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Clemens Brüser

Publications and source records attributed to Clemens Brüser.

4 recordsLinked to original sources
arXiv2026

The Geometry of Polycons and a Counterexample to Wachspress' Conjecture

Polycons, initially introduced by Wachspress in 1975 as a tool in finite element methods, are generalizations of polygons in that they allow conic boundary components. We are interested in the adjoint curve of a given polycon, i.e. the unique curve of minimal degree vanishing in the so-called residual arrangement. It was conjectured by Wachspress that under some regularity assumptions this curve does not vanish in the interior of its defining polycon. However, until recently the only class of polycons for which this was proven were convex polygons. We present a polycon bounded by three conics that constitutes a counterexample to Wachspress' conjecture. The origin of this counterexample reveals some beautiful geometry of polycons. Replacing one degree two boundary component of a polycon with a line produces a new polycon. We show that the adjoint of the latter is a contact curve to the adjoint of the former. This naturally leads to the consideration of symmetric linear determinantal representations of adjoints, which lets us explicitly describe the fibers of the adjoint map in the case of polycons bounded by three conics. As a corollary we prove that generically the adjoint of a polycon bounded by three conics is smooth.

math.AG↗
arXiv2025

Geometry of Adjoint Hypersurfaces for Polytopes

In this article we prove that the adjoint polynomial of arbitrary convex polytopes is up to scaling uniquely determined by vanishing to the right order on the polytopes residual arrangement. This answers a problem posed by Kohn and Ranestad and generalizes their main theorem to non-simple polytopes. We furthermore prove that the adjoint polynomial is already characterized by vanishing to the right order on a zero-dimensional subset of the residual arrangement.

math.CO↗
arXiv2025

Adjoints of Polytopes: Determinantal Representations and Smoothness

In this article we study determinantal representations of adjoint hypersurfaces of polytopes. We prove that adjoint polynomials of all polygons can be represented as determinants of tridiagonal symmetric matrices of linear forms with the matrix size being equal to the degree of the adjoint. We prove a sufficient combinatorial condition for a surface in the projective three-space to have a determinantal representation and use it to show that adjoints of all three-dimensional polytopes with at most eight facets and a simple facet hyperplane arrangement admit a determinantal representation. This includes all such polytopes with a smooth adjoint. We demonstrate that, starting from four dimensions, adjoint hypersurfaces may not admit linear determinantal representations. Along the way we prove that, starting from three dimensions, adjoint hypersurfaces are typically singular, in contrast to the two-dimensional case. We also consider a special case of interest to physics, the ABHY associahedron. We construct a determinantal representation of its universal adjoint in three dimensions and show that in higher dimensions a similarly structured representation does not exist.

math.AG↗