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Computing Riemann theta functions. (English) Zbl 1092.33018
This paper is based on Computer Algebra Systems (Java, Maple, with an excellent two-part appendix providing the necessary protocol for both implementations); the central problem is approximating values of theta functions (with characteristics) and their derivatives. Theta functions are quasi-periodic analytic functions of $g$ complex variables, where $g$ is any positive integer (the genus, in the case when the periods are complete integrals over the first homology of a Riemann surface). The importance of this line of work is underscored by the rapid development, over the past three decades, of the area of “integrable equations”. The results obtained in this paper are very strong, will be useful, and are especially beautiful in their mathematical basis. Much of algebraic geometry is being substantially advanced by computational methods (cf. systems like MACAULAY, SINGULAR or MAGMA). Two among the present authors already provided Maple-based techiniques to compute the genus and approximate the period lattice of plane curves [cf. {\it B. Deconinck} and {\it M. van Hoeij}, Phys. D 152--153, 28--46 (2001; Zbl 1054.14079)]. The key results on which this work is based are the Fourier expansion of theta functions and a Taylor-polynomial-type error estimate; an error estimate based on the length of the eigenvectors of the period matrix (“Fill Factor Error”); modular transformations that, without changing the theta function, bring the period matrix into `minimal’ form (in particular a theorem of C. L. Siegel, revisited and exploited in this setting); and hard estimates to obtain both pointwise and uniform approximations. This is clever and strenuous work which will be valuable in applications to physics and PDE theory, as well as pure mathematics to study issues of special symmetry and reduction for period matrices. One aspect that is not treated and perhaps could be, within this work, is the (modular) dependence of theta on the period matrix, linked to the analytic variables by the heat equations; in that context, estimates for the size of the period lattice are available [{\it P. Buser} and {\it P. Sarnak}, Invent. Math. 117, No. 1, 27--56 (1994; Zbl 0814.14033)] and have great mathematical depth, being related to the spectrum of the Laplacian [{\it A. Kokotov} and {\it D. Korotkin}, Lett. Math. Phys. 71, No.3, 241--242 (2005; Zbl 1084.58505)].

33F05Numerical approximation and evaluation of special functions
11F30Fourier coefficients of automorphic forms
65F15Eigenvalues, eigenvectors (numerical linear algebra)
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