The classifiction of M-curves of bidegree (d,3) in the torus
| dc.contributor.committeeChair | Korchagin, Anatoly | |
| dc.contributor.committeeMember | Weinberg, David A. | |
| dc.contributor.committeeMember | Wang, Alex | |
| dc.creator | Williams, Lina Mabel | |
| dc.date.accessioned | 2016-11-14T23:13:03Z | |
| dc.date.available | 2012-06-01T15:42:28Z | |
| dc.date.available | 2016-11-14T23:13:03Z | |
| dc.date.issued | 2005-05 | |
| dc.degree.department | Mathematics | |
| dc.description.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. | |
| dc.format.mimetype | application/pdf | |
| dc.identifier.uri | http://hdl.handle.net/2346/1000 | |
| dc.language.iso | eng | |
| dc.rights.availability | Unrestricted. | |
| dc.subject | Nonsingular | |
| dc.subject | Isotopy | |
| dc.subject | Branch | |
| dc.subject | Ovals | |
| dc.subject | M-Curve | |
| dc.subject | Bidegree | |
| dc.title | The classifiction of M-curves of bidegree (d,3) in the torus | |
| dc.type | Thesis |