CIRM - Videos & books Library - The existential closedness problem for analytic solutions of difference equations

Multi angle

Authors : Padgett, Adele (Author of the conference)
CIRM (Publisher )

Loading the player...

Abstract : The existential closedness problem for a function $f$ is to show that a system of complex polynomials in $2 n$ variables always has solutions in the graph of $f$, except when there is some geometric obstruction. Special cases have be proven for exp, Weierstrass $\wp$ functions, the Klein $j$ function, and other important functions in arithmetic geometry using a variety of techniques. Recently, some special cases have also been studied for well-known solutions of difference equations using different methods. There is potential to expand on these results by adapting the strategies used to prove existential closedness results for functions in arithmetic geometry to work for analytic solutions of difference equations.

Keywords : existential closedness; $\Gamma $-function

MSC Codes :
03C05 - Equational classes, universal algebra, See also {08Axx}
11U09 - Model theory, See also {03Cxx}
30C15 - "Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral), {For algebraic theory, See 12D10; for real methods, See 26C10}"
32A60 - Zero sets of holomorphic functions
33B15 - Gamma, beta and polygamma functions

Information on the Video

Film maker :

Petit, Jean

Language :

Available date :

Conference Date :

Subseries :

arXiv category :

Mathematical Area(s) :

Format :

Video Time :

Targeted Audience :

Download :

https://videos.cirm-math.fr/2023_06-02_Padgett_1.mp4

Information on the Event

Event Title : Model theory of valued fields / Théorie des modèles des corps valués
Event Organizers : Chatzidakis, Zoé ; Jahnke, Franziska ; Rideau-Kikuchi, Silvain
Dates : 29/05/2023 - 02/06/2023
Event Year : 2023
Event URL : https://conferences.cirm-math.fr/2761.html

Citation Data

DOI : 10.24350/CIRM.V.20053103
Cite this video as: Padgett, Adele (2023). The existential closedness problem for analytic solutions of difference equations. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20053103
URI : http://dx.doi.org/10.24350/CIRM.V.20053103

[Multi angle] Beautiful pairs revisited / Author of the conference Ye, Jinhe.
[Multi angle] An Imaginary Ax-Kochen/Ershov principle: the equicharacteristic zero case / Author of the conference Vicaria, Mariana.
[Multi angle] Existential closedness of $\mathbb{Q}^{alg}$ as a globally valued field / Author of the conference Szachniewicz, Michal.
[Multi angle] Interpretable, definably semisimple groups in various valued fields / Author of the conference Peterzil, Ya'acov.
[Multi angle] Multi topological fields and NTP2 / Author of the conference Montenegro Guzman, Samaria.
[Multi angle] A Hasse principle over Berkovich analytic curves / Author of the conference Mehmeti, Vlerë.
[Multi angle] Definable convex and henselian valuations on ordered fields / Author of the conference Krapp, Lothar Sebastian.
[Multi angle] Beyond the Fontaine-Wintenberger theorem / Author of the conference Kartas, Konstantinos.
[Multi angle] Around NIP Noetherian domains / Author of the conference Johnson, Will.
[Multi angle] An invitation to globally valued fields / Author of the conference Hrushovski, Ehud.
[Multi angle] Lang-Weil type bounds in finite difference fields / Author of the conference Hils, Martin.
[Multi angle] Residue field domination / Author of the conference Haskell, Deirdre.
[Multi angle] Motivic integration with pseudo-finite residue field and fundamental lemma / Author of the conference Forey, Arthur.
[Multi angle] Tropical functions on skeletons / Author of the conference Ducros, Antoine.
[Multi angle] Contracting endomorphisms of valued fields / Author of the conference Dor, Yuval.
[Multi angle] Ax-Kochen-Ershov principles for finitely ramified henselian fields / Author of the conference Dittmann, Philip.
[Multi angle] Group construction in $C$-minimal structures / Author of the conference Delon, Françoise.
[Multi angle] Homology groups in algebraically closed valued fields / Author of the conference Cubides Kovacsics, Pablo.
[Multi angle] Existential uniform p-adic integration and descent for integrability and largest poles / Author of the conference Cluckers, Raf.
[Multi angle] Polynomials that vanish on many sets of codimension 2 / Author of the conference Ben Yaacov, Itaï.
[Multi angle] The model theory of Hardy fields and linear differential equations / Author of the conference Aschenbrenner, Matthias.
[Multi angle] Transfer of decidability for existential theories of (valued) fields / Author of the conference Anscombe, Sylvy.

Bibliography

ASLANYAN, Vahagn, KIRBY, Jonathan, et MANTOVA, Vincenzo. A geometric approach to some systems of exponential equations. International Mathematics Research Notices, 2023, vol. 2023, no 5, p. 4046-4081. - https://doi.org/10.1093/imrn/rnab340

ETEROVIĆ, Sebastian et HERRERO, Sebastián. Solutions of equations involving the modular 𝑗 function. Transactions of the American Mathematical Society, 2021, vol. 374, no 6, p. 3971-3998. - http://dx.doi.org/10.1090/tran/8244

LI, Bao Qin et STEUDING, Jörn. Fixed points of the Riemann zeta function and dirichlet series. Monatshefte für Mathematik, 2022, vol. 198, no 3, p. 581-589. - http://dx.doi.org/10.1007/s00605-022-01709-x

Padgett.jpg

Bookmarks Report an error

TRUSTEES

INSTITUTIONAL PARTNERS

Destination de la recherche

Raccourcis

See Also