Counting real curves with passage/tangency conditions (Q2890314)

The authors solve the following real enumerative problem: how many rational curves of a given degree \(d\) in the real affine plane pass through a configuration of \(3d-2\) real points and are tangent to a given immersed oriented curve \(\Gamma\)? A similar problem was considered by \textit{J.-Y. Welschinger} [Geom. Funct. Anal. 16, No. 5, 1157--1182 (2006; Zbl 1107.53059)], who in the case of a smooth \(\Gamma\) found an invariant way to count curves in question together with real rational cuspidal curves and reducible curves passing through the given configuration of \(3d-2\) points, all counted with weights \(\pm1\). The paper under review suggests an original approach to the problem: using simple differential-geometric tools, the authors count curves in question with certain signs and provide an explicit formula for their algebraic number. The formula involves certain finite type invariants as well as Welschinger invariants of the plane (corresponding to the point constraints). In addition, it gives an explicit values of jumps of the computed number when one crosses the discriminant in the space of the given data.