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