Rosenlicht fields

Shackell, John

doi:10.1090/s0002-9947-1993-1085945-5

Cited by 7 publications

(15 citation statements)

References 2 publications

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“…A function (or germ) u has level if u ∈ I and there are k, s ∈ Z with k+s • u ∼ k ; we then say that u has level s and write level(u) = s. Basic properties of level can be found in [23]. See Shackell [24] for a detailed examination (from a somewhat different point of view) and Marker and Miller [14] for some modeltheoretic considerations.…”

Section: Conventionsmentioning

confidence: 99%

Differential equations over polynomially bounded o-minimal structures

Lion

Miller

Speissegger

2002

Proc. Amer. Math. Soc.

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Abstract. We investigate the asymptotic behavior at +∞ of non-oscillatory solutions to differential equations y = G(t, y), t > a, where G : R 1+l → R l is definable in a polynomially bounded o-minimal structure. In particular, we show that the Pfaffian closure of a polynomially bounded o-minimal structure on the real field is levelled.A classical topic in asymptotic analysis is the study of the behavior of solutions at +∞ to equations P (t, y(t), . . . , y (n) (t)) = 0, where P : R 2+n → R is a nontrivial polynomial function. It was conjectured by E. Borel that the growth of such solutions should be bounded at +∞ by the (n + 1)-st compositional iterate of e x . This turned out to be false-see Boshernitzan [2] for a detailed account-unless infinitely oscillating solutions are somehow ruled out. But with such assumptions, sharper estimates are obtained that rule out transexponential growth as well as certain kinds of intermediate asymptotic behavior (e.g., that of solutions to the functional equation g • g = e x ; see Rosenlicht [23, §2]). In this paper, we extend these results to (systems of) first-order ODEs defined over polynomially bounded o-minimal structures on the real field. Examples of large classes of maps that generate such structures are given by van den Dries and Speissegger [9, 10] and Rolin et al. [20].Some of our results hold in o-minimal expansions of arbitrary ordered fields, while some appear to make sense only over the real numbers. All but one we prove by so-called standard methods (that is, without model theory). We organize the exposition as follows: First, results are stated over R in the form of an extended abstract, followed by proofs. Then we indicate those results that, after appropriate modification, hold in o-minimal expansions of arbitrary ordered fields.

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Section: Conventionsmentioning

confidence: 99%

Differential equations over polynomially bounded o-minimal structures

Lion

Miller

Speissegger

2002

Proc. Amer. Math. Soc.

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“…On the other hand, exponentials will frequently be applied to a wide range of arguments, and so for example, e\(x(l + e~x)) stands for exp(a;(l + e~x)). The following lemma, which formed part of Lemma 3 of [14], will be required. Proof of Lemma, 1.…”

Section: Theorem 2 -Let F Be a Hardy Field And Let F And G Be Polynmentioning

confidence: 99%

“…In [14] and [16] a generalisation of the concept of an asymptotic expansion was developed. Let T be a Hardy field and let 0 be a positive element of T which tends either to zero or infinity.…”

Section: Theorem 2 -Let F Be a Hardy Field And Let F And G Be Polynmentioning

confidence: 99%

“…This includes further discussion of Hardy fields, definitions of "nested form" and "nested expansion" and a more detailed description of the key results from [14] and [16]. In addition, we prove a lemma, which will be needed later, concerning the level of an element of a Hardy field.…”

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confidence: 98%

“…In [14] it was shown firstly that the asymptotic behaviour of such a function could be represented by a nested form^ which is a particular type of expression constructed using exponentials and logarithms. Secondly, it was shown that only a restricted number of these are possible for solutions of an equation of given order.…”

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Growth orders occurring in expansions of Hardy-field solutions of algebraic differential equations

Shackell¹

1995

Annales De L’institut Fourier

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Symbolic Asymptotics: Functions of Two Variables, Implicit Functions

Salvy

Shackell

1998

Journal of Symbolic Computation

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A number of recent papers have been concerned with algorithms to decide the limiting behaviour of functions of a single variable. Here we make a corresponding study of a class of functions of two variables, namely the exp-log functions. As in the one-variable case, we need to make certain assumptions regarding the handling of constants.Two of the main tools in the one-variable case are Hardy fields and nested forms. Here, we show how to compute some asymptotic estimates for two-variable exp-log functions. This method is then used to give an algorithm for computing the nested forms of real implicit functions.c Academic Press LimitedThis work is part of a global effort to automate the formal aspects of asymptotic expansions. It is possible to mechanize some techniques of asymptotics and build a computer algebra toolbox of these. A lot of work in symbolic asymptotics follows this approach and most existing facilities for asymptotic expansions in computer algebra systems have been obtained in this way. An alternative approach aims at studying the asymptotics of whole classes of problems, investigating all the possible asymptotic scales that may occur. The main tools here are nested forms and expansions, zero-equivalence methods and the theory of Hardy fields. The present paper follows this path. Nested forms and nested expansions were introduced by Shackell (1993a). A formal definition is given in Section 1. An example of a nested form is e log 2 xe √ log log x(c+φ 1 (x)), where c is a real constant and φ 1 (x) tends to 0 when x tends to infinity. In some cases, one can compute the nested form of φ 1 , introducing a new function φ 2 and then repeat the process, thus generating a sequence of nested forms; this sequence is called a nested expansion.The field H(x) of exp-log functions of a single variable, x, is formed of expressions built from x and the field K of real elementary constants (in the sense of Richardson, 1994), by means of arithmetic operations and the operations:In previous works (Shackell, 1990(Shackell, , 1993a(Shackell, , 1995(Shackell, , 1996Gruntz, 1996;Richardson et al., 1996) an algorithmic treatment in terms of nested expansions was given for the 0747-7171/98/030329 + 21 $25.00/0 sy970179 c 1998 Academic Press Limited 330 B. Salvy and J. Shackell asymptotics of (i) exp-log functions; (ii) Liouvillian functions and (iii) Hardy-field solutions of algebraic differential equations. All these algorithms require the use of a method for deciding zero equivalence in the class of functions concerned. This brings particular difficulties regarding constants. We discuss this matter more fully in Section 1.4. Inverse functions have long been problematic in asymptotics (Hardy, 1911;De Bruijn, 1981). However Salvy and Shackell (1992) gave an algorithm for inverting nested forms which solves the problem of expressing the asymptotic behaviour of inverse functions. In the present paper we treat the more general problem of implicit functions. More precisely, let H(x, y) denote the field obtained by closing K(x, y) ...

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Rosenlicht fields

Cited by 7 publications

References 2 publications

Differential equations over polynomially bounded o-minimal structures

Differential equations over polynomially bounded o-minimal structures

Growth orders occurring in expansions of Hardy-field solutions of algebraic differential equations

Symbolic Asymptotics: Functions of Two Variables, Implicit Functions

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