Arf invariants of real algebraic curves (Q952943)

If a link in the three-sphere is proper, that is, each component has even linking number with the other components, one can define its Arf invariant as the Arf invariant of an associated quadratic form. The author generalized it to links in rational homology three-spheres [\textit{P. M. Gilmer}, J. Knot Theory Ramifications 2, No.3, 285--320 (1993; Zbl 0797.57003)]. In this paper the author studies real algebraic curves with only real nodal singularities, where a nodal singularity is a transverse crossing of two branches. Given such a curve \(C\) one can associate a link \(\mathcal{T}(C)\) in the tangent circle bundle over the real projective plane \(\mathbb{RP}(2)\). Then, since the circle bundle over \(\mathbb{RP}(2)\) is a rational homology three-sphere (in fact it is homeomorphic to the lens space of type \((4,3)\) [\textit{P. Gilmer}, Pac. J. Math. 153, No.1, 31--69 (1992; Zbl 0784.57002)], one can define the Arf invariant of \(\mathcal{T}(C)\). A real algebraic curve of degree \(m\) is called an \(M\)-curve if it has \((m-1)(m-2)/2+1\) components, which is the maximum number of components of such curves by Harnack's inequality. For a singular \(M\)-curve \(C\) of degree \(2k\), the author proves congruence relations modulo eight between \(k\) and numbers defined by the placement on \(\mathbb{RP}(2)\) of the components of \(C\) by using the Arf invariant of the associated link \(\mathcal{T}(C)\). This generalizes congruences of [\textit{T. Fiedler}, Sov. Math., Dokl. 27, 566--568 (1983; Zbl 0541.14022); Sov. Math., Dokl. 33, 262--266 (1986; Zbl 0608.14025)].