The classifiction of M-curves of bidegree (d,3) in the torus

The classification, up to homeomorphism, of real algebraic curves in the projective plane was the first part of Hilbert's sixteenth problem. We provide a classification for a new family of curves in the torus.

More precisely, a real homogeneous polynomial f(u,v,x,y) is said to be of bidegree (d,e) if it is homogeneous of degree d (resp. e) with respect to the variables (u,v) (resp. (x,y)). Such polynomials then have naturally defined zero sets on the torus T, provided one realizes T as the product of two real projective lines. The real zero set of f in T is then said to be an M-curve of bidegree (d,e) if it has maximally many real connected components.

We completely classify all M-curves of bidegree (d,3) on the torus. In particular, we show that for any integer d (with d>=2), there are M-curves of bidegree (d,3) realizing the class 2(d-1) O + <a+nb> in H_1(T), where O is homologous to 0, a and b are the generators of H_1(T), and n<=d is any integer with the same parity as d.