D. Roy - Criteria of algebraic independence and approximation by hypersurfaces

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D. Roy - Criteria of algebraic independence and approximation by hypersurfaces

D. ROY, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada

Criteria of algebraic independence and approximation by hypersurfaces

Given a point $\theta$ in $\bbd C^m$ , a fundamental problem is how close one can approximate $\theta$ by a point of an algebraic variety of dimension d, defined over $\bbd Q$ , with degree $\le D$ and logarithmic height $\le T$ . The problem has a different flavor whether, for a fixed d, one wants an estimate valid for a pair (D,T) or for infinitely many pairs (D_n,T_n) chosen from a given non-decreasing sequence of positive integers $(D_n)_{n\ge 1}$ , and a given non-decreasing unbounded sequence of positive real numbers $(T_n)_{n\ge 1}$ with $T_n\ge D_n$ for each $n\ge 1$ . In a joint work with Michel Laurent, we analyze the second type of problem when d=m-1. We show that, for infinitely many indices n, there exists a nonzero polynomial $P\in \bbd Z[X_1,\dots,X_m]$ of degree $\le D_n$ whose coefficients have absolute value $\le \exp(T_n)$ , such that Padmits at least one zero $\alpha$ in $\bbd C^m$ with

$\begin{displaymath}\Vert\theta-\alpha\Vert \le \exp\bigl(-c(m) D_{n-1}^m T_{n-1}\bigr) \end{displaymath}$

for some positive constant c(m) which depends only on m. This follows from a new criteria of algebraic independence with multiplicities. The object of the talk is to explain the link between the two results.

Next: Gary Walsh - Old Up: Number Theory / Théorie Previous: Gael Rémond - Theta