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Abstract

Abstract. A theorem of Bôcher and Grace states that the critical points of a cubic polynomial are the foci of an ellipse tangent to the sides of the triangle joining the zeros. A more general result of Siebert and others states that the critical points of a polynomial of degree N are the algebraic foci of a curve of class N − 1 which is tangent to the lines joining pairs of zeroes. We prove the analogous results for hyperbolic polynomials, that is, for Blaschke products with N roots in the unit disc.

Orthogonal Families of Bicircular Quartics, Quadratic Differentials, and Edwards Normal Form

“…Likewise, quadruple foci, etc., are foci (though such higher order branch points of π I do not occur in our examples). 1 m j z−z j as the foci of a related conic (see [37] for the hyperbolic case). But his result on quartics and elliptic functions also deserves to be well known-hence, this brief appendix on his paper [3].…”

Hyperbolic monopoles from hyperbolic vortices

“…(57) Using (55) in (54) and changing variables again gives the relation (33). In fact, there are certain vortex configurations, in particular those with an infinite number of vortex zeros, [21], which have only been constructed by Witten's method (12) and for which no JNR function ψ is known.…”

“…It is also clear that there are precisely N spectral lines for each choice of ζ on the boundary, and that any geodesic with ζ ∈ R also has η ∈ R. Specialising to N = 2 with ζ ∈ R leads to an interesting geometric picture in terms of Poncelet's theorem, which has already given insight into the geometry of instantons [15] and indeed hyperbolic monopoles [17]. We will work through the details explicitly in our case, making use of various theorems of Daepp-Gorkin-Mortini [11] and Singer [33].…”

“…The prescribed Blaschke product (64) then generates all of the spectral lines (with ζ ∈ R) and hence a family of ideal triangles corresponding to the gauge freedom in the JNR data. It was shown in [33] that the envelope of this family of triangles is a hyperbolic ellipse (the locus of points for which the sum of the geodesic distances from the foci is constant) whose foci are at the critical points of f, i.e. at the vortex zeros 6 .…”

Confocal Families of Hyperbolic Conics via Quadratic Differentials

“…this is the unique isometry from the disc with the Klein metric to the disc with the Poincaré metric that keeps the ideal boundary pointwise fixed [7]; it takes the straight line segment between points on the boundary to the geodesic between the same two points in the Poincaré disc. As we have chosen the focus inside the disc to be at the origin, the rays from the focus coincide (but not pointwise).…”