Quintic function

About: Quintic function is a research topic. Over the lifetime, 1677 publications have been published within this topic receiving 26780 citations.

Stationary localized solutions in the subcritical complex Ginzburg-Landau equation

Abstract: It is shown that pulses in the complete quintic one-dimensional Ginzburg–Landau equation with complex coefficients appear through a saddle-node bifurcation which is determined analytically through a suitable approximation of the explicit form of the pulses The results are in excellent agreement with direct numerical simulations

Resolution of algebraic equations by theta constants

Abstract: The history of algebraic equations is very long. The necessity and the trial of solving algebraic equations existed already in the ancient civilizations. The Babylonians solved equations of degree 2 around 2000 B.C. as well as the Indians and the Chinese. In the 16th century, the Italians discovered the resolutions of the equations of degree 3 and 4 by radicals known as Cardano’s formula and Ferrari’s formula. However in 1826, Abel [1] (independently about the same epoch Galois [7]) proved the impossibility of solving general equations of degree ≥ 5 by radicals. This is one of the most remarkable event in the history of algebraic equations. Was there nothing to do in this branch of mathematics after the work of Abel and Galois? Yes, in 1858 Hermite [8] and Kronecker [15] proved that we can solve the algebraic equation of degree 5 by using an elliptic modular function. Since \( \sqrt[n]{a} = \exp \left( {\left( {{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern- ulldelimiterspace} n}} \right)\log a} \right) \) which is also written as exp((1/n) ∫ 1 a (1/x)dx), to allow only the extractions of radicals is to use only the exponential. Hence under this restriction, as we learn in the Galois theory, we can construct only compositions of cyclic extensions, namely solvable extentions. The idea of Hermite and Kronecker is as follows; if we use another transcendental function than the exponential, we can solve the algebraic equation of degree 5. In fact their result is analogous to the formula \( \sqrt[n]{a} = \exp (1/n)\int_1^a {(1/x)dx).} \) . In the quintic equation they replace the exponential by an elliptic modular function and the integral ∫(1/x)dx by elliptic integrals. Kronecker [15] thought the resolution of the equation of degree 5 by an elliptic modular function would be a special case of a more general theorem which might exist. Kronecker’s idea was realized in few cases by Klein [11], [13]. Jordan [10] showed that we can solve any algebraic equation of higher degree by modular functions. Jordan’s idea is clarified by Thomae’s formula, 8 Chap, m (cf. Lindemann [16]). In this appendix, we show how we can deduce from Thomae’s formula the resolution of algebraic equations by a Siegel modular function which is explicitely expressed by theta constants (Theorem 2). Therefore Kronecker’s idea is completely realized. Our resolution of higher algebraic equations is also similar to the formula \( \sqrt[n]{a} = \exp (1/n)\int_1^a {(1/x)dx).} \) In our resolution the exponential is replaced by tne Siegel modular function and the integral ∫(1/x)dx is replaced by hyperelliptic integrals. The existance of such resolution shows that the theta function is useful not only for non-linear differential equations but also for algebraic equations.

Spin curves and Scorza quartics

Abstract: In the paper (Takagi and Zucconi in “On blow-ups of the quintic del Pezzo 3-fold and varieties of power sums of quartic hypersurfaces”, pp 1–40, preprint, submitted, 2008), we construct new subvarieties in the varieties of power sums for certain quartic hypersurfaces. In this paper, we show that these quartics coincide with the Scorza quartics of general pairs of trigonal curves and ineffective theta characteristics. Among other applications, we give an affirmative answer to the conjecture of Dolgachev and Kanev on the existence of the Scorza quartics for any general pairs of curves and ineffective theta characteristics. We also give descriptions of the moduli spaces of trigonal even spin curves.