Guillaume Rond

The analogue of Izumi's Theorem for Abhyankar valuations

A well known theorem of Shuzo Izumi, strengthened by David Rees, asserts that all the divisorial valuations centered in an analytically irreducible local noetherian ring are linearly comparable to each other. In the present paper we generalize this theorem to the case of Abhyankar valuations with archimedian value semigroup. Indeed, we prove that in a certain sense linear equivalence of topologies characterizes Abhyankar valuations with archimedian semigroups, centered in analytically irreducible local noetherian rings. Then we show that some of the classical results on equivalence of topologies in noetherian rings can be strengthened to include linear equivalence of topologies. We also prove a new comparison result between the Krull topology and the topology defined by the symbolic powers of an arbitrary ideal.

LINEAR NESTED ARTIN APPROXIMATION THEOREM FOR ALGEBRAIC POWER SERIES

We give an elementary proof of the nested Artin approximation theorem for linear equations with algebraic power series coefficients. Moreover, for any Noetherian local subring of the ring of formal power series, we clarify the relationship between this theorem and the problem of the commutation of two operations for ideals: the operation of replacing an ideal by its completion and the operation of replacing an ideal by one of its elimination ideals. In particular we prove that a Grothendieck conjecture about morphisms of analytic/formal algebras and Artin’s question about linear nested approximation problem are equivalent.

Support of Laurent series algebraic over the field of formal power series: SUPPORT OF ALGEBRAIC LAURENT 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.

HIGHER ORDER APPROXIMATION OF ANALYTIC SETS BY TOPOLOGICALLY EQUIVALENT ALGEBRAIC SETS

It is known that every germ of an analytic set is homeomorphic to the germ of an algebraic set. In this paper we show that the homeomorphism can be chosen in such a way that the analytic and algebraic germs are tangent with any prescribed order of tangency. Moreover, the space of arcs contained in the algebraic germ approximates the space of arcs contained in the analytic one, in the sense that they are identical up to a prescribed truncation order.

Asymptotic behaviour of standard bases

We prove that the elements of any standard basis of In, where I is an ideal of a Noetherian local ring and n is a positive integer, have order bounded by a linear function in n. We deduce from this that the elements of any standard basis of In in the sense of Grauert-Hironaka, where I is an ideal of the ring of power series, have order bounded by a polynomial function in n. The aim of this paper is to study the growth of the orders of the elements of a standard basis of I, where I is an ideal of a Noetherian local ring. Here we show that the maximal order of an element of a standard basis of I is bounded by a linear function in n. For this we prove a linear version of the strong Artin-Rees lemma for ideals in a Noetherian ring. The main result of this paper is Theorem 3. First we prove the following proposition inspired by Corollary 3.3 of [4]: Proposition 1. Let A be a Noetherian ring and let I and J be ideals of A. There exists an integer λ ≥ 0 such that ∀x ∈ A, ∀n,m ∈ N, n ≥ λm, (x) ∩ (J + I) = ((x) ∩ (J + Im))(xn−λm). Proof. Let B := A/J . By Theorem 3.4 of [5], there exists λ such that for any m ≥ 1, there exists an irredundant primary decomposition I = Q 1 ∩ · · · ∩Q (m) r such that if P (m) i := √ Q (m) i , then (P (m) i ) λm ⊂ Q i for 1 ≤ i ≤ m. We denote by Q (m) i the image of Q (m) i in A/(J + I ) for 1 ≤ i ≤ r. We denote by P i the inverse image of P (m) i in A, for 1 ≤ i ≤ r. Let x ∈ A. If x ∈ P i , then x ∈ (P (m) i ) n and (Qi (m) : x) = A/(J + I) for any n ≥ λm. If x / ∈ P i , then x / ∈ (P (m) i ) n and (Q (m) i : x ) = Q (m) i for any n ≥ λm. Thus, for any n ≥ λm, ( 0A/(J+Im) : x n ) = (⋂