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Patrick Speissegger

Publications and source records attributed to Patrick Speissegger.

14 recordsLinked to original sources
arXiv2026

Definability of complex functions in o-minimal structures

We prove that some holomorphic continuations of functions in the classes $\mathbf{an}^*$ and $\mathcal{G}$ are definable in the o-minimal structures $\mathbb{R}_{\mathrm{an}^*}$ and $\mathbb{R}_{\mathcal{G}}$ respectively. More specifically, we give complex domains on which the holomorphic continuations are definable, and show they are optimal. As an application, we describe optimal domains on which the Riemann $ζ$ function is definable in o-minimal expansions of $\mathbb{R}_{\mathrm{an}^*,\exp}$ and on which the $Γ$ function is definable in o-minimal expansions of $\mathbb{R}_{\mathcal{G},\exp}$.

math.LO↗
arXiv2024

Transasymptotic expansions of o-minimal germs

Given an o-minimal expansion $\mathbb{R}_{\mathcal{A}}$ of the real ordered field, generated by a generalized quasianalytic class $\mathcal{A}$, we construct an explicit truncation closed ordered differential field embedding of the Hardy field of the expansion $\mathbb{R}_{\mathcal{A},\exp}$ of $\mathbb{R}_{\mathcal{A}}$ by the unrestricted exponential function, into the field $\mathbb{T}$ of transseries. We use this to prove some non-definability results. In particular, we show that the restriction to the positive half-line of Euler's Gamma function is not definable in the structure $\mathbb{R}_{\text{an}^{*},\exp}$, generated by all convergent generalized power series and the exponential function, thus establishing the non-interdefinability of the restrictions to a neighbourhood of $+\infty$ of Euler's Gamma and of the Riemann Zeta function.

math.LO↗
arXiv2023

Multisummability for generalized power series

We develop multisummability, in the positive real direction, for generalized power series with natural support, and we prove o-minimality of the expansion of the real field by all multisums of these series. This resulting structure expands both $\mathbb{R}_{\mathcal{G}}$ and the reduct of $\mathbb{R}_{\mathrm{an}^*}$ generated by all convergent generalized power series with natural support; in particular, its expansion by the exponential function defines both the Gamma function on $(0,\infty)$ and the Zeta function on $(1,\infty)$.

math.CA↗
arXiv2019

Expansions of the real field by canonical products

We consider expansions of o-minimal structures on the real field by collections of restrictions to the positive real line of the canonical Weierstrass products associated to sequences such as $(-n^s)_{n>0}$ (for $s>0$) and $(-s^n)_{n>0}$ (for $s>1$), and also expansions by associated functions such as logarithmic derivatives. There are only three possible outcomes known for the resulting structures: (i)~o-minimality; (ii)~d-minimality (but not o-minimality); (iii)~definability of $\mathbb{Z}$.

math.LO↗
arXiv2019

Ilyashenko algebras based on transserial asymptotic expansions

We construct a Hardy field that contains Ilyashenko's class of germs at infinity of almost regular functions as well as all log-exp-analytic germs. In addition, each germ in this Hardy field is uniquely characterized by an asymptotic expansion that is an LE-series as defined by van den Dries et al. As these series generally have support of order type larger than that of the set of natural numbers, the notion of asymptotic expansion itself needs to be generalized.

math.LO↗
arXiv2018

Analytic continuations of log-exp-analytic germs

We describe maximal, in a sense made precise, analytic continuations of germs at infinity of unary functions definable in the o-minimal structure R_an,exp on the Riemann surface of the logarithm. As one application, we give an upper bound on the logarithmic-exponential complexity of the compositional inverse of an infinitely increasing such germ, in terms of its own logarithmic-exponential complexity and its level. As a second application, we strengthen Wilkie's theorem on definable complex analytic continuations of germs belonging to the residue field of the valuation ring of all polynomially bounded definable germs.

math.LO↗
arXiv2017

Trajectories in interlaced integral pencils of 3-dimensional analytic vector fields are o-minimal

Let X be an analytic vector field defined in a neighborhood of the origin of R^3, and let I be an analytically non-oscillatory integral pencil of X; that is, I is a maximal family of analytically non-oscillatory trajectories of X at the origin all sharing the same iterated tangents. We prove that if I is interlaced, then for any trajectory T in I, the expansion of the structure generated over the real field by T and all globally subanalytic sets is model-complete, o-minimal and polynomially bounded.

math.CA↗
arXiv2017

Quasianalytic Ilyashenko algebras

I construct a quasianalytic field $\mathcal F$ of germs at $+\infty$ of real functions with logarithmic generalized power series as asymp\-totic expansions, such that $\mathcal F$ is closed under differentiation and $\log$-composition; in particular, $\mathcal F$ is a Hardy field. Moreover, the field $\mathcal F \circ (-\log)$ of germs at $0^+$ contains all transition maps of hyperbolic saddles of planar real analytic vector fields.

math.CV↗
arXiv2011

Hausdorff limits of Rolle leaves

Let R be an o-minimal expansion of the real field. We introduce a class of Hausdorff limits, the T-infinity limits over R, that do not in general fall under the scope of Marker and Steinhorn's definability-of-types theorem. We prove that if R admits analytic cell decomposition, then every T-infinity limit over R is definable in the pfaffian closure of R.

math.LO↗
arXiv2010

The theorem of the complement for nested subpfaffian sets

Let R be an o-minimal expansion of the real field, and let L(R) be the language consisting of all nested Rolle leaves over R. We call a set nested subpfaffian over R if it is the projection of a boolean combination of definable sets and nested Rolle leaves over R. Assuming that R admits analytic cell decomposition, we prove that the complement of a nested subpfaffian set over R is again a nested subpfaffian set over R. As a corollary, we obtain that if R admits analytic cell decomposition, then the pfaffian closure P(R) of R is obtained by adding to R all nested Rolle leaves over R, a one-stage process, and that P(R) is model complete in the language L(R).

math.DG↗
arXiv2009

Transition maps at non-resonant hyperbolic singularities are o-minimal

We construct a model complete and o-minimal expansion of the field of real numbers such that, for any planar analytic vector field X and any isolated, non-resonant hyperbolic singularity p of X, a transition map for X at p is definable in this structure. This structure also defines all convergent generalized power series with natural support and is polynomially bounded.

math.DS↗
arXiv2007

An ordered structure of rank two related to Dulac's problem

For a vector field F on the Euclidean plane we construct, under certain assumptions on F, an ordered model-theoretic structure associated to the flow of F. We do this in such a way that the set of all limit cycles of F is represented by a definable set. This allows us to give two restatements of Dulac's Problem for F--that is, the question whether F has finitely many limit cycles--in model-theoretic terms, one involving the recently developed notion of thorn-rank and the other involving the notion of o-minimality.

math.LO↗
arXiv1997

The Pfaffian closure on an o-minimal structure

Every o-minimal expansion R-tilde of the real field has an o-minimal expansion P(R-tilde) in which the solutions to Pfaffian equations with definable C^1 coefficients are definable.

math.RA↗