Lojasiewicz exponent at infinity in C[x, y, z]

Abstract

We consider the set \mathscr{F}={p(z)x+q(y, z), p∈C[z]{\backslash}{0}, q∈C[y, z]}. We connect algebraic properties of a polynomial f∈\mathscr{F}, such that f is a variable in C[x, y, z] or f is a tame variable in C[z][x, y] with the Lojasiewicz exponent at infinity of f. We compute this exponent for some polynomials of \mathscr{F}.