Algebraic interaction strength for translation surfaces with multiple singularities

Abstract: We compute the maximal ratio of the algebraic intersection of two closed curves on two families of translation surfaces with multiple singularities. This ratio, called the interaction strength, is difficult to compute for translation surfaces with several singularities as geodesics can change direction at singularities. The main contribution of this paper is to deal with this type of surfaces. Namely, we study the interaction strength of the regular $n-$gons for $n \equiv 2 \pmod 4$ and the Bouw-M\"oller surfaces $S_{m,n}$ with $1 < \gcd(m,n) < n$. This answers a conjecture of the author from (Boulanger, Algebraic intersection, lengths and Veech surfaces, arXiv:2309.17165). and it completes the study of the algebraic interaction strength KVol on the regular polygon Veech surfaces. Our results on Bouw-M\"oller surfaces extends the results of (Boulanger-Pasquinelli, Algebraic intersections on Bouw-M\"oller surfaces, and more general convex polygons, arXiv:2409.01711). This is also the first exact computation of KVol on translation surfaces with several singularities, and the pairs of curves that achieve the best ratio are singular geodesics made of two saddle connections with different directions.