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.

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

The Abhyankar-Jung Theorem

We show that every quasi-ordinary Weierstrass polynomial P(Z) = Z^d+a_1 (X) Z^{d-1}+...+a_d(X) \in \K[[X]][Z] , $X=(X_1,..., X_n)$, over an algebraically closed field of characterisic zero \K, and satisfying $a_1=0$, is $\nu$-quasi-ordinary. That means that if the discriminant \Delta_P \in \K[[X]] is equal to a monomial times a unit then the ideal $(a_i^{d!/i}(X))_{i=2,...,d}$ is principal and generated by a monomial. We use this result to give a constructive proof of the Abhyankar-Jung Theorem that works for any Henselian local subring of \K[[X]] and the function germs of quasi-analytic families.Comment: 14 pages. The toric case has been added. To be published in Journal of Algebr

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