserved, whether the whole or
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publisher, the amount of the fee determined to be with agreement by the publisher.
© by Spdnger-Verlag Berlin Heidelberg 1980
Printed ni Germany
Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
2141/3140-543210
FOREWORD
The main part of this volume constitutes the Proceedings of the
Conference on Applications of Logic to Algebra and Arithmetic held
at the mountain resort of Bierutowice - Karpacz in Poland,
September I - 7 , 1979. The volume contains papers contributed by
the invited speakers and a few by other participants. Some papers
by mathematicians who were invited but could not come have also
been included.
The abstracts of all contributed papers will appear in the
Journal of Symbolic Logic.
The conference was the fourth in the series Set Theory and
Hierarchy Theory organized by the Institute of Mathematics of the
Technical University of Wroc~aw. The conference was attended by
80 registrated participants from 12 countries.
The organizing committee consisted of A.Macintyre /Yale Univer-
sity/, L.Pacholski /Polish Academy of Sciences, Wroc~aw/,
Z.Szczepaniak /Technical University, Wroc~aw/ and J.Wierzejewski
/Technical University, Wroc~aw; chairman/.
On behalf of the organizing committee we wish to thank the
Technical University of Wroc~aw and all the people who by their
help contributed to the success of the conference and its good
scientific and friendly atmosphere.
L.Pacholski
J.Wierzejewski
A.Wilkie
TABLE OF CONTENTS
J.BECKER, J.DENEF and L.LIPSHITZ I
Further remarks on the elementary theory of formal power
series rings
CHANTAL BERLINE !0
Elimination of quantifiers for non semi-simplm rings of
characteristic p
M.BOFFA, A.MACINTYRE and F.POINT 20
The quantifier elimination problem for rings without
nilpotent elements and for semi-simple rings
ELISABETH BOUSCAREN 13
Existentially closed modules : types and prime models
GREGORY CHERLiN 44
Rings of continuous functions : decision problems
PETER CLOTE 92
Weak partition relations, finite games, and independence
results in Peano arithmetic
FRAN~OISE DELON 108
Hensel fields in equal characteristic p >0
M.A.DICKMANN 117
On polynomials over real closed rings
JEAN-LOUIS DURET 136
Les corps faiblement alg~briquement clos non s@parablement
clos ont la propri~tS d'ind~psndance
ULRICH FELGNER 163
Horn-theories of abelian groups
PETR H~JEK and PAVEL PUDLAtK 174
Two orderings of the class of all countable models
of Peano arithmetic
ANGUS ~CINTYRE 186
Ramsey quantifiers in arithmetic
IV
KENNETH L.~L~NDERS 211
Computational complexity of decision problems in elementary
number theory
KENNETH McKENNA 228
Some diophantine Nullstellens~tze
GEORGE MILLS 2~8
A tree analysis of unprovable combinatorial statements
J .B. PARIS 312
A hierarchy of cuts in models of arithmetic
C.SMORYNSKI and J .STAVI 338
Cofinal extension preserves recursive saturation
J~OL VAN DEN DRIES ~6
Some model theory and number theory for models of weak
systems of arithmetic
A Jo .WILKIE 363
Applications of complexity theory to ~0 - definability
problems in arithmetic
GEORGE WILMERS 370
Minimally saturated models
B.I.ZILBER 381
Totally categorical theories : structural properties
and the non-finite axiomatizability
FURTHER REMARKS ON THE ELEMENTARY
THEORY OF FORMAL POWER SERIES RINGS
J. Becker~:~ J. Denef "~;:~ and L. Lipshitz
INTRODUCTION. |§ contains an elementary proof that the theory of the field F ((t))
P
of formal power series over the p element field F with cross section is
P
undecidable, and some extensions of this. This result is due to J. Ax (unpublished).
The authors learned it from B. Jacob who had independently rediscovered it. The
previous proofs were not elementary making use of a norm form and properties of the
norm residue symbol. §2 contains some results on the existential theories of power
series rings in n ~ 2 variables and §3 contains a result on definability in the
rings ~{X} of convergent power series.
§|. Let F be a field of characteristic p and let K = F((t)), the field of
formal power series over F . Define
Sol(s) +-4 ~x,y],...,yp_| C K (~ p=-xx +ty~ +t2y2 p ""+ tP-lYPp-|)" +
I.I. LEMMA. Suppose that F is perfect and let ~ = ~ t.~( i a. @ F . Then
iE77
i~>n
Sol(~) ++ 3x 0 ~ F (~0 = xP-xo)"
PROOF. Since (I aitl) p = I aPtiP we certainly have that if Sol(s) then
i.
0~( = x~-Xo ' where x = [ xitl . For the converse notice that Sol(cO is additive
in ~ (i.e. Sol(CO, Sol(E) ÷ SoI(c~+8)) so it suffices to prove
co
(a) Sol( [ c~itl) and )b( S-okl (at ) for all a E F k , E ~, k > O. For (a) if
i=!
oo . 2
= i I]= ~'tll set x = -~ * (-CO p + (-(~)P + ..• and y! = Y2 .... = Yp-1 = 0.
(b) Case .I p~k. Then k = pq-j with j E {l,2,...,p-l}. Since F is perfect
there is a w ~ F such that p = w a . Hence at -k = (wt-q)Pt j so we can set
:= Supported in part by N.S.F. 8002789.
~=~: Supported by the National Science Foundation of Belgium.
x = 0 , yj = wt -q and Yi = 0 for i # j.
(b) General case. The proof is by induction on k . k = I follows from case .I
Suppose Sol(wt -q) for all 0 < q < k , q ~ ~ , and w E F. We must prove
Sol(at-k). If p ~ k we are in case ,1 so suppose that k = qp . Let p = w a .
We have
at = [(wt-q) p wt -q] + wt -q
Now Sol[(wt-q) p - wt -q] (Set x = wt -q and Yi = OVi) and by the induction
hypothesis Sol(wt -q) so the result follows by the additivity of Sol. |
Define the following predicates (on F((t)))
Cros(u) ~-+ u ~ {t,t-l,t2,t-2,t3,t-3,...}
Con(x) +-+ x E F
Zer(~) ~-+ the constant term of ~ is zero
Int(x) +~+ x = ~ aitZ i ~ , a F (i.e. ord(x) >~ 0).
i>~O
We shall consider fields F which satisfy:
I )I F is perfect of characteristic p ¢ 0.
1
2) F is not closed under Artin-Schreier extensions (i.e. 3~ E F such
that p x -x = ~ has no solution in F ),
1.2. LEMMA. If F satisfies (x) then Zer(~) is definable in (F((t));Con,t>.
PROOF. We have by Lemma 1.1 that
Zer(~) ~+V6(Con(B) + Sol($~)).
1.3. LEMMA. Int is definable in F((t)).
PROOF. See [2].
1.4. LEMMA. F[t] is definable in (F((t));Zer,Cros>.
PROOF. We have
x @ Fit] ~-+ Int(x) A ~sVw[(Cros(w) A Int(~)) ÷ Zer(wx)].
Io5. LEMMA. If F satisfies (x) then (F((t));Cros,Con) is undecidable.
PROOF. This follows immediately from Lenm~a 1.2 and 1.4 and the fact that F[t] is
undecidable for any field F . (Notice that t is definable from Cros.)
1.6. THEOREM. If F is a finite field then <F((t));Cros) is undecidable.
PROOF. Let q = the cardinality of F . Then Con(x) +-+ q x -x = 0. |
Next we sharpen Lemma 1.5 to get the undecidability of (F((t));Cros) for any
field F which satisfies (x).
Define
)x(rTg-z ~ V B[(Vw(Cros(w) ÷ sol(~w))) ÷ Sol(~x)].
1.7. LEMMA. If F satisfies (~=) then
)a( Zer(x) is definable in <F((t));Cros> ,
(b) Zer(x) + Zer(x),
)c( [x e F ((t)) A Zer(x)] ÷ Zer(x).
P
PROOF. (b) Zer(x) ÷ VB(Con(~) ÷ Sol(Bx)).
)c( Let x = ~ aitl i C , F a 0 = , a 0 and let B e F((t)) be such that
P
Sol(Bt i) for all i # .0 We must show that Sol(~x). Let ord(B) > -n. Then
ord($ ~ aiti) > 0 and hence SoI(B ~ aiti). By the additivity of Sol we
i~>n i~>n
also have Sol( ~ ait3t~-), since a. ~ ~t , and hence So1(5 ~ aitl).|
i~<n 1 p
1.8. LEMMA. Suppose F satisfies (x). Then F ((t)) is definable in
P
<F((t));Cros).
PROOF. x E Fp((t)) +-+ Vw]c e Fp[(Cros(w) v w= )I ~ Zer(wx-c)]. |
From Theorem 1.6 and Lemma 1.8 we have:
1.9. THEOREM. If F satisfies (x) then (F((t));Cros) is undeeidable.
1.I0. THEOREM. For any prime p there are two perfect fields I F and 2 F of
characteristic p such that 2 ~ I F F but (Fl((t));Cros,Con) ~ (F2((t));Cros,Con).
PROOF, If F satisfies (~:) then F [t] and Fit] are definable in
P
(F((t));Cros,Con) by a formula depending only on the characteristic p of F .
The following formula ~ also depends only on p :
Vx3P ~Q(Con(x) ÷ P @ F [t] A Q e F[t] A p = (t-x)Q).
P
F((t)) satisfies ~ iff F is algebraic over F . Let I F be an infinite
P
algebraic extension of F which is not closed under Artin-Schreier extensions.
P
Let 2 F be any field which is elementarily equivalent with F] but not algebraic
over F . Then F ((t)) =] ~ and F2((t)) =] ~.
p I
REMARK. It is well known [3] that <F((t));Cros> is decidable when F is a
decidable field of characteristic zero.
QUESTION. Is F ((t)) (without crossseetion) decidable?
P
§2. In this section we shall extend some of the results of [6] and [5] about
the existential theories of power series rings. £ will denote the complex
numbers and F an arbitrary field. F~X~ is the local ring of formal power
series in the variables X = ,)nX,...,]X( and )X( denotes the maximal ideal of
F~X~ . F<X) is the subring of F~X~ of all power series algebraic over FIx].
If F is a valued field (eg. ¢) then F{X} is the ring of all convergent power
series (i.e. convergent on some neighbourhood of 0 ). In §2 and §3 we will always
assume that there are symbols for ...,2X,IX and Xn in the first order language.
The following results follow immediately from the extension of the Artin
Approximation Theorem given in §4 of [5]:
)]( <F<X) ; F<X|>) ~ <F~X~;F~ F]~) i.e. if fi(X,Y), g(X,Y) ~ F<X)[Y],
Y = )mY,...,IY( and ~ = 3Y],...,Ym{Aifi(X,Y) = 0 A g(X,Y) ¢ 0 A y1,...,yr depend
only on X]} then F~X~ ~ ~ implies F<X) ~ ¢ .
(2) (i) is also true with > < replaced by { } and F by { . This re-
quires the results of §5 of [8].
(3) Let fi(X,Y) C FIx,Y], Y = (YI,...,Ym). If the system
" fi(X,Y) = 0 A yl,...,y depend only on 1 X "
r
has a solution Y mod(X) k for all k , then it has a solution Y @ F~X~ .
(4) If the existeztial theory of F is decidable then so is the positive
existential theory of <F~X~ ;F[XI~> and hence by )I( and )2( also of
<F<X);F<XI )) and (F{X};F{XI}) (if F is a valued field). See [5], §6,
Remark (i).
On the other hand it was shown in [6] and [9] that:
(5) <¢<x1,x2>; ~<x1> , ¢<X2>> and (6[TXI,X2]]){~XIT] 7~X2~> have
different positive existential theories, i.e. there is a system
A.I )Y'X(1'f = 0 A lrY,...,]Y depend only on ]X
A Y ... depend only on 2 X
' r1+! ,Yr2
(with fi(X,Y) @ FIx,Y] Y , = ))mY,...,IY( which has a solution in ¢~XI,X2]] but
not in ~<X|,X2>. (The similar result is true with either ( ) or ~ ~ replaced
by { } and can be easily established by the same method.)
)6( <¢{X,Y};¢{Y})~ <~X,Y~ 7¢~X~) where X = (XI,X )2 and Y = .)4Y,-..,|Y(
This uses a counter example of Gabrielov (see [6]).
2,1. PROPOSITION. Let F be an arbitrary field.
)i( The existential theory of <F~X|,X2~ ;F~XI~ ~F~X2~) is undecidable,
i,e. formulas of the form
mY,..o,IY~ C F~X],X2~ (A fi(X,Y) = 0 g(X,Y) A # 0
A YI'""Yr e I F~XI~ A 2rY,...,l+lry E F~X2~ )
are undecidable (fi(X,Y),g(X,Y) E FIx,Y]).
(ii) The existential theory of (F~ XI,X2,X3~ ;F~ X I~ jF~XI,X2~> si undecid-
able i.e, formulas of the form
)7( ~Y~ ..... Ym ~ F~XI'X2'X3~ (~ (x'Y) = 0 fi
A g(X,Y) # 0 A Y|, ... ,Yrl ~ F~X|~ Yr1+l A 2r~ Y,,o,, F~ XI,X2~)
are undecidable, (The same results hold with ~ ~ replaced by ( ) or { ).}
PROOF. )i( We shall represent <~;+," ) in F~X]~- }0{ by the correspon-
dence ~ n +---- [Xln] = {gXln I s si a unit in F~XI~ .} Addition in ~ corre-
sponds to multiplication in F~X|~ and multiplication in ~ to exponentiation
in . F[[X]]] Since multiplication si positive existentially definable from 2
squaring it is sufficient to show that from A = IX[ n] we can define B = IX[ n ]
by a formula of the correct kind, The result will then follow from the undecid-
ability of Hilbert's Tenth problem.
Notice: )a( If A = [x1n] and m] B = IX2 and A ~ B mod(X I -X )2 then m =n
(i.e. if A(X )2 = B then m=n).
)b( If A E [xln] and ~] ~ B [X2 and C @ IX m I ] and ~ B C mod X 2-A
then m=n 2 (i.e. if B(A) =C then m=n2). The only inequalities which we needed
were of the form .Y # 0 because we represent ~ in F~XI~ - {0}.
(ii) We shall do the same as )i( but using formulas of the required form.