Mario Kummer

Two results on the Convex Algebraic Geometry of sets with continuous symmetries

Renato G. Bettiol, Mario Kummer, Ricardo A. E. Mendes

We prove two results on convex subsets of Euclidean spaces invariant under an orthogonal group action. First, we show that invariant spectrahedra admit an equivariant spectrahedral…

math.AG2013

Hyperbolic polynomials, interlacers, and sums of squares

Mario Kummer, Daniel Plaumann, Cynthia Vinzant

Hyperbolic polynomials are real polynomials whose real hypersurfaces are nested ovaloids, the inner most of which is convex. These polynomials appear in many areas of mathematics,…

math.AG2023

A signed count of 2-torsion points on real abelian varieties

Mario Kummer

We prove that a natural signed count of the $2$ -torsion points on a real principally polarized abelian variety $A$ always equals to $2^{g}$ where $g$ is the dimension of $A$ . When…

On Huisman's conjectures about unramified real curves

Mario Kummer, Dimitri Manevich

Let $X \subset P^{n}$ be an unramified real curve with $X (R) \neq = \emptyset$ . If $n \geq 3$ is odd, Huisman conjectures that $X$ is an $M$ -curve and that every br…

math.HO2026

Benchmarks in Leipzig

Andrei Balakin, MiklÃ³s BÃ³na, Marie-Charlotte Brandenburg +45

Between April 1 and May 15, 2026, a group of 49 mathematicians compiled a dataset of research-level mathematics questions with known answers. Most of the work was done during the 3…

math.NT2016

Eigenvalues of symmetric matrices over integral domains

Mario Kummer

Given an integral domain A we consider the set of all integral elements over A that can occur as an eigenvalue of a symmetric matrix over A. We give a sufficient criterion for bein…

math.AG2015

Two results on the size of spectrahedral descriptions

Mario Kummer

A spectrahedron is a set defined by a linear matrix inequality. Given a spectrahedron we are interested in the question of the smallest possible size $r$ of the matrices in the des…

math.DG2025

Geography of pinched four-manifolds

Renato G. Bettiol, Mario Kummer, Ricardo A. E. Mendes

We prove several new restrictions on the Euler characteristic and signature of oriented 4-manifolds with (positively or negatively) pinched sectional curvature. In particular, we s…

math.GR2023

Some thoughts and experiments on Bergman's compact amalgamation problem

Michael Joswig, Mario Kummer, Andreas Thom +1

We study the question whether copies of $S^{1}$ in $SU (3)$ can be amalgamated in a compact group. This is the simplest instance of a fundamental open problem in the theory o…

math.AG2024

(Positive) Quadratic Determinantal Representations of Quartic Curves and the Robinson Polynomial

Clemens BrÃ¼ser, Mario Kummer

We prove that every real nonnegative ternary quartic whose complex zero set is smooth can be represented as the determinant of a symmetric matrix with quadratic entries which is ev…

math.CO2016

Interlacing Ehrhart Polynomials of Reflexive Polytopes

Akihiro Higashitani, Mario Kummer, Mateusz MichaÅek

It was observed by Bump et al. that Ehrhart polynomials in a special family exhibit properties similar to the Riemann Î¶ function. The construction was generalized by Matsui et al.…

math.DG2021

Convex Algebraic Geometry of Curvature Operators

Renato G. Bettiol, Mario Kummer, Ricardo A. E. Mendes

We study the structure of the set of algebraic curvature operators satisfying a sectional curvature bound under the light of the emerging field of Convex Algebraic Geometry. More p…

math.AG2019

The separating semigroup of a real curve

Mario Kummer, Kristin Shaw

We introduce the separating semigroup of a real algebraic curve of dividing type. The elements of this semigroup record the possible degrees of the covering maps obtained by restri…

math.AG2022

Equivariant algebraic and semi-algebraic geometry of infinite affine space

Mario Kummer, Cordian Riener

We study $Sym (\infty)$ -orbit closures of not necessarily closed points in the Zariski spectrum of the infinite polynomial ring $\mathbb{C}[x_{ij}:\, i\in\mathbb{N},\,j\in[…

Positive Ulrich Sheaves

Christoph Hanselka, Mario Kummer

We provide a criterion for a coherent sheaf to be an Ulrich sheaf in terms of a certain bilinear form on its global sections. When working over the real numbers we call it a positi…

math.MG2021

Iso Edge Domains

Mathieu Dutour SikiriÄ, Mario Kummer

Iso-edge domains are a variant of the iso-Delaunay decomposition introduced by Voronoi. They were introduced by Baranovskii & Ryshkov in order to solve the covering problem in dime…

math.AG2026

Ulrich sheaves and determinantal representations for higher secant varieties of curves

Daniele Agostini, Mario Kummer, Jinhyung Park

We show that higher secant varieties of smooth projective curves have symmetric admissible determinantal representations with symmetric Ulrich sheaves of rank one if the embedding…

Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment

Grigoriy Blekherman, Mario Kummer, Raman Sanyal +2

A linear principal minor polynomial or lpm polynomial is a linear combination of principal minors of a symmetric matrix. By restricting to the diagonal, lpm polynomials are in bije…

A generalization of the space of complete quadrics

Abeer Al Ahmadieh, Mario Kummer, Miruna-Stefana Sorea

To any homogeneous polynomial $h$ we naturally associate a variety $I^c ◯_{h}$ which maps birationally onto the graph $I^_{h}$ of the gradient map $\nabla h$ and which agrees with the spa…

math.FA2021

The multidimensional truncated Moment Problem: CarathÃ©odory Numbers from Hilbert Functions

Philipp J. di Dio, Mario Kummer

In this paper we improve the bounds for the CarathÃ©odory number, especially on algebraic varieties and with small gaps (not all monomials are present). We provide explicit lower a…

Rational functions with only real periodic points

Khazhgali Kozhasov, Mario Kummer

We study self-morphisms of smooth real projective algebraic curves that have only real periodic points. In the case of the projective line we provide a convenient characterization…

math.AG2016

The geometry of rank-one tensor completion

Thomas Kahle, Kaie Kubjas, Mario Kummer +1

The geometry of the set of restrictions of rank-one tensors to some of their coordinates is studied. This gives insight into the problem of rank-one completion of partial tensors.…

math.AG2015

Determinantal Representations and BÃ©zoutians

Mario Kummer

We show that for every smooth hyperbolic polynomial h there is another hyperbolic polynomial q such that qh has a definite determinantal representation. This is proved by consideri…

math.AG2015

A Note on the Hyperbolicity Cone of the Specialized VÃ¡mos Polynomial

Mario Kummer

The specialized VÃ¡mos polynomial is a hyperbolic polynomial of degree four in four variables with the property that none of its powers admits a definite determinantal representati…

math-ph2024

Higher Dimensional Fourier Quasicrystals from Lee-Yang Varieties

Lior Alon, Mario Kummer, Pavel Kurasov +1

In this paper, we construct Fourier quasicrystals with unit masses in arbitrary dimensions. This generalizes a one-dimensional construction of Kurasov and Sarnak. To do this, we em…

math.NA2018

Generalized eigenvalue methods for Gaussian quadrature rules

Grigoriy Blekherman, Mario Kummer, Cordian Riener +2

A quadrature rule of a measure $I^¼$ on the real line represents a convex combination of finitely many evaluations at points, called nodes, that agrees with integration against $I^¼$ …

math.CO2025

Lorentzian polynomials and matroids over triangular hyperfields 1: Topological aspects

Matthew Baker, June Huh, Mario Kummer +1

Lorentzian polynomials serve as a bridge between continuous and discrete convexity, connecting analysis and combinatorics. In this article, we study the topology of the space $\mat…

math.AG2026

Periodic Hypersurfaces and Lee-Yang Polynomials

Lior Alon, Mario Kummer

We study periodic measures on $R^{n}$ whose Fourier transform is confined to a proper double cone, in the sense of Meyer's notion of lighthouse measures. Lee--Yang polynomia…

Nodes on quintic spectrahedra

Taylor Brysiewicz, Khazhgali Kozhasov, Mario Kummer

We classify transversal quintic spectrahedra by the location of 20 nodes on the respective real determinantal surface of degree 5. We identify 65 classes of such surfaces and find…

math.RA2022

Spectrahedral Shadows and Completely Positive Maps on Real Closed Fields

Manuel Bodirsky, Mario Kummer, Andreas Thom

In this article we develop new methods for exhibiting convex semialgebraic sets that are not spectrahedral shadows. We characterize when the set of nonnegative polynomials with a g…

math.AC2016

On the real rank of monomials

Enrico Carlini, Mario Kummer, Alessandro Oneto +1

In this paper we study the real rank of monomials and we give an upper bound for the real rank of all monomials. We show that the real and the complex ranks of a monomial coincide…

math.CO2023

Matroids on Eight Elements with the Half-plane Property and Related Concepts

Mario Kummer, BÃ¼Åra Sert

We classify all matroids with at most 8 elements that have the half-plane property, and we provide a list of some matroids on 9 elements that have, and that do not have the half-pl…

math.AG2017

On the connectivity of the hyperbolicity region of irreducible polynomials

Mario Kummer

We give an elementary proof for the fact that an irreducible hyperbolic polynomial has only one pair of hyperbolicity cones.

Ulrich sheaves, the arithmetic writhe and algebraic isotopies of space curves

Daniele Agostini, Mario Kummer

We establish a connection between the theory of Ulrich sheaves and $A^{1}$ -homotopy theory. For instance, we prove that the $A^{1}$ -degree of a morphism between proje…

Hyperbolic Secant Varieties of M-Curves

Mario Kummer, Rainer Sinn

We relate the geometry of curves to the notion of hyperbolicity in real algebraic geometry. A hyperbolic variety is a real algebraic variety that (in particular) admits a real fibe…

math.AG2024

Maximal Mumford Curves from Planar Graphs

Mario Kummer, Bernd Sturmfels, Raluca Vlad

A curve of genus g is maximal Mumford (MM) if it has g+1 ovals and g tropical cycles. We construct full-dimensional families of MM curves in the Hilbert scheme of canonical curves.…

Adjoints of Polytopes: Determinantal Representations and Smoothness

Clemens BrÃ¼ser, Mario Kummer, Dmitrii Pavlov

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 o…

math.CO2024

Three results related to the half-plane property of matroids

Mario Kummer, David Sawall

We settle three problems from the literature on stable and real zero polynomials and their connection to matroid theory. We disprove the weak real zero amalgamation conjecture by S…

math.AG2018

Real Fibered Morphisms and Ulrich Sheaves

Mario Kummer, Eli Shamovich

In this paper we define and study real fibered morphisms. Such morphisms arise in the study of real hyperbolic hypersurfaces in projective space and other hyperbolic varieties. We…

math.AG2019

On Deformations of Hyperbolic Varities

Mario Kummer, Eli Shamovich

In this paper we study flat deformations of real subschemes of $P^{n}$ , hyperbolic with respect to a fixed linear subspace, i.e. admitting a finite surjective and real fiber…

math.CO2025

Representation theory for polymatroids

Matthew Baker, June Huh, Donggyu Kim +2

We develop a theory of representations of (discrete) polymatroids over tracts in terms of PlÃ¼cker coordinates and suitable PlÃ¼cker relations. As special cases, we recover polymat…

math.FA2026

Non-negative polynomials on generalized elliptic curves

Mario Kummer, AljaÅ¾ Zalar

We study the cone of non-negative polynomials on generalized elliptic curves. We show that the zero set of every extreme ray has dense real points. If a generalized elliptic curve…

math.CO2026

Lorentzian polynomials and matroids over triangular hyperfields 2: Analytic aspects

Matthew Baker, June Huh, Mario Kummer +1

BrÃ¤ndÃ©n and Huh showed that Lorentzian polynomials unify Hodge-Riemann relations in combinatorics: their supports are M-convex, and every M-convex set supports a Lorentzian polyn…

math.AG2016

The Chow form of a reciprocal linear space

Mario Kummer, Cynthia Vinzant

A reciprocal linear space is the image of a linear space under coordinate-wise inversion. These fundamental varieties describe the analytic centers of hyperplane arrangements and a…

cs.CL2026

Soohak: A Mathematician-Curated Benchmark for Evaluating Research-level Math Capabilities of LLMs

Guijin Son, Seungone Kim, Catherine Arnett +73

Following the recent achievement of gold-medal performance on the IMO by frontier LLMs, the community is searching for the next meaningful and challenging target for measuring LLM…

math.AG2018

Spectrahedral representations of plane hyperbolic curves

Mario Kummer, Simone Naldi, Daniel Plaumann

We describe a new method for constructing a spectrahedral representation of the hyperbolicity region of a hyperbolic curve in the real projective plane. As a consequence, we show t…

Spectral linear matrix inequalities

Mario Kummer

We prove, under a certain representation theoretic assumption, that the set of real symmetric matrices, whose eigenvalues satisfy a linear matrix inequality, is itself a spectrahed…

math.AG2018

Schottky Algorithms: Classical meets Tropical

Lynn Chua, Mario Kummer, Bernd Sturmfels

We present a new perspective on the Schottky problem that links numerical computing with tropical geometry. The task is to decide whether a symmetric matrix defines a Jacobian, and…

math.AG2019

Totally real theta characteristics

Mario Kummer

A totally real theta characteristic of a real curve is a theta characteristic which is linearly equivalent to a sum of only real points. These are closely related to the facets of…

Real-fibered morphisms of del Pezzo surfaces and conic bundles

It goes back to Ahlfors that a real algebraic curve admits a real-fibered morphism to the projective line if and only if the real part of the curve disconnects its complex part. In…