J. van der Hoeven. Automatic asymptotics. PhD thesis, #cole polytechnique, France, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Ecole Polytechnique 91128, Palaiseau France Email: vdhoeven lix.polytechnique.fr Web : http: lix.polytechnique.fr:80 vdhoeven February 15, 1996 1 Introduction In the last years, several asymptotic expansion algorithms have appeared (see [Sh 90] Sh 91] GoGr 92] RSSV 96] VdH 96b] etc. These algorithms are have the property that they can deal with very general types of singularities, such as singularities arising in the study of certain algebraic differential equations. However, attention has been restricted so far to functions with strongly monotonic asymptotic ....

....functions at infinity, originally due to Shackell (see [Sh 91] For this, we assume the existence of an oracle for deciding whether an explog function is zero in a neighbourhood of infinity. This problem has been reduced to the corresponding problem for exp log constants in [VdH 96a] resp. VdH 96b] A solution to the constant problem was given by Richardson in [Rich 94] modulo Schanuel s conjecture: Conjecture 1. Schanuel) If ff 1 ; Delta Delta Delta ; ff n are Q linearly independent complex numbers, then the transcendence degree of Q [ff 1 ; Delta Delta Delta ; ff n ; e ] ....

[Article contains additional citation context not shown here]

J. van der Hoeven. Automatic asymptotics. PhD. thesis, ' Ecole Polytechnique, France (in preparation).

No context found.

J. van der Hoeven. Automatic asymptotics. PhD thesis, #cole polytechnique, France, 1997.

No context found.

J. van der Hoeven. Automatic asymptotics . PhD thesis, #cole polytechnique, France, 1997.

....more standard setting of di erential algebra. We believe it to be more ecient. With some more work, it might be possible to give complexity bounds for the algorithm (or a modi ed version of it) along the same lines as [12] Such bounds are also interesting in relation to witness conjectures [17, 13, 16, 8]. On the longer run, the algorithm might generalize to the multivariate setting of partial di erential equations with initial conditions on a subspace of dimension 0. Throughout the paper, we will assume that the reader is familiar with di erential algebra and the notations used in this eld; ....

....expansions [3, 14] for implementing a zerotest. However, such a zero test might be slow for expressions which can be quickly rewritten to zero (like x x, where x is a complicated expression) Also, if we want the approach to be ecient, good bounds (such as the ones predicted by witness conjectures [17, 13, 16, 8]) would be necessary. At the moment, we only have Khovanskii type bounds in the case of Pfaan functions. A new strategy for obtaining bounds, which might generalize to higher order equations by adapting the algorithm in this paper, has been proposed in [12] However, the obtained bounds are still ....

[Article contains additional citation context not shown here]

van der Hoeven, J. Automatic asymptotics. PhD thesis,  Ecole polytechnique, France, 1997.

....we show how to determine the solutions of an arbitrary algebraic dioeerential equation over the complex transseries. We will show that such equations always admit complex transseries solutions. However, the eld of complex transseries is not dioeerentially algebraically closed. 1 Introduction In [vdH97], we have studied the asymptotic behaviour of solutions to algebraic dioeerential equations in the setting of strongly monotonic or real transseries. We have given a theoretical algorithm to nd all such solutions, which is actually eoeective for suitable subclasses of transseries. More recently, ....

....dene complex transseries. The diOEculty is that it is not clear a priori whether an expression like e should be seen as an innitely large or an innitely small transmonomial. Several approaches can be followed. A rst approach, based on pointwise algebras, was already described in chapter 6 of [vdH97]. However, this approach has the drawback that it is not easy to compute with complex transseries. A second more computational approach is described in section 3. Roughly speaking, it is based on the observation that all computations with complex transseries can be done in a similar way as in the ....

[Article contains additional citation context not shown here]

J. van der Hoeven. Automatic asymptotics . PhD thesis, #cole polytechnique, France, 1997.

....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 log x Xi ; e x 2 e = Gamma(x) 2 xlog x Gammax Gamma log x In [vdH97], we have shown how to dioeerentiate, integrate and compose such transseries, and how to solve algebraic dioeerential equations (whenever possible) In this paper, we will be concerned with the development of an abstract operator theory for generalized power series, in the setting of partially ....

....plugging in the left hand side of (6.4) into the right hand side. Again, the solution may be expressed naturally in terms of the coeOEcients of the equation. Example 6.7. Let T = C[ M] be the eld of transseries in x, whose logarithmic and exponential depths are bounded by some integer d 2 N [vdH97]. The transseries e Gammae x x Xi is an example of an element in T if d = 2. Now consider the integral equation ; 6:5) for f ; g 2 T and where f ; g OE e . Taking N= fm2 Mjm OE e g we may consider the operator Phi: C[ N] Theta C[ N] C[ N] f ; g) Delta g ....

J. van der Hoeven. Automatic asymptotics . PhD thesis, #cole polytechnique, France, 1997.

....de l equation fonctionnelle E(x 1) e E(x) qui cro t plus vite que toute exponentielle it er ee. Il conjectura egalement l inexistence d une L fonction asymptotique a l inverse fonctionnelle de log x log log x. Dans ce r esum e nous esquissons une d emonstration de cette conjecture [8] (qui fut d emontr ee ind ependamment par Marker, Macintyre et van den Dries [5] voir aussi [7] Pour ce faire, nous commen cons par donner un algorithme th eorique pour calculer un d eveloppement asymptotique d une L fonction. Nous utiliserons egalement un r esultat d u a Liouville [4] ....

....moyen simple pour d evelopper une L s erie lexicographiquement par rapport a b n ; b 1 , les coecient it er es etant a nouveau des L s eries. Remarque. Pour rendre ces d eveloppements totalement e ectifs, il faut remplacer R par un sous corps e ectif R, stable par exp et log. On montre [8], qu il existe alors un test a z ero pour les L s eries. Souvent, on prend pour R le corps des fonctions exp logs, c. a d. le plus petit sous corps de R stable par exp et log. Richardson [6] a d emontr e que ce corps est e ectif, si la conjecture de Schanuel est vraie. 4 Bases canoniques Soit S ....

J. van der Hoeven. Automatic asymptotics. PhD thesis,  Ecole polytechnique, France, 1997. 6

....the path z z in uence the complexity of e ective analytic continuation In particular, what happens if the path approaches a singularity The remainder of the introduction is devoted to a brief discussion of these questions. We notice that much of the material presented here also appeared in [15], but we think that the presentation in the present paper is more elegant. The section 4.1 and algorithm B from section 2.2 are new. 1.1 E ective bounds Since all our analytic continuation algorithms will be based on power series evaluations, question Q1 reduces to the problem of computing ....

J. van der Hoeven. Automatic asymptotics. PhD thesis,  Ecole polytechnique, France, 1997. 21

....like (2) if N is a formal parameter. In this article, we propose a new algorithm in order to deal with these problems and a generalization to the case of parameterized exp log functions. In section 2, we first recall the basic expansion algorithm from [25] using the terminology from our thesis [16]. In section 3, we introduce the important concept of Cartesian representations in automatic asymptotics: we show that, given the germ at infinity of an exp log function f , their exist infinitesimal elements z 1 ; Delta Delta Delta ; z k of a suitable asymptotic scale, such that f can be ....

....asymptotic zero test for exp log functions. ffl Several efficient algorithms can be used for the manipulation of Laurent series in several variables [18, 1, 2, 17] ffl Cartesian representations are essentially needed for a further development of automatic asymptotics: see section 5 and [16]. Although we did not perform a detailed complexity analysis of our expansion algorithm which uses Cartesian representations, we have implemented a prototype of it in C . This program takes a few seconds to compute several terms of the asymptotic expansion of a typical exp log function, and, we ....

[Article contains additional citation context not shown here]

J. van der Hoeven. Automatic asymptotics. PhD thesis, ' Ecole polytechnique, France, 1997.

....this neighbourhood. Introduction 1 Eoeective asymptotic analysis. Transseries also implicitly appeared during the research of algorithms for doing asymptotic analysis [Sha90, Sal91, GG92] In the formal context of transseries, we were able to do such eoeective computations in a more systematic way [vdH97]. There is no doubt that the combination of the techniques from these three dioeerent areas will lead to an extremely powerful theory, whose development is far from nished. A nice feature of such a theory will be that it will both have theoretical and practical aspects (we expect eoeective ....

....in T. Our proof is based on a dioeerential version of the Newton polygon method, which will be sketched in section 3. Using a variant of Stevin s dichotomic method to nd roots of continuous functions, we next indicate (section 4) how to nd a solution h of P (h) 0. For full proofs, we refer to [vdH97, vdH00, vdH01]. In section 5, we will nally discuss some perspectives for the resolution of more general algebraic dioeerential equations using complex transseries. 2 The eld of real, grid based transseries 2.1 Generalized power series Let C be a constant eld and (M; a totally ordered, multiplicative, ....

[Article contains additional citation context not shown here]

J. van der Hoeven. Automatic asymptotics . PhD thesis, #cole polytechnique, France, 1997.

....zero equivalent since f 3 Gamma f 4 is. However, Shackell s algorithm needs O(M ) steps to conclude this, whereas ours terminates after one step: 1 P is pseudo reduced to zero by P . For more details about zero equivalence algorithms, and alternative tests, we refer to [P el 95] P el 97] and [VdH 97] 3 ....

J. van der Hoeven. Automatic asymptotics. PhD. Thesis, ' Ecole polytechnique, France.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC