On the {\L}ojasiewicz exponent

Abstract

Let $\Bbb K$ be an algebraically closed field and let $X\subset \Bbb K^l$ be an $n-$dimensional affine variety of degree $D.$ We give a sharp estimation of the degree of the set of non-properness for generically-finite separable and dominant mapping $f=(f_1,...,f_n): X\to \Bbb K^n$. We show that such a mapping must be finite, provided it has a sufficiently large geometric degree. Moreover, we estimate the \L ojasiewicz exponent at infinity of a polynomial mapping $f: X\to \Bbb K^m$ with a finite number of zeroes.