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Abstract: We study the following question: given a set P of 3d-2 points and an immersedcurve G in the real plane R^2, all in general position, how many real rationalplane curves of degree d pass through these points and are tangent to thiscurve. We count each such curve with a certain sign, and present an explicitformula for their algebraic number. This number is preserved under smallregular homotopies of a pair P, G, but jumps in a well-controlled way whenin the process of homotopy we pass a certain singular discriminant. We discussthe relation of such enumerative problems with finite type invariants. Ourapproach is based on maps of configuration spaces and the intersection theoryin the spirit of classical algebraic topology.



Author: Sergei Lanzat, Michael Polyak

Source: https://arxiv.org/



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