Abstract. Let f be a real analytic function defined in a neighborhood of 0 ∈ R n such that f −1 (0) = {0}. We describe the smallest possible exponents α, β, θ for which we have the following estimates:Mathematics Subject Classification (1991). Primary 14B05; Secondary 32S05.

Let f (x, y) = 0 and l(x, y) = 0 be respectively a singular and a regular analytic curve defined in the neighborhood of the origin of the complex plane. We study the family of analytic curves f (x, y) − t l(x, y) M = 0, where t = 0 is a complex parameter. For all but a finite number of parameters the curves of this family have the same embedded topological type. The exceptional parameters are called special values. We show that the number of special values does not exceed the number of components of the curve f (x, y) = 0 counted without multiplicities. Then we apply this result to estimate the number of critical values at infinity of complex polynomials in two variables.d . A generic curve of this pencil consists of d straight lines crossing at the origin. The curve f t0 (x, y) = 0 is special if and only if t 0 is a critical value of the polynomial H; then f t0 (x, y) = 0 consists of d − 2 single straight lines and one double line. Example 1.3 Let H(Z) = d i=1 (Z − a i ) mi be a complex polynomial with d − 1 nonzero critical values different from H(0). Consider the family of quasihomogeneous polynomials f t (x, y) = d i=1 (x 2 − a i y 3 ) mi − ty 3 deg H . For t = 0 the curve f t (x, y) = 0 consists of d multiple cusps (x 2 − a i y 3 ) mi = 0. For t = H(0) the curve f t (x, y) = 0 consists of deg H −1 cusps and one double line x 2 = 0. If t is a nonzero critical value of H then the curve f t (x, y) = 0 has deg H −2 single cusps and one double cusp. For all other t ∈ C the curve f t (x, y) = 0 decomposes into deg H pairwise different cusps of the form x 2 − b i y 3 = 0 for i = 1, . . . , deg H.This example shows that the bound for the number of special values given in Theorem 1.1 is sharp.

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Abstract. We prove that every polynomial P (x, y) of degree d has at most 2(d + 2) 12 zeros on the curve y = e x + sin(x), x > 0. As a consequence we deduce that the existence of a uniform bound for the number of zeros of polynomials of a fixed degree on an analytic curve does not imply that this curve belongs to an o-minimal structure. Main resultThe aim of this note is to estimate the number of real solutions of the systemwhere P (x, y) is a non-zero polynomial of degree d. As we prove below we have the following bound. Theorem 1. The number of solutions of the system (1) is not greater than A(d) = 2(d + 2)12 .

We show that every local polynomial diffeomorphism (f, g) of the real plane such that deg f ≤ 3, deg g ≤ 3 is a global diffeomorphism.