A global Łojasiewicz inequality for algebraic varieties

Abstract:

Let $X$ be the locus of common zeros of polynomials ${f_1}, \ldots ,{f_k}$ in $n$ complex variables. A global upper bound for the distance to $X$ is given in the form of a Lojasiewicz inequality. The exponent in this inequality is bounded by ${d^{\min (n,k)}}$ where $d = \max (3,\deg {f_i})$. The estimates are also valid over an algebraically closed field of any characteristic.