Intersecting curves

Consider the graphs of n distinct polynomials of a real variable intersecting at some point. In the neighborhood of this point, the qualitative picture is described by some permutation of {1, . . . , n}. We describe the permutations that occur in such a situation. In this note, we shall examine the relative positions of the graphs of several functions in the neighborhood of a point where they intersect. In order to keep the discussion as elementary as possible, we shall restrict ourselves to polynomials in the real variable x f(x) = a0 + a1x+ . . .+ adx , where the coefficients ai are real numbers. As usual, one says that the valuation of f at 0 is the integer k ≤ d such that a0 = a1 = . . . = ak−1 = 0 and ak 6= 0 (and ∞ if f = 0). It is well known that the function f changes sign in the neighborhood of the origin if and only if its valuation is odd. Two curves Assume now that the graphs of two distinct polynomials f1, f2 intersect in some point, say the origin (0, 0). Then the relative position of the graphs of f1 and f2 in the neighborhood of this point is easy to describe. If the valuation of f1 − f2 is odd, these graphs cross each other. Otherwise they touch without crossing. Figure 1: Two intersecting curves