Combinatorial analysis of the period mapping: the topology of 2D fibres

Abstract

We study the period mapping from the moduli space of real hyperelliptic curves with marked point on an oriented oval to the euclidean space. The mapping arises in the analysis of Chebyshev construction used in the constrained optimization of the uniform norm of polynomials and rational functions.The decomposition of the moduli space into polyhedra labeled by planar graphs allows to investigate the global topology of low dimensional fibers of the period mapping. * Supported by RFBR grants 16-01-00568, and RAS Program "Modern problems of theoretical mathematics"

Coordinate spaces of graphs: approaching interior faces

“…It turns out that for the moduli space H 1 2 there are only nine "stable" topological types of the graphs Γ which do not change their combinatorial structure under arbitrary small perturbation of a curve M (E), they are listed e.g. in [8]. Two of such graphs Γ + and Γ − are shown in the left and the right panels of Fig.…”

“…This Riemann surface may be glued from a finite number of stripes in a way determined by combinatorics and weights of the graph. The detailed instruction for this assembly is given in [4,7,8]. Unfortunately, this approach to the reconstruction of the curve cannot be called efficient and it does not give any quantitative characteristics of the embedding of a coordinate space to the moduli space.…”

“…This author claimed [8] that the embedding of a positive codimension coordinate space like A[Γ 0 ] coincides with the continuation of the embedding of suitable full dimensional polyhedra A[Γ]…”

“…[14,17,10]. The study of the period mapping lying in the core of the Chebyshev construction for the solutions of uniform rational approximation problems [3,7] also grounds on a pictorial technique [4,8]. Pictorial technique proved to be extremely useful in the study of mesoscale geometry (as opposed to the differential and the global ones) of various moduli spaces and related structures.…”

Degeneration of a graph describing conformal structure

“…○ 2023 Russian Academy of Sciences, Steklov Mathematical Institute of RAS used for the solution of problems in uniform rational approximation [13], [14] is also grounded on graphical techniques [15], [16]. Other closely related topics include flat surfaces [17], triangulated surfaces [18], the complex Pell-Abel equations [19]; also see [20] and [21] for other examples.…”