A proof of A. Gabrielov’s rank theorem

Autor: André Belotto da Silva, Guillaume Rond, Octave Curmi

Publikováno v: Journal de l'École polytechnique — Mathématiques
Journal de l'École polytechnique — Mathématiques, École polytechnique, 2021, 8, pp.1329-1396. ⟨10.5802/jep.173⟩

This article contains a complete proof of Gabrielov's rank Theorem, a fundamental result in the study of analytic map germs. Inspired by the works of Gabrielov and Tougeron, we develop formal-geometric techniques which clarify the difficult parts of

Externí odkaz: https://explore.openaire.eu/search/publication?articleId=doi_dedup___::974bb84d6219e96aef7a40c651d5b0c4
https://doi.org/10.5802/jep.173

Artin approximation over Banach spaces

Autor: Guillaume Rond

Publikováno v: Communications in Algebra
Communications in Algebra, Taylor & Francis, 2020, 48 (10), pp.4214-4223. ⟨10.1080/00927872.2020.1758937⟩
Communications in Algebra, 2020, 48 (10), pp.4214-4223. ⟨10.1080/00927872.2020.1758937⟩

We give examples showing that the usual Artin Approximation theorems valid for convergent series over a field are no longer true for convergent series over a commutative Banach algebra. In particular we construct an example of a commutative integral

Externí odkaz: https://explore.openaire.eu/search/publication?articleId=doi_dedup___::b1b9106d48a48f3fb6e559a64156f1f4
http://arxiv.org/abs/2004.08877

Algebraic varieties are homeomorphic to varieties defined over number fields

Autor: Adam Parusinski, Guillaume Rond

Publikováno v: Commentarii Mathematici Helvetici
Commentarii Mathematici Helvetici, 2020, 95, pp.339--359. ⟨10.4171/CMH/490⟩

We show that every affine or projective algebraic variety defined over the field of real or complex numbers is homeomorphic to a variety defined over the field of algebraic numbers. We construct such a homeomorphism by choosing a small deformation of

Externí odkaz: https://explore.openaire.eu/search/publication?articleId=doi_dedup___::8e1ff2131faf77ef942e358c55713843
https://hal.science/hal-02536240

Multiparameter perturbation theory of matrices and linear operators

Autor: Adam Parusinski, Guillaume Rond

Publikováno v: Transactions of the American Mathematical Society
Transactions of the American Mathematical Society, American Mathematical Society, 2020, 373 (4), pp.2933-2948. ⟨10.1090/tran/8061⟩
Transactions of the American Mathematical Society, 2020, 373 (4), pp.2933-2948. ⟨10.1090/tran/8061⟩

We show that a normal matrix $A$ with coefficient in $\mathbb C[[X]]$, $X=(X_1, \ldots, X_n)$, can be diagonalized, provided the discriminant $\Delta_A $ of its characteristic polynomial is a monomial times a unit. The proof is an adaptation of the a

Externí odkaz: https://explore.openaire.eu/search/publication?articleId=doi_dedup___::4ebe822c68a6f743fd880d09fc834c11
http://arxiv.org/abs/1807.04242

Higher order approximation of analytic sets by topologically equivalent algebraic sets

Autor: Adam Parusinski, Guillaume Rond, Marcin Bilski, Krzysztof Kurdyka

Publikováno v: Mathematische Zeitschrift
Mathematische Zeitschrift, 2018, 288 (3-4), pp.1361-1375. ⟨10.1007/s00209-017-1937-5⟩
Mathematische Zeitschrift, Springer, 2018, 288 (3-4), pp.1361-1375. ⟨10.1007/s00209-017-1937-5⟩

It is known that every germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that the homeomorphism can be chosen in such a way that the analytic and algebraic germs are tangent with any prescribed order of ta

Externí odkaz: https://explore.openaire.eu/search/publication?articleId=doi_dedup___::9b98ca0b1950e298f238e5f4d8c294bc
https://ruj.uj.edu.pl/xmlui/handle/item/54339