Browsing by Subject "M-Curve"
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Item The classifiction of M-curves of bidegree (d,3) in the torus(2005-05) Williams, Lina Mabel; Korchagin, Anatoly; Weinberg, David A.; Wang, AlexThe 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 + 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.Item The classifiction of m-curves of bidegree (d,3) on Torus(Texas Tech University, 2005-05) Williams, Lina Mabel; Korchagin, Anatoly; Weinberg, David A.; Wang, XiaochangThe 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 + 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.