On analytic equivalence of functions at infinity

In this paper we define the relation of analytic equivalence of functions at infinity. We prove that if the Łojasiewicz exponent at infinity of the gradient of a polynomial f∈R[x1,…,xn] is greater or equal to k−1, then there exists ε>0 such that for every polynomial P∈R[x1,…,xn] of degree less or equal to k, whose coefficients of monomials of degree k are less or equal ε, the polynomials f and f+P are analytically equivalent at infinity.