Generic differential equations are strongly minimal

DeVilbiss, Matthew; Freitag, James

doi:10.48550/arxiv.2106.02627

Cited by 4 publications

(5 citation statements)

References 21 publications

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“…Establishing the notion has been the key step to resolving a number of longstanding open conjectures [6,49]. Despite these factors, there are few enough equations for which the property has been established that a comprehensive list of such equations appears in [13]. In this manuscript, we generalize results of Poizat [57] and Brestovski [4] by showing that Theorem.…”

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confidence: 79%

“…First, once strong minimality of an equation is established, the trichotomy theorem, a model theoretic classification result, along with other model theoretic results can often be employed in powerful ways [27,49]. Second, among nonlinear differential equations, the property seems to hold rather ubiquitously; in fact there are theorems to this effect in various settings [13,28]. Even for equations which are not themselves minimal, there is a well-known decomposition technique, semi-minimal analysis 1 [46], which often allows for the reduction of questions to the minimal case.…”

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confidence: 99%

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On the equations of Poizat and Liénard

Freitag¹,

Jaoui²,

Marker³

et al. 2022

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We study the structure of the solution sets in universal differential fields of certain differential equations of order two, the Poizat equations, which are particular cases of Liénard equations. We give a necessary and sufficient condition for strong minimality for equations in this class and a complete classification of the algebraic relations for solutions of strongly minimal Poizat equations. We also give an analysis of the non strongly minimal cases as well as applications concerning the Liouvillian and Pfaffian solutions of some Liénard equations.

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confidence: 79%

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confidence: 99%

On the equations of Poizat and Liénard

Freitag¹,

Jaoui²,

Marker³

et al. 2022

Preprint

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“…In the theory of differentially closed fields of characteristic zero, Freitag and Moosa [8] give an upper bound for the degree of nonminimality in terms of the U-rank of the type. In [4], DeVilbiss and Freitag use the bounds of [8] along with computations involving the Lascar rank of underdetermined systems of differential equations to give a proof of the strong minimality of generic differential equations of sufficiently high degree. One of the challenging aspects of generalizing the proofs of [4] to various other classes of equations is the complexity of the series of algebraic reductions used to calculate the rank of a certain associated linear system.…”

Section: Nonminimalitymentioning

confidence: 99%

Strong minimality of triangle functions

Casale¹,

DeVilbiss²,

Freitag³

et al. 2022

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“…These results are especially pertinent because many of the number theoretic functions to which the Pila-Wilkie theorem has been applied have been shown to be strongly minimal in recent years. For these as well as various other strong minimality results on nonlinear higher order differential equations, see [7,8,10,11,15,19,28,34]. Besides results for specific equations, as [11,19] indicate, strong minimality is a pervasive condition for nonlinear differential equations of order at least two -it holds generically both in the space of constant coefficient equations as well as the space of nonconstant equations.…”

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confidence: 93%

Not Pfaffian

Freitag

2021

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This short note describes the connection between strong minimality of the differential equation satisfied by an complex analytic function and the real and imaginary parts of the function being Pfaffian. This connection combined with a theorem of Freitag and Scanlon ( 2017) provides the answer to a question of Binyamini and Novikov (2017). We also answer a question of Bianconi (2016). We give what seem to be the first examples of functions which are definable in o-minimal expansions of the reals and are differentially algebraic, but not Pfaffian. Why Pfaffian?Pfaffian functions were introduced in [17] where Khovanskii showed that the class has strong finiteness properties. For instance, Khovanskii exhibits an effective bound on the number of zeros in a system of equations involving Pfaffian functions. Later, the strong finiteness properties of this class of functions played an important role in model completeness results for o-minimal expansions of the real field [37,25]. Any algebraic function is Pfaffian on a suitable domain, but there are two well-known ways to see that a transcendental function is not Pfaffian:• The function is differentially transcendental. 1• The function violates the strong finiteness properties of the class. 2 This manuscript introduces a third way of showing a function is not Pfaffian. Our work is based on a very simple idea -Pfaffian functions are built using solutions to certain order one differential equations, while solutions to higher order strongly minimal differential equations can not satisfy order one differential equations. For the notion of C-Pfaffian, this summary gives nearly a complete indication of the proof that a solution to a higher order strongly minimal equation is not C-Pfaffian. There are only slightly more complications to describe when considering if the real and imaginary parts of the function are Pfaffian. These complications can already be seen in the work of Macintyre [24], who shows that Date: September 21, 2021. J. Freitag is partially supported by NSF CAREER award 1945251.1 Sometimes this property is also called being hypertranscendental or transcendentally transcendental [35]. The Gamma-function is differentially transcendental by a classical theorem of Hölder [16] and its restriction to (0, ∞) is definable in an o-minimal expansion of the reals [12].

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Generic differential equations are strongly minimal

Cited by 4 publications

References 21 publications

On the equations of Poizat and Liénard

On the equations of Poizat and Liénard

Strong minimality of triangle functions

Not Pfaffian

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