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            <title xml:lang="en">Embeddings of Danielewski hypersurfaces</title>
            <title xml:lang="fr">Sur les plongements des hypersurfaces de Danielewski</title>
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                <forename type="first">Pierre-Marie</forename>
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                <title xml:lang="en">Embeddings of Danielewski hypersurfaces</title>
                <title xml:lang="fr">Sur les plongements des hypersurfaces de Danielewski</title>
                <author role="aut">
                  <persName>
                    <forename type="first">Pierre-Marie</forename>
                    <surname>Poloni</surname>
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                <authority type="institution">Université de Bourgogne</authority>
                <authority type="supervisor">Lucy Moser-Jauslin</authority>
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                <term xml:lang="fr">automorphismes polynomiaux</term>
                <term xml:lang="fr">dérivations localement nilpotentes</term>
                <term xml:lang="fr">problème de l'équivalence stable</term>
                <term xml:lang="fr">polynômes équivalents</term>
                <term xml:lang="fr">hypersurfaces de Danielewski</term>
                <term xml:lang="fr">surfaces de Danielewski</term>
                <term xml:lang="fr">polynomial automorphisms.</term>
                <term xml:lang="fr">locally nilpotent derivations</term>
                <term xml:lang="fr">stable equivalence problem</term>
                <term xml:lang="fr">equivalent polynomials</term>
                <term xml:lang="fr">Danielewski hypersurfaces</term>
                <term xml:lang="fr">Danielewski surfaces</term>
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              <p>In this thesis, we study a class of hypersurfaces in $\mathbb{C}^3$, called \emph{Danielewski hypersurfaces}. This means hypersurfaces $X_{Q,n}$ defined by an equation of the form $x^ny=Q(x,z)$ with $n\in\mathbb{N}_{\geq1}$ and $\deg_z(Q(x,z))\geq2$. We give their complete classification, up to isomorphism, and up to equivalence via an automorphism of $\mathbb{C}^3$. In order to do that, we introduce the notion of standard form and show that every Danielewski hypersurface is isomorphic (by an algorithmic procedure) to a Danielewski hypersurface in standard form. This terminology is relevant since every isomorphism between two standard forms can be extended to an automorphism of the ambiant space. (We show that this property does not hold for general Danielewski hypersurfaces.)&lt;br&gt;Problems of stable equivalence and analytic equivalence are also studied. We construct examples of polynomials $P,Q\in\mathbb{C}[x,y,z]$ such that there does not exist an algebraic automorphism of $\mathbb{C}[x,y,z]$ which sends $P$ to $Q$, whereas these polynomials are equivalent via an automorphism of $\mathbb{C}[x,y,z,w]$.&lt;br&gt;Most of these results are based on a precise picture of the sets of locally nilpotent derivations of the algebras of regular functions on the hypersurfaces $X_{Q,n}$, obtained using techniques developed by Makar-Limanov.</p>
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              <p>Dans cette thèse, nous étudions une classe d'hypersurfaces de $\mathbb{C}^3$, dites \emph{hypersurfaces de Danielewski}. Ce sont les hypersurfaces $X_{Q,n}$ définies par une équation de la forme $x^ny=Q(x,z)$ avec $n\in\mathbb{N}_{\geq1}$ et $\deg_z(Q(x,z))\geq2$. Nous établissons leurs classifications complètes à isomorphisme près, et à équivalence via un automorphisme de $\mathbb{C}^3$ près. Pour cela, nous introduisons le concept de forme standard et montrons que toute hypersurface de Danielewski est isomorphe, par un procédé algorithmique, à une hypersurface sous forme standard. Cette terminologie est justifiée par le fait que tout isomorphisme entre deux formes standards s'étend en un automorphisme de l'espace ambiant (ce qui n'est pas&lt;br&gt;vrai pour des hypersurfaces de Danielewski quelconques).&lt;br&gt;Nous étudions aussi les problèmes de l'équivalence stable et de l'équivalence analytique. Nous construisons notamment des exemples de polynômes $P,Q\in\mathbb{C}[x,y,z]$ pour lesquels il n'existe aucun automorphisme algébrique de $\mathbb{C}[x,y,z]$ qui envoie $P$ sur $Q$, bien que ces polynômes soient équivalents via un automorphisme de $\mathbb{C}[x,y,z,w]$.&lt;br&gt;La plupart de ces résultats reposent sur la description précise, grâce aux techniques développées par Makar-Limanov, des dérivations localement nilpotentes sur les algèbres des fonctions régulières des hypersurfaces $X_{Q,n}$.</p>
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