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Based on a recently developed rank Theorem for Eisenstein power series, we provide new proofs of the following two results of W. Pawlucki: I) The non regular locus of a complex or real analytic map is an analytic set. II) The set of semianalytic or Nash points of a subanalytic set X is a subanalytic set, whose complement has codimension two in X., Comment: Important: Our original pre-print arXiv:2205.03079 had two set of distinct results. We have divided that pre-print in two. This paper contains the second set of results ; v2 of the original submission contains the first set of results. We have divided our pre-print

We address the following question of partial desingularization preserving normal crossings. Given an algebraic (or analytic) variety X in characteristic zero, can we find a finite sequence of blowings-up preserving the normal-crossings locus of X, after which the transform X' of X has only singularities from an explicit finite list of minimal singularities, which we define using the determinants of circulant matrices. In the case of surfaces, for example, the pinch point or Whitney umbrella is the only singularity needed in addition to normal crossings. We develop techniques for factorization (splitting) of a monic polynomial with regular (or analytic) coefficients, satisfying a generic normal crossings hypothesis, which we use together with resolution of singularities techniques to find local circulant normal forms of singularities. These techniques in their current state are enough for a positive answer to the question above, for dim X up to 4, or in arbitrary dimension if we preserve normal crossings only of order at most three. In these cases, minimal singularities have smooth normalization., Comment: 47 pages; revision including remarks on further progress

We prove a generalization of Gabrielov's rank theorem for families of rings of power series which we call W-temperate. Examples include the families of complex analytic functions and of Eisenstein series. As a Corollary, we provide rank Theorems for convergent series in general characteristic zero complete valued fields (not necessarily algebraically closed, nor archimedean)., Comment: Two important changes from previous version: 1) The previous paper had two sets of distinct results. We have divided the paper in two, and this version contains the first set of results. The second part will appear in a new arXiv submission. The title has changed. 2) Corollary 1.2 has been added

We provide examples of vector fields on $(\mathbb{C}^3, 0)$ admitting a formal first integral but no holomorphic first integral. These examples are related to a question raised by D. Cerveau and motivated by the celebrated theorems of Malgrange and Mattei-Moussu.

We prove that the topological type of a normal surface singularity $(X,0)$ provides finite bounds for the multiplicity and polar multiplicity of $(X,0)$, as well as for the combinatorics of the families of generic hyperplane sections and of polar curves of the generic plane projections of $(X,0)$. A key ingredient in our proof is a topological bound of the growth of the Mather discrepancies of $(X,0)$, which allows us to bound the number of point blowups necessary to achieve factorization of any resolution of $(X,0)$ through its Nash transform. This fits in the program of polar explorations, the quest to determine the generic polar variety of a singular surface germ, to which the final part of the paper is devoted., Comment: 20 pages, 3 figures. arXiv admin note: text overlap with arXiv:2006.01773

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 the original proof. These techniques are of independent interest, and we illustrate this by adding a new (very short) proof of the Abhyankar-Jung Theorem. We include, furthermore, new extensions of the rank Theorem (concerning the Zariski main Theorem and elimination theory) to commutative algebra., Comment: Final version, 62 pages, to appear in J. Ec. polytech. Math

If F is an infinitely differentiable function whose composition with a blowing-up belongs to a Denjoy-Carleman class C_M (determined by a log convex sequence M=(M_k)), then F, in general, belongs to a larger shifted class C_N, where N_k = M_2k; i.e., there is a loss of regularity. We show that this loss of regularity is sharp. In particular, loss of regularity of Denjoy-Carleman classes is intrinsic to arguments involving resolution of singularities., Comment: V2: minor corrections. To appear in IMRN. 16 pages

We undertake a systematic study of Lipschitz Normally Embedded normal complex surface germs. We prove in particular that the topological type of such a germ determines the combinatorics of its minimal resolution which factors through the blowup of its maximal ideal and through its Nash transform, as well as the polar curve and the discriminant curve of a generic plane projection, thus generalizing results of Spivakovsky and Bondil that were known for minimal surface singularities. In an appendix, we give a new example of a Lipschitz Normally Embedded surface singularity., Comment: v4: New appendix with the proof of a folklore result. A few corrections made, exposition improved. 34 pages, 2 figures. To appear in Compositio Mathematica

We prove a monomialization theorem for mappings in general classes of infinitely differentiable functions that are called quasianalytic. Examples include Denjoy-Carleman classes, the class of $\cC^\infty$ functions definable in a polynomially bounded $o$-minimal structure, as well as the classes of real- or complex analytic functions, and algebraic functions over any field of characteristic zero. The monomialization theorem asserts that a mapping in a quasianalytic class can be transformed to a mapping whose components are monomials with respect to suitable local coordinates, by sequences of simple modifications of the source and target -- local blowings-up and power substitutions in the real cases, in general, and local blowings-up alone in the algebraic or analytic cases. Monomialization is a version of resolution of singularities for a mapping. We show that it is not possible, in general, to monomialize by global blowings-up, even in the real-analytic case., Comment: 66 pages; revised version, theorems unchanged; to appear in Ann. Sci. Ecole Norm. Sup

Given a complex analytic germ $(X, 0)$ in $(\mathbb C^n, 0)$, the standard Hermitian metric of $\mathbb C^n$ induces a natural arc-length metric on $(X, 0)$, called the inner metric. We study the inner metric structure of the germ of an isolated complex surface singularity $(X,0)$ by means of an infinite family of numerical analytic invariants, called inner rates. Our main result is a formula for the Laplacian of the inner rate function on a space of valuations, the non-archimedean link of $(X,0)$. We deduce in particular that the global data consisting of the topology of $(X,0)$, together with the configuration of a generic hyperplane section and of the polar curve of a generic plane projection of $(X,0)$, completely determine all the inner rates on $(X,0)$, and hence the local metric structure of the germ. Several other applications of our formula are discussed in the paper., Comment: Proposition 5.3 strengthened, exposition improved, some typos corrected, references updated. 42 pages and 10 figures. To appear in Geometry & Topology

First order optimization algorithms play a major role in large scale machine learning. A new class of methods, called adaptive algorithms, were recently introduced to adjust iteratively the learning rate for each coordinate. Despite great practical success in deep learning, their behavior and performance on more general loss functions are not well understood. In this paper, we derive a non-autonomous system of differential equations, which is the continuous time limit of adaptive optimization methods. We prove global well-posedness of the system and we investigate the numerical time convergence of its forward Euler approximation. We study, furthermore, the convergence of its trajectories and give conditions under which the differential system, underlying all adaptive algorithms, is suitable for optimization. We discuss convergence to a critical point in the non-convex case and give conditions for the dynamics to avoid saddle points and local maxima. For convex and deterministic loss function, we introduce a suitable Lyapunov functional which allow us to study its rate of convergence. Several other properties of both the continuous and discrete systems are briefly discussed. The differential system studied in the paper is general enough to encompass many other classical algorithms (such as Heavy ball and Nesterov's accelerated method) and allow us to recover several known results for these algorithms.

We introduce a notion of generalized Flow-Box property valid for general singular distributions and sub-varieties (based on a dynamical interpretation). Just as in the usual Flow-Box Theorem, we characterize geometrical and algebraic conditions of (quasi) transversality in order for an analytic sub-variety $X$ (not necessarily regular) to be a section of a line foliation. We also discuss the case of more general foliations. This study is originally motivated by a question of Jean-Francois Mattei (concerning the strengthening of a Theorem of Mattei) about the existence of local slices for a (non-compact) Lie group action., Comment: Changes in Section 4

We prove two main results on Denjoy-Carleman classes: (1) a composite function theorem which asserts that a function f(x) in a quasianalytic Denjoy-Carleman class Q, which is formally composite with a generically submersive mapping y=h(x) of class Q, at a single given point in the source (or in the target) of h, can be written locally as f(x) = g(h(x)), where g(y) belongs to a shifted Denjoy-Carleman class Q' ; (2) a statement on a similar loss of regularity for functions definable in the o-minimal structure given by expansion of the real field by restricted functions of quasianalytic class Q. Both results depend on an estimate for the regularity of an infinitely differentiable solution g of the equation f(x) = g(h(x)), with f and h as above. The composite function result depends also on a quasianalytic continuation theorem, which shows that the formal assumption at a given point in (1) propagates to a formal composition condition at every point in a neighbourhood., Comment: 13 pages

We characterize the limit periodic sets of families of polynomial planar vector fields up to homeomorphisms. We show that any limit periodic set is topologically equivalent to a compact and connected semialgebraic set of the sphere with empty interior. Conversely, we show that any compact and connected semialgebraic set of the sphere with empty interior can be realized as a limit periodic set., Comment: 15 pages, 3 figures

Given a totally nonholonomic distribution of rank two on a three-dimensional manifold we investigate the size of the set of points that can be reached by singular horizontal paths starting from a same point. In this setting, the Sard conjecture states that that set should be a subset of the so-called Martinet surface of 2-dimensional Hausdorff measure zero. We prove that the conjecture holds in the case where the Martinet surface is smooth. Moreover, we address the case of singular real-analytic Martinet surfaces and show that the result holds true under an assumption of non-transversality of the distribution on the singular set of the Martinet surface. Our methods rely on the control of the divergence of vector fields generating the trace of the distribution on the Martinet surface and some techniques of resolution of singularities., Comment: 4 figures

The article develops techniques for solving equations G(x,y)=0, where G(x,y)=G(x_1,...,x_n,y) is a function in a given quasianalytic class (for example, a quasianalytic Denjoy-Carleman class, or the class of infinitely differentiable functions definable in a polynomially-bounded o-minimal structure). We show that, if G(x,y)=0 has a formal power series solution y=H(x) at some point a, then H is the Taylor expansion at a of a quasianalytic solution y=h(x), where h(x) is allowed to have a certain controlled loss of regularity, depending on G. Several important questions on quasianalytic functions, concerning division, factorization, Weierstrass preparation, etc., fall into the framework of this problem (or are closely related), and are also discussed., Comment: revised version, 25 pages, to appear in Selecta Math

The main problem studied is resolution of singularities of the cotangent sheaf of a complex- or real-analytic variety Y (or of an algebraic variety Y over a field of characteristic zero). Given Y, we ask whether there is a global resolution of singularities s: X -> Y such that the pulled-back cotangent sheaf of Y is generated by differential monomials in suitable coordinates at every point of X ("Hsiang-Pati coordinates''). Desingularization of the cotangent sheaf is equivalent to monomialization of Fitting ideals generated by minors of a given order of the logarithmic Jacobian matrix of s. We prove resolution of singularities of the cotangent sheaf in dimension up to three. It was previously known for surfaces with isolated singularities (Hsiang-Pati 1985, Pardon-Stern 2001). Consequences include monomialization of the induced Fubini-Study metric on the smooth part of a complex projective variety Y; there have been important applications of the latter to L2-cohomology., Comment: 42 pages. Minor revision of original posting (including change of title)

Let $M$ be a complex- or real-analytic manifold, $\theta$ be a singular distribution and $\mathcal{I}$ a coherent ideal sheaf defined on $M$. We prove the existence of a local resolution of singularities of $\mathcal{I}$ that preserves the class of singularities of $\theta$, under the hypothesis that the considered class of singularities is invariant by $\theta$-admissible blowings-up. In particular, if $\theta$ is monomial, we prove the existence of a local resolution of singularities of $\mathcal{I}$ that preserves the monomiality of the singular distribution $\theta$., Comment: Updated version. See full reviewed version in the Journal