Algebraic curve

Monopoles, minimal surfaces and algebraic curves , Seminaire de Mathematiques... 

At still smaller values of

horizontal asymptote for the curve (5.6) and from the fact that at small Tc the exponent (5.7) decreases more rapidly than the algebraic curve (5.6) which has a vertical asymptote u = 0 in common with the exponent. 

The Jacobian has always been the corner-stone in the analysis of algebraic curves and compact Riemann surfaces. Its power lies in the fact that it abelianizes the curve and is a reification of Hi, e.g.,... 

Then by analytic continuation, the 71,..., 7d are part of an algebraic curve T C Pn of degree d (possibly reducible), so that in some neighbourhood U of H, T 0... 

D. Mumford, Theta characteristics of an algebraic curve (Annales Ecole Norm. Sup., 4 (1971), p. 181). 

Y. Manin, Rational points on algebraic curves over function fields, (Izvestija Akad. Nauk, 27 (1963)). 

S. Ju. Arakelov, Families of algebraic curves with fixed degeneracies, (Izvest, Akad. Nauk 35 (1971)). 

A.N. Parsin, Algebraic curves over function fields, (Izvest. Akad. Nauk, 32 (1968)). 

G. Kempf, Schubert methods with an application to algebraic curves (Stichting Math. Centrom, Amsterdam, 1971). 

A. Mattuck, A. Mayer, The Riemann-Roch theorem of algebraic curves, (Annali Sc. Norm. Pisa, 17 (1963)). 

A. Andreotti, A. Mayer, On period relations for Abelian integrals on algebraic curves (Ann. Scu. Norm. Sup. Pisa (1967)). 

F. Oort and J. Steenbrink, The local Torelli problem for algebraic curves, Journees de Geometrie Algebrique d Angers (A. Beauville ed.) Suthoff Noordhoff, 1980, 157-204. 

The operators act on the space of smooth vector functions, and the Lie algebra Afn i is realized by square zero-trace matrices. The matrices a and b are diagonal with distrinct diagonal elements. According to the finite-zoned integration theory (see [77]), the commutativity equations [Lai Aa] = 0 are integrated by means of the theta-functions of the Riemann surface of the algebraic curve Q W A) = det(lV — X - Aa) = 0. 

Corollary 4.2.3. The solutions of the Euler equations with the metrics ahD on the Lie algebra sl(m, C) are rational functions of the exponents and restrictions of the theta-functions of the algebraic curves Q(Wf A) = 0 to rectilinear windings of the Jacobian tori J T) of these curves. 

Krichever, I. M. "Algebraic curves and nonlinear differential equations. Uspekhi Mat. Nauk 33 (1978), No. 4, 215-216. 

McKean, H. Integrable systems and algebraic curves. In Global Analysis. Lecture Notes in Math., v. 755 (1976), 83-200. Springer, New York. 

Moser, J. Various aspects of integrable Hamiltonian systems. In Dynamical Systems. Prog, in Math. 8. Birkhauser-Boston Cambridge, Mass. (1980). Van Moerbeke, P., and Mumford, D., The spectrum of difference operators and algebraic curves. Acta Math. 148 (1979), 93-154. 

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