# Graph Theory and Applications, Proceedings of the First by J. Akiyama, Y. Egawa and H. Enomoto (Eds.)

By J. Akiyama, Y. Egawa and H. Enomoto (Eds.)

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**Sample text**

Gatellier et al. The last case, which remains to be treated, is the case where L = V = W is the set of points with the same y-coordinate as the x-critical point c. We deﬁne p as the point of L with greatest y-coordinate among those with greatest z-coordinate and q as the point with greatest z-coordinate. If q = c, then p has to be connected to q . Otherwise p ∈ L is connected to p ∈ L and q ∈ L connected to q ∈ L , with p = q, p = q . Then we have z(p ) < z(q ). By lemma 14 we deduce z(p) = z(q) and as p has the greatest z-coordinate: z(p) > z(q), which contradicts the fact that C is smooth above ]αi , αi+1 [.

Notice, that G(x, y) is not necessarily a squarefree polynomial. Consider for instance the case P1 = x2 +y 2 −1, P2 = x2 +y 2 +z 2 −2, where g(x, y) = (x2 +y 2 −1)2 . In this case, there are generically two (complex) points of C above a point of C . We can easily compute the gcd of G(x, y) and ∂y G(x, y) (using proposition 4), in order to get the squarefree part g(x, y) = G(x, y)/gcd(G(x, y), ∂y G(x, y)) of G(x, y). Similarly, for the projection C of C on the (x, z)-plane, we compute H(x, z) = Resy (P1 , P2 ) and its square-free part h(x, z) from the gcd of H(x, z) and ∂z H(x, z).

As a consequence, we give theoretical results to guarantee that an algebraic curve lies in the offset region of another at some small distance. Furthermore, we see how these results particularize to the univariate case giving bounds that ﬁt properly with the classical results in numerical analysis. This paper is structured as follows. In sect. 2 we introduce the basic notions and we state some preliminary properties of –points and –roots. e. for –roots. Section 4 focusses on the corresponding analysis of distance properties for the bivariate case.