B. Dahn and P. Goring, Notes on exponential-logarithmic terms, Fund. Math. 127 (1986), 45-50.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....states that, given a eld C and a totally ordered monomial group M, the set C[ M] of series f : C M with well ordered support in M carries a natural eld structure. This result was generalized by Higman [Hig52] to the case of partially ordered monomial monoids M. More recently, Dahn and G#ring [DG86] and #calle [#92] constructed so called elds of itransseriesj, which are elds of generalized power series C[ M] in the sense of Hahn, with additional structure, such as exponentiation, dioeerentiation, integration, composition, etc. Examples of transseries are = x log x log log x log log ....

B. I. Dahn and P. G#ring. Notes on exponential-logarithmic terms. Fundamenta Mathematicae, 127:4550, 1986.

....are f 4 = i(x) and f 5 . Notice that f 5 satises the functional equation f 5 (x ) f 5 (e log 2 Historically speaking, transseries appeared independently in at least three dioeerent contexts: Model theory. The rst construction of the eld of transseries goes back to Dahn and G#ring [DG86], who were interested in non standard models for the theory of real numbers with exponentiation. Much recent progress on this subject has been made through works by Wilkie, van den Dries, Macintyre and others. The theory of transseries also bears many similarities with Conway s theory of surreal ....

B. I. Dahn and P. G#ring. Notes on exponential-logarithmic terms. Fundamenta Mathematicae, 127:45 50, 1986.

....In this paper we concentrate on developing the formalism of LE series. Section 1 establishes notation and also contains some useful facts on generalized series fields k( G) for which we do not know a convenient reference. In section 2 we follow in essence the original method of Dahn and Goring [3, 4] in constructing the field R( x Gamma1 ) LE , but present it, we hope, in a more intuitive way. In addition it is notationally more natural than in our earlier paper [6] We also show in section 2 that each LE series over R contains only countably many monomials. In section 3 we define ....

....LE series. As this extra generality could be useful in questions about dependence on parameters and doesn t cost any extra effort, we actually work in this somewhat more general setting. It seems that the interest in logarithmic exponential series has two rather different origins. Dahn and Goring [2, 3] were influenced by Tarski s problem about the real exponential field, while Ecalle [8] and Il yashenko [14] encountered LE series in their work on the Dulac problem. There are also potential connections with the theory of surreal numbers of Conway and Kruskal, and super exact asymptotics . The ....

B. Dahn and P. Goring, Notes on exponential logarithmic terms, Fund. Math 127 (1986), 45-50.

....In this paper we concentrate on developing the formalism of LE series. Section 1 establishes notation and also contains some useful facts on generalized series elds k( G) for which we do not know a convenient reference. In section 2 we follow in essence the original method of Dahn and G oring [3, 4] in constructing the eld R( x 1 ) LE , but present it, we hope, in a more intuitive way. In addition it is notationally more natural than in our earlier paper [6] We also show in section 2 that each LE series over R contains only countably many monomials. In section 3 we de ne the ....

....LE series. As this extra generality could be useful in questions about dependence on parameters and doesn t cost any extra e ort, we actually work in this somewhat more general setting. It seems that the interest in logarithmic exponential series has two rather di erent origins. Dahn and G oring [2, 3] were in uenced by Tarski s problem about the 4 L. VAN DEN DRIES, A. MACINTYRE AND D. MARKER real exponential eld, while Ecalle [8] and Il yashenko [14] encountered LE series in their work on the Dulac problem. There are also potential connections with the theory of surreal numbers of Conway ....

B. Dahn and P. Goring, Notes on exponential logarithmic terms, Fund. Math 127 (1986), 45-50.

....will define an exponential E so that E1) E2) E4) and E5) are satisfied. This step of the construction is due to Dahn ( 3] The second step of the construction is to extend R( t) E to R( t) LE by adding logarithms. Our methods here considerably simplify earlier work of Dahn and Goring ([4]) This construction is carried out in x2. In x3 we prove some technical results about truncations of series. These results are crucial for the applications in x4 and x5. In x4 we answer Hardy s question, while in x5 we prove the undefinability of certain natural integrals and the zeta function ....

B. Dahn and P. Goring, Notes on exponential logarithmic terms, Fund. Math 127 (1986), 45-50.

.... is obtained by repeated application of logarithm, exponential and field operations starting from one variable x and rational numbers, with the restriction that any subexpression must be real for sufficiently large x) was studied from an asymptotic point of view in [7] However, it was only in [3] that the comparison problem for exp log functions was shown to be reducible to constant comparisons, and in [15] that an effective algorithm was produced to perform this reduction. Although no complexity study of this comparison algorithm has been done, it is rather expensive in practice. We now ....

Dahn, B. I., and Goring, P. (1986) `Note on exponential-logarithmic terms', Fundamenta Mathematicae, 127, 45--50.

....at x = 1. The problem here is that if the exponentials are expanded, and terms collected, the powers of x Gamma1 dominate the exponentials but forever cancel out. There is thus a risk of non termination if such examples are not handled with care. It was shown in 1984 by B. Dahn and P. Goring [4] that limit computation could be reduced to the so called constant problem; that is to say the problem of finding an algorithm to determine the signs of constant expressions. Then in [21] an actual algorithm was given to perform limit computation (modulo the constant problem) The underlying ....

Dahn, B. I., and G oring, P. Note on exponential-logarithmic terms. Fundamenta Mathematicae 127 (1986), 45--50.

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B. Dahn and P. Goring, Notes on exponential-logarithmic terms, Fund. Math. 127 (1986), 45-50.

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B.I. Dahn and P. Goring. Notes on exponential-logarithmic terms. Fundamenta Mathematicae, 127:45--50, 1986.

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