Local topological algebraicity with algebraic coefficients of analytic sets or functions

We prove that any complex or real analytic set or function germ is topologically equivalent to a germ defined by polynomial equations whose coefficients are algebraic numbers.Comment: 16 pages. To appear in Algebra & Number Theor

Local zero estimates and effective division in rings of algebraic power series

We give a necessary condition for algebraicity of finite modules over the ring of formal power series. This condition is given in terms of local zero estimates. In fact we show that this condition is also sufficient when the module is a ring with some additional properties. To prove this result we show an effective Weierstrass Division Theorem and an effective solution to the Ideal Membership Problem in rings of algebraic power series. Finally we apply these results to prove a gap theorem for power series which are remainders of the Grauert-Hironaka-Galligo Division Theorem.Comment: Final version - 48 pp - to appear in J. Reine Angew. Mat

About the algebraic closure of the field of power series in several variables in characteristic zero

We construct algebraically closed fields containing an algebraic closure of the field of power series in several variables over a characteristic zero field. Each of these fields depends on the choice of an Abhyankar valuation and are constructed via the Newton-Puiseux method. Then we study more carefully the case of monomial valuations and we give a result generalizing the Abhyankar-Jung Theorem for monic polynomials whose discriminant is weighted homogeneous.Comment: final versio

Transcendental holomorphic maps between real algebraic manifolds in a complex space

We give an example of a real algebraic manifold embedded in a complex space that does not satisfy the Nash-Artin approximation Property. This Nash-Artin approximation Property is closely related to the problem of determining when the biholomorphic equivalence for germs of real algebraic manifolds coincides with the algebraic equivalence. This example is an elliptic Bishop surface, and its construction is based on the functional equation satisfied by the generating series of some walks restricted to the quarter plane.Comment: to appear in Proceedings of the A.M.

Doctors at War: Life and Death in a Field Hospital

[Excerpt from jacket] Doctors at War is a candid account of a trauma surgical team based, for a tour of duty, at a field hospital in Helmand, Afghanistan. Mark de Rond tells of the highs and lows of surgical life in hard-hitting detail, bringing to life a morally ambiguous world in which good people face impossible choices and in which routines designed to normalize experience have the unintended effect of highlighting war\u27s absurdity. With stories that are at once comical and tragic, de Rond captures the surreal experience of being a doctor at war. He lifts the cover on a world rarely ever seen, let alone written about, and provides a poignant counterpoint to the archetypical, adrenaline-packed, macho tale of what it is like to go to war

Remarks on Artin approximation with constraints

We study various approximation results of solutions of equations $f(x,Y)=0$ where $f(x,Y)\in\mathbb K[[x]][Y]^r$ and $x$ and $Y$ are two sets of variables, and where some components of the solutions $y(x)\in\mathbb K[[x]]^m$ do not depend on all the variables $x_j$. These problems have been highlighted by M. Artin.Comment: 9 pages. To appear in Osaka J. Mat

Support of Laurent series algebraic over the field of formal power series

This work is devoted to the study of the support of a Laurent series in several variables which is algebraic over the ring of power series over a characteristic zero field. Our first result is the existence of a kind of maximal dual cone of the support of such a Laurent series. As an application of this result we provide a gap theorem for Laurent series which are algebraic over the field of formal power series. We also relate these results to diophantine properties of the fields of Laurent series.Comment: 31 pages. To appear in Proc. London Math. So