Properties of O-minimal Exponential Fields

Transcription

1 1 Universität Konstanz, Fachbereich Mathematik und Statistik 11 September th ÖMG Congress and Annual DMV Meeting 1 Supported by DVMLG-Nachwuchsstipendium

2 1 Preliminary Notions 2 Decidability of R exp 3 Residue Exponential Fields

3 1 Preliminary Notions 2 Decidability of R exp 3 Residue Exponential Fields

4 Model Theoretic Setting

5 Model Theoretic Setting fixed language of ordered exponential fields L exp = (+,, 0, 1, <, exp)

6 Model Theoretic Setting fixed language of ordered exponential fields L exp = (+,, 0, 1, <, exp) L exp -structures e.g. (R, +,, 0, 1, <, exp)

7 Model Theoretic Setting fixed language of ordered exponential fields L exp = (+,, 0, 1, <, exp) L exp -structures e.g. (R, +,, 0, 1, <, exp) L exp -formulas and L exp -sentences e.g. y x > exp(y) or x y x > exp(y)

8 Model Theoretic Setting fixed language of ordered exponential fields L exp = (+,, 0, 1, <, exp) L exp -structures e.g. (R, +,, 0, 1, <, exp) L exp -formulas and L exp -sentences e.g. y x > exp(y) or x y x > exp(y) notion of satisfiability e.g. (R, +,, 0, 1, <, exp) = x y x > exp(y)

9 Ordered Exponential Fields

10 Ordered Exponential Fields Definition Let (K, +,, 0, 1, <) be an ordered field. A unary function exp which is an order-preserving isomorphism from (K, +, 0, <) to (K >0,, 1, <) is called an exponential on (K, +,, 0, 1, <). The L exp -structure (K, +,, 0, 1, <, exp) is called an ordered exponential field.

11 Ordered Exponential Fields Definition Let (K, +,, 0, 1, <) be an ordered field. A unary function exp which is an order-preserving isomorphism from (K, +, 0, <) to (K >0,, 1, <) is called an exponential on (K, +,, 0, 1, <). The L exp -structure (K, +,, 0, 1, <, exp) is called an ordered exponential field. Most prominent example: R exp = (R, +,, 0, 1, <, exp) the real exponential field.

12 O-minimality

13 O-minimality Definition An ordered structure (M, <,...) is called o-minimal if every parametrically definable subset of M is a finite union of points and open intervals in M.

14 O-minimality Definition An ordered structure (M, <,...) is called o-minimal if every parametrically definable subset of M is a finite union of points and open intervals in M. Theorem (Wilkie 1996) The real exponential field R exp is o-minimal.

15 O-minimality Definition An ordered structure (M, <,...) is called o-minimal if every parametrically definable subset of M is a finite union of points and open intervals in M. Theorem (Wilkie 1996) The real exponential field R exp is o-minimal. Example: The formula y x 2 > exp(y) + π parametrically defines the set {x R y x 2 > exp(y) + π} = (, π) ( π, ) over R exp.

16 1 Preliminary Notions 2 Decidability of R exp 3 Residue Exponential Fields

17 Decidability

18 Decidability Definition An L-structure M is called decidable if there exists an algorithm that determines whether for a given L-sentence ϕ we have M = ϕ or M = ϕ.

19 Decidability Definition An L-structure M is called decidable if there exists an algorithm that determines whether for a given L-sentence ϕ we have M = ϕ or M = ϕ. Th(M) the theory of M, i.e. the set of all L-sentences which M satisfies.

20 Decidability Definition An L-structure M is called decidable if there exists an algorithm that determines whether for a given L-sentence ϕ we have M = ϕ or M = ϕ. Th(M) the theory of M, i.e. the set of all L-sentences which M satisfies. Theorem An L-structure M is decidable if and only if Th(M) is recursively axiomatizable.

21 Schanuel s Conjecture

22 Schanuel s Conjecture Schanuel s Conjecture Let α 1,..., α n C be linearly independent over Q. Then td Q (Q(α 1,..., α n, e α 1,..., e αn )) n.

23 Schanuel s Conjecture Schanuel s Conjecture Let α 1,..., α n C be linearly independent over Q. Then td Q (Q(α 1,..., α n, e α 1,..., e αn )) n. Schanuels Conjecture would, for instance, imply the algebraic independence of e and π.

24 Schanuel s Conjecture Schanuel s Conjecture Let α 1,..., α n C be linearly independent over Q. Then td Q (Q(α 1,..., α n, e α 1,..., e αn )) n. Schanuels Conjecture would, for instance, imply the algebraic independence of e and π. Real Schanuel s Conjecture (SC) Let α 1,..., α n R be linearly independent over Q. Then td Q (Q(α 1,..., α n, e α 1,..., e αn )) n.

25 Decidability of the Real Exponential Field

26 Decidability of the Real Exponential Field Theorem (Macintyre, Wilkie 1996) Assume (SC). Then R exp is decidable.

27 Decidability of the Real Exponential Field Theorem (Macintyre, Wilkie 1996) Assume (SC). Then R exp is decidable. Macintyre and Wilkie construct a recursive subtheory of Th(R exp ) which, under the assumption of (SC), is complete.

28 Transfer Conjecture

29 Transfer Conjecture Elementary equivalence: Two L-structures M and N are elementarily equivalent if they satisfy exactly the same L-sentences. We write M N.

30 Transfer Conjecture Elementary equivalence: Two L-structures M and N are elementarily equivalent if they satisfy exactly the same L-sentences. We write M N. EXP: L exp -sentence stating that the differential equation exp = exp holds.

31 Transfer Conjecture Elementary equivalence: Two L-structures M and N are elementarily equivalent if they satisfy exactly the same L-sentences. We write M N. EXP: L exp -sentence stating that the differential equation exp = exp holds. Transfer Conjecture (TC) Let K exp be an o-minimal EXP-field. Then K exp R exp.

32 Transfer Conjecture Elementary equivalence: Two L-structures M and N are elementarily equivalent if they satisfy exactly the same L-sentences. We write M N. EXP: L exp -sentence stating that the differential equation exp = exp holds. Transfer Conjecture (TC) Let K exp be an o-minimal EXP-field. Then K exp R exp. Theorem (Berarducci, Servi 2004) Assume (TC). Then R exp is decidable.

33 Transfer Conjecture Elementary equivalence: Two L-structures M and N are elementarily equivalent if they satisfy exactly the same L-sentences. We write M N. EXP: L exp -sentence stating that the differential equation exp = exp holds. Transfer Conjecture (TC) Let K exp be an o-minimal EXP-field. Then K exp R exp. Theorem (Berarducci, Servi 2004) Assume (TC). Then R exp is decidable. Open question: Does (SC) imply (TC)?

34 Transfer Conjecture Elementary equivalence: Two L-structures M and N are elementarily equivalent if they satisfy exactly the same L-sentences. We write M N. EXP: L exp -sentence stating that the differential equation exp = exp holds. Transfer Conjecture (TC) Let K exp be an o-minimal EXP-field. Then K exp R exp. Theorem (Berarducci, Servi 2004) Assume (TC). Then R exp is decidable. Open question: Does (SC) imply (TC)? This question motivates the study of o-minimal EXP-fields.

35 1 Preliminary Notions 2 Decidability of R exp 3 Residue Exponential Fields

36 Natural Valuation

37 Natural Valuation Let K be an ordered field. We define an equivalence relation on K by a b if and only if there exists n Z such that a < n b and b < n a. The equivalence class of a given a K is called the archimedean equivalence class of a.

38 Natural Valuation Let K be an ordered field. We define an equivalence relation on K by a b if and only if there exists n Z such that a < n b and b < n a. The equivalence class of a given a K is called the archimedean equivalence class of a. Let G = {[a] a K \ {0}} and define on G addition by [a] + [b] = [ab] and an order by [a] < [b] if and only if a > b and a b. Then (G, +, <) is an ordered group with neutral element 0 = [1]. It is called the valuation group of K under the natural valuation.

39 Natural Valuation Let K be an ordered field. We define an equivalence relation on K by a b if and only if there exists n Z such that a < n b and b < n a. The equivalence class of a given a K is called the archimedean equivalence class of a. Let G = {[a] a K \ {0}} and define on G addition by [a] + [b] = [ab] and an order by [a] < [b] if and only if a > b and a b. Then (G, +, <) is an ordered group with neutral element 0 = [1]. It is called the valuation group of K under the natural valuation. Set v : K G { } by v(a) = [a] for a K \ {0} and v(0) =. v is called the natural valuation on K.

40 Residue Exponential Field

41 Residue Exponential Field Definition Let K be an ordered field. Let O = {x K v(x) 0} and I = {x K v(x) > 0}. Then K = O/I defines an archimedean field. This is called the residue field of K.

42 Residue Exponential Field Definition Let K be an ordered field. Let O = {x K v(x) 0} and I = {x K v(x) > 0}. Then K = O/I defines an archimedean field. This is called the residue field of K. Proposition Let K exp be an o-minimal EXP-field. Then exp : K K >0, a exp(a) defines an exponential on K. Moreover, K exp = (K, +,, 0, 1, <, exp) is an archimedean EXP-field.

43 Residue Exponential Field Definition Let K be an ordered field. Let O = {x K v(x) 0} and I = {x K v(x) > 0}. Then K = O/I defines an archimedean field. This is called the residue field of K. Proposition Let K exp be an o-minimal EXP-field. Then exp : K K >0, a exp(a) defines an exponential on K. Moreover, K exp = (K, +,, 0, 1, <, exp) is an archimedean EXP-field. We call K exp the residue exponential field of K exp.

44 Archimedean O-minimal Exponential Fields

45 Archimedean O-minimal Exponential Fields Theorem (Laskowski, Steinhorn 1995) Let K exp be an archimedean o-minimal EXP-field. Then K exp R exp.

46 Archimedean O-minimal Exponential Fields Theorem (Laskowski, Steinhorn 1995) Let K exp be an archimedean o-minimal EXP-field. Then K exp R exp. Approach towards (TC): Show that any o-minimal EXP-field has an archimedean prime model.

47 Approach through Residue Exponential Field

48 Approach through Residue Exponential Field Theorem Let K exp be an exponential field such that K exp R exp. Then K exp K exp.

49 Approach through Residue Exponential Field Theorem Let K exp be an exponential field such that K exp R exp. Then K exp K exp. Assertion Let K exp be an o-minimal EXP-field. Then (SC) implies K exp K exp.

50 Embeddability of the Residue Field

51 Embeddability of the Residue Field Theorem Let K exp be a sufficiently saturated o-minimal EXP-field. Then K exp = R exp. Moreover, assuming (SC), we have R exp K exp.

52 Embeddability of the Residue Field Theorem Let K exp be a sufficiently saturated o-minimal EXP-field. Then K exp = R exp. Moreover, assuming (SC), we have R exp K exp. Corollary Assume (SC). Let K exp be an o-minimal EXP-field. Then K exp = Th (R exp ).

53 Embeddability of the Residue Field Theorem Let K exp be a sufficiently saturated o-minimal EXP-field. Then K exp = R exp. Moreover, assuming (SC), we have R exp K exp. Corollary Assume (SC). Let K exp be an o-minimal EXP-field. Then K exp = Th (R exp ). Theorem Let K exp be an o-minimal EXP-field. Then the following are equivalent: 1 Th (R exp ) = Th (K exp ) and K exp is model complete. 2 (TC): K exp R exp.

54 T -convexity

55 T -convexity Theorem Let K exp be an o-minimal EXP-field and T = Th(K exp ). Then the following are equivalent: 1 O is T -convex. (i.e. for any continuous 0-definable function f on K we have f (O) O). 2 Any continuous 0-definable function f : K K is exponentially bounded (i.e. for any continuous 0-definable function f on K and sufficiently large x we have f (x) < exp(exp(... exp(x)))) and bounded by an integer on [0, 1]. 3 (TC): K exp R exp.

56 T -convexity Theorem Let K exp be an o-minimal EXP-field and T = Th(K exp ). Then the following are equivalent: 1 O is T -convex. (i.e. for any continuous 0-definable function f on K we have f (O) O). 2 Any continuous 0-definable function f : K K is exponentially bounded (i.e. for any continuous 0-definable function f on K and sufficiently large x we have f (x) < exp(exp(... exp(x)))) and bounded by an integer on [0, 1]. 3 (TC): K exp R exp. This draws a connection between (SC) and the open question whether an o-minimal structure which is not exponentially bounded exists.

57 References [1] A. Berarducci and T. Servi, An effective version of Wilkie s theorem of the complement and some effective o-minimality results, Ann. Pure Appl. Logic 125 (2004) [2] L. van den Dries and A. H. Lewenberg, T -convexity and Tame Extensions, J. Symb. Log. 60 (1995) [3] S. Kuhlmann, Ordered Exponential Fields, Fields Inst. Monogr. 12 (Amer. Math. Soc., Providence, RI, 2000). [4] A. Macintyre and A. Wilkie, On the decidability of the real exponential field, Kreiseliana: about and around Georg Kreisel (ed. P. Odifreddi; A. K. Peters, Wellesley, MA, 1996) [5] A. Tarski, A decision method for elementary algebra and geometry (RAND Corporation, Santa Monica, CA, 1948). [6] A. Wilkie, Model completeness results for expansions of the ordered Field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc. 9 (1996)