Pacific Journal of Mathematics EXPONENTIAL RINGS, EXPONENTIAL POLYNOMIALS AND EXPONENTIAL FUNCTIONS LOU VAN DEN DRIES Vol. 113, No. 1 March 1984 PACIFIC JOURNAL OF MATHEMATICS Vol. 113, No 1,1984 EXPONENTIAL RINGS, EXPONENTIAL POLYNOMIALS AND EXPONENTIAL FUNCTIONS Lou VAN DEN DRIES In this paper we define the category of exponential rings and develop some of its basic properties. Introduction. An exponential ring, or E-ring for short, is a pair (i?, E) with R a ring—in this paper always commutative with 1—and E a morphism of the additive group of R into the multiplicative group of units of R, that is, E(x + y) = E(x)E(y) for all x9 y in i?, and E(0) = 1. Examples are (R, ax), a any positive real, and (C, ex). Of course, any ring R can be expanded to an £-ring (R, E) by putting E(x) — 1 for all x\ such brings will be called trivial. Ken Manders observed that an Zί-ring whose underlying ring has no nilpotents φ 0 and has characteristic a prime/? > 0 is trivial: in such a ring each x satisfies 1 = E(0) — E(px) = E{x)p, so (E(x) - \y = 0, which implies E(x) = 1. Related notions of exponential ring have been considered by M. Beeson, by B. Dahn and Wolter, and by A. Wilkie, all in connection with the longstanding open problem of A. Tarski on the decidability of the field of reals with exponentiation. An effective positive solution to this problem seems unlikely without major advances in transcendental number theory: such a solution would give us a decision method to answer any question: is ee = p/qΊ>> where/?, q are positive integers. Of course there is such a decision method, but, as we don't know yet whether ee is rational, we don't know how it works. Now in mathematical practice it is less the effectiveness of Tarski's decision method for the real field which matters—though this aspect is interesting—but rather the information the method provides on the algebraic-topological nature of the definable sets in Rm, and on the asymptotic behavior of definable functions. For example in semi-algebraic and real algebraic geometry this use is formalized in the Tarski-Seidenberg theorem (in an inconstructive version) and in a result like the finiteness of the number of connected components of a semi-algebraic set. Parts of this use of Tarski's work on the elementary theory of the reals offer more hope of being generalized to the E-ήng (R, ex). The following 51 52 LOU VAN DEN DRIES conjecture seems to me justified: Each subset of Rm definable (with parameters) in the language of E-rίngs has finitely many connected components. In fact, what I have in mind is a stronger conjecture, that definable sets have cylindrical decompositions, a little bit similar to those for semi-algebraic sets constructed by G. E. Collins [C]. This stronger conjec- ture has the advantage that one only needs to prove it for quantifier free definable sets. For m = 1 this has been done long ago by Hardy [H], with effective versions more recently given by Richardson [R] and Macintyre [M]. (In the case m = 1 the components are of course intervals.) The case m — 2 will be treated in [D]. To prepare the ground for [D] we solve here two basic problems: (1) What is the structure of the free E-rings, respectively of the free E-algebras over a given E-ringΊ (2) When is the map p \-+p (assigning to an E-polynomial in Xl9... 9Xm over (R, E) the corresponding R-valuedfunction in m variables) infective! The answer to (1) is completely satisfactory, see §1 and §2, in particular (2.4). For (2) we have a sufficient condition in terms of the existence of derivations of certain type—see §3 and (4.1)—and we get positive answers in the cases (R, ex) and (C, ex). The case (R, ex) gener- alizes in fact to brings (/?, E) where R is an ordered field and E(x) > 1 + x for all x. If we add to this inequality the assumption that £-poly- nomial functions in one variable over R have a maximum on each segment [a, b]9 then we can also give effective finite bounds on the number of roots such functions can have in R, see (4.11). (This is related to results for R due to Richardson [R] and Macintyre [M].) After a first draft of this paper was written we learned that the result in (4.6) for (C, ex) was also proved by W. Henson and L. Rubel by completely different methods, namely Nevanlinna theory, see Notices Amer. Math. Soc, November 1981. (In contrast, we only use elementary algebra.) Let us also mention here that the one variable case of (4.6) had been obtained before by A. Wilkie using differential algebra (unpublished). 1. Polynomial brings. (1.1) In the following let R be an E-ήng with exponential map denoted by E, and m a positive integer. We are going to define the ring of E-polynomials in the indeterminates E Xl9...9Xm over R, denoted by R[Xl9... ,Xm] It will have the structure of a group ring over the ordinary polynomial ring R[XX9...9Xm]. Its additive EXPONENTIAL RINGS 53 group will be constructed as a direct sum and for each k >: — 1, the additive group R ® Ao ® ®Ak is equipped with a multiplication making it a ring Rk\ this is done in such a way that R_ j = R as a ring and i?^ is a subring of i?^+ j. Further, for each k > — 1 we define a morphism £^ of the additive group of Rk into the group of units of Rk+\\ again Ek+λ will extend £Λ. Then the underlying ring of E R[XΪ9...,Xm] will be taken as lim Rk — U Rk, and its exponential map will be given by E(x) = £Λ( x) if ?E Λ*. The construction of the Ak9 Rk, and Ek is done by recursion: Put R_x = R, Ro = R[Xλ9. 9Xm] and Ao = the ideal (*,,. ,XJ Λ[A\, »-^m]ϊ *^s §iyes RQ = ^ ® ^o (as additive groups); further 2?_! is the compositioni?_! = i? -^i? =» i?[^i,.. .,^m] = i?0 Suppose fc >: 0 and Λ^-^ i?^, Ak and J5'Λ:_ x have been defined in accordance with the description above: Rk = Rk-\ ® Ak, and £^_! is a morphism Rk_x ^> group of units of Rk. Then we form a multiplicative copy exp(v4£) of the additive group Ak9 i.e., exp(Ak) is a multiplicative group and an isomorphism exp: Ak — exp(^l^) is given. We put Rk+X = i?^[exρ(^l^)] (group ring of exp(^) over Rk), and Ak+ι — i?rsubmodule of Rk+λ (freely) generated by the exp(α) with a E Ak, aΦO. Obviously Rk+ι — Rk ® Ak+ι as additive groups. Write r E Rk as r — r' + a r with r E Rk-γ, a E Ak, and define .E^: Rk -» 2?^+! by ^(r) = Vi(Op() E This completes the definition of R[Xl9.. .,Xm] . Note that exp: E Ak -> exρ(^4^) is a restriction of the exponential map £Όf i?[^,... ,Xm] \ therefore we will write from now on E(Ak)9 E(p) instead of exρ(^), E exp(/?). R[Xl9...,Xm] is obviously an .E-ring extension of R. (1.2) PROPOSITION. Given an E-ring S andsu...9sm E S, each E-ring morphism <j>: R -» S has a unique extension to an E-ring morphism ψ: Proof. Use the corresponding universal property of R[Xλ9... ,Xm] = Ro and extend stepwise to each Rk, k>0. D E (1.3) We call the members of R[XX9... ,Xm] exponential polynomi- als, or ^-polynomials for short, in the variables Xl9...9Xm over R. 54 LOU VAN DEN DRIES E Note that R[Xl9...,Xm] is generated as an £-ring by Xλ9...,Xm over R. More generally if S is an jE-ring extension of R and x,,...,xw6S, E then we write R[xl9... ,xm] for the j?-subring of S generated by xl9...9xm over R. (1.4) If φ: 2? -* S is an Zί-ring morphism, then, according to (1.2), φ E has a unique extension to an is-ring morphism R[Xl9...9Xm] -* E S[Xλ9...9Xm] sending Xt to X{9 this morphism we will indicate by P I-+P*. (For ordinary polynomials it replaces the coefficients of/? by their φ-images.) If S is an is-ring extension of R and φ is the inclusion R-* S9 then ψ p H>/?Φ is 1-1, and we will identify p and /? , as is common for ordinary E polynomials. So R[Xl9...,XJ* C S[Xl9... 9Xm] . (1.5) A set / gives rise to the iϊ-ring Rr of functions I -> R where the operations are defined pointwise. For /^0 we identify the constant functions with their values in R, so R C Rτ. If / = Rm, then the coordi- nate functions are denoted by xl9...9xm where Xi(rl9...9rm) — η. The E (ΛW) i?-ring morphism R[Xl9... ,Xm] -^ i? fixing i? and sending each Jζ to Λ:,- will be indicated by p \-*β. Them's are called ^-polynomial functions (in m variables). In §4 we will give conditions under which p \->p is a 1-1 correspon- dence. (1.6) PROPOSITION. If R is an integral domain of characteristic 0 then E R[Xλ,...,Xm] is an integral domain whose units are of the form uE(p), u a E unitofR9p(ΞR[Xu...,Xm] .