Computation of conformal representations of compact Riemann surfaces

Abstract

Abstract. We find a system of two polynomial equations in two unknowns, whose solution allows us to give an explicit expression of the conformal representation of a simply connected three-sheeted compact Riemann surface onto the extended complex plane. This function appears in the description of the ratio asymptotic of multiple orthogonal polynomials with respect to so-called Nikishin systems of two measures.

Asymptotics of multiple orthogonal polynomials for a system of two measures supported on a starlike set

“…Using Theorem 3.1 from [11], we can easily describe the cubic algebraic equation solved by ψ. The coefficients of this equation can be computed exclusively in terms of the endpoints of the intervals ∆ 1 and ∆ 2 .…”

“…χ also satisfies χ (z) = z + O(1) as z → ∞ (1) , and has a simple zero at ∞ (0) ∈ S. Observe that χ (∞ (2) ) = −ψ(∞ (0) ) (the reader is cautioned that in this relation, ∞ (2) ∈ S and ∞ (0) ∈ R). χ and S are the types of conformal mappings and Riemann surfaces analyzed in [11]. It follows from [11, Theorem 3.1] that χ (∞ (2) ) = 2/H (β), where H and β are described in the statement of Proposition 1.8 (the uniqueness of β and γ is justified in [11]).…”

“…χ and S are the types of conformal mappings and Riemann surfaces analyzed in [11]. It follows from [11, Theorem 3.1] that χ (∞ (2) ) = 2/H (β), where H and β are described in the statement of Proposition 1.8 (the uniqueness of β and γ is justified in [11]). So χ(z) = ψ(−a 3 (1 + z)/2) + 2/H (β).…”

Riemann-Hilbert analysis for a Nikishin system

“…A way to obtain such a mapping was described in [27]. Using an affine transformation, if necessary, without loss of generality we can assume that with constants A, B and h given by Figure 4: The image of R through the mapping ψ for the intervals [−5, −1] and [1,2] whereβ,α,â,b (β < −1 <α <â < 1 <b) are the critical points of H, which are univocally determined as solutions of some algebraic equations depending solely on µ and λ. Specifically,β andb are the solutions of the quadratic equation…”

“…have equal length, these equations reduce substantially and can be solved exactly with radicals, see [27]). In other words, we can take ψ as the solution of the cubic equation…”