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Lichtenbaum proved that index and period coincide for a curve of genus one over a $p$-adic field. Salberger proved that the Hasse principle holds for a smooth complete intersection of two quadrics $X \subset P^n$ over a number field, if it contains a conic and if $n\geq 5$. Building upon these two results, we extend recent results of Creutz and Viray (2021) on the existence of a quadratic point on intersections of two quadrics over $p$-adic fields and number fields. We then recover Heath-Brown's theorem (2018) that the Hasse principle holds for smooth complete intersections of two quadrics in $P^7$. We also give an alternate proof of a theorem of Iyer and Parimala (2022) on the local-global principle in the case $n=5$., Comment: Paper in French, appendix by A. Kuznetsov in English. Version v2 takes into account the remarks of a referee. Final version, to appear in Journal f\"ur die reine und angewandte Mathematik (Crelle)

Quantifying the propagation of landslides is a key step for analyzing gravitational risks. In this context, thin-layer models have met a growing success over the past years to simulate the dynamics of gravitational flows, such as debris flows. They are easier to use and require less computing ressources than 3D models, and provide more detailed estimations of flow thicknesses and velocities than purely empirical methods. In this literature review, we present the main rheologies used to model homogeneous gravitational flows, and describe a practical case study with debris flow modeling in the Pr{\^e}cheur River (Martinique, Lesser Antilles). Then, we discuss possible developments for operational hazard and risk assessment with thin-layer models., Comment: in French language

In this article, we introduce a notion of polygraphic homology of a strict $\omega$-category with coefficients in a local system, generalizing the polygraphic homology with coefficients in $\mathbb Z$, introduced by Fran\c{c}ois M\'etayer. We show that the homology of a simplicial set with coefficients in a local system coincides with the polygraphic homology of its image by the left adjoint of the Street nerve with coefficients in the corresponding local system. We define in this framework a comparison morphism between the polygraphic homology of a strict $\omega$-category and the homology of its Street nerve, and we show that this morphism is an isomorphism for (1-)categories. This is not true for an arbitrary $\omega$-category. Nevertheless, we conjecture that for an analogous construction in the framework of weak $\omega$-categories ``\`a la Grothendieck'' we would always obtain an isomorphism., Comment: 56 pages, in French language. Minor corrections

It is generally accepted that the phonon gas of a superfluid always enters a weak coupling regime at sufficiently low temperatures, whatever the strength of the interactions between the underlying particles (constitutive of the superfluid). Thus, in this limit, we should always be able to calculate the damping rate of thermal phonons by applying Fermi's golden rule to the Hamiltonian $H_3$ of cubic phonon-phonon coupling taken from quantum hydrodynamics, at least in the case of a convex acoustic branch and in the collisionless regime (where the eigenfrequency of the considered phonons remains much greater than the gas thermalization rate). Using the many-body Green's function method, we predict that, unexpectedly, this is not true in two dimensions, contrary to the three-dimensional case. We confirm this prediction with classical phonon-field simulations and a non-perturbative theory in $H_3$, where the fourth order is regularized by hand, giving a complex energy to the virtual phonons of the four-phonon collisional processes. For a weakly interacting fluid and a phonon mode in the long-wavelength limit, we predict a damping rate about three times lower than that of the golden rule., Comment: In French (43 pages) and in English (42 pages) (published version)

We give give an elementary and constructive version of the theory of "Pr\"ufer v-Multiplication Domains" (which we call "anneaux \`a diviseurs" in the paper) and Krull Domains. The main results of these theories are revisited from a constructive point of view, following the Bishop style, and without assuming properties of complete factorizations. Nous pr\'esentons dans cet article une approche constructive, dans le style de Bishop, de la th\'eorie des diviseurs et des anneaux de Krull. Nous accordons une place centrale aux "anneaux \`a diviseurs," appel\'es PvMD dans la litt\'erature anglaise. Les r\'esultats classiques sont obtenus comme r\'esultats d'algorithmes explicites sans faire appel aux hypoth\`eses de factorisation compl\`ete., Comment: in French, W.r.t. the published version we have improved the references, we have fixed some typos, proof of 1.19 and 1.20 are best explained and the proof og Theorem 1.29 is shortened. A table of contents is now added at the end of the paper

Over the function field of a complex algebraic curve, strong approximation off a non-empty finite set of places holds for the complement of a codimension $2$ closed subset in a homogeneous space under a semisimple algebraic group, and for the complement of a codimension $2$ closed subset in an affine smooth complete intersection of low degree., Comment: 24 pages. The article is in french. v2: minor modifications; definition of strong approximation has been moved to the introduction; few references have been added. Final version

We generalize the analog of Buchberger's first criterion, stated by Boulier et al., for detecting useless S-polynomials reductions in the computation of characteristic sets of differential ideals. The original version assumes linear polynomials; this result is here extended to a product of linear differential polynomials depending of the same arbitrary differential polynomial., Comment: 12 pages. In French. With an extended abstract in English