We say that a line in Pⁿ⁺¹_k is osculating to a hypersurface Y if they meet with contact order n+1. When k=C, it is known that through a fixed point of Y, there are exactly n! of such lines. Under some parity condition on n and deg(Y), we define a quadratically enriched count of these lines over any perfect field k. The count takes values in the Grothendieck--Witt ring of quadratic forms over k and depends linearly on deg(Y).

The author thanks Ethan Cotterill, Gabriele Degano, Stephen McKean, Kyler Siegel, and Israel Vainsencher for many useful discussions. The author also thank Michael Stillman and Matthias Zach for their computational help, and the Reviewers for taking the necessary time and effort to review the manuscript. This work is supported by FCT - Fundaco para a Ciencia e a Tecnologia, under the project: UIDP/04561/2020 (https://doi.org/10.54499/UIDP/04561/2020). The author is a member of GNSAGA (INdAM).