Expoente de Lojasiewicz: aplicação à relação de equivalência de polinômios no infinito

Abstract

We present the Lojasiewicz exponent at infinity, we show that this is a rational number for polynomial maps F: Cn → Cm, n ≥ 2. We also show that, for this case, the exponent is attained in the set of z ∈ Cn, such that one of its coordinate functions is zero. Furthermore, we define the analytic equivalence relation of functions at infinity and use the exponent of Lojasiewicz to show that in the complement of a proper algebraic set of polynomials of degree less than or equal to a certain fixed degree, there are only a finite number of equivalence classes.