arXiv subjects

Julian Weigert

Publications and source records attributed to Julian Weigert.

12 recordsLinked to original sources

arXiv2026

Nash Loci

In the study of Nash equilibria of finite-player games, one often seeks equilibria that are compatible with predetermined constraints, either determined by the players or by an external agent. We discuss the algebraic loci, called Nash loci, of games whose Nash equilibrium scheme intersects a fixed algebraic variety in a product of projective spaces. We determine their dimensions and multidegrees in multiprojective space, and their equations for two-player games and for small multiple-player games. The multilinear equations defining the Nash equilibrium scheme allow us to describe Nash loci in the language of Grassmannians and Pl\"{u}cker coordinates. Motivated by this fact, we relate Nash loci to multigraded associated varieties, which are subvarieties in products of Grassmannians that generalize the multigraded Cayley-Chow hypersurfaces of Osserman and Trager.

math.AG↗

arXiv2026

Polar Degrees of Matroids

We show that the polar degrees of the coordinate-wise inverse of a linear subspace $L \subseteq \mathbb{P}^n$ are given by the coefficients of a substitution of the reduced characteristic polynomial of the associated matroid $\mathrm{M}(L)$. Our proof connects the geometry of conormal varieties of reciprocal linear spaces to the combinatorial conormal fan of $\mathrm{M}(L)$. As a corollary, we settle two open conjectures regarding matroid discriminants.

math.CO↗

arXiv2026

Benchmarks in Leipzig

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-day workshop *Benchmarks in Leipzig* with 35 participants at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany. We present the resulting collection of 100 questions. We evaluated these questions in three stages: a single attempt by five state-of-the-art LLMs, followed by a 20-runs-per-model evaluation with three of these models, and finally a 3-run attempt with two heavy-thinking models. After Stage 1, 41 questions remained completely unsolved; after Stage 2, this count dropped to 16; and we concluded Stage 3 with only 2 unsolved questions. This demonstrates that the mathematical reasoning capabilities of LLMs are becoming impressive.

math.HO↗

arXiv2025

Nonlinear Kalman varieties

We study the locus of square matrices having at least one eigenvector on a prescribed algebraic variety $X$. When $X$ is a linear subspace, this data locus is known as the Kalman variety of $X$ and was studied first by Ottaviani and Sturmfels. Motivated by recent applications to quantum chemistry and optimization, in this work, we focus on nonlinear Kalman varieties, that is, Kalman varieties relative to arbitrary projective varieties $X$. We study the basic invariants of these varieties, such as their dimensions, degrees, and singularities. Furthermore, Ottaviani and Sturmfels provide determinantal equations in the linear case. We generalize their result to Kalman varieties of hypersurfaces by providing a determinantal-like description of their equation.

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

Toric extensions of P\'olya's theorem

The classical version of P\'olya's theorem provides a simple method for certifying that a homogeneous polynomial of degree d is strictly copositive, that is, it takes only positive values on the nonnegative real orthant. However, this method might fail to detect copositivity of polynomials that are missing certain degree d monomials. In this paper, we present extensions and converses to P\'olya's theorem for sparse polynomials, using techniques from positive toric geometry. Furthermore, we explore how this method can be used to study the convergence of Feynman integrals in particle physics.

math.AG↗

arXiv2025

Nonlinear Rayleigh quotient optimization

Rayleigh quotient minimization deals with optimizing a quadratic homogeneous function over a sphere. Its critical points correspond to the normalized eigenvectors of the symmetric matrix associated with the quadratic form. In this paper, we consider a homogeneous polynomial objective function $f$ over a sphere, a projective algebraic variety $X$, and we study the $X$-eigenpoints of $f$, which are classes of critical points of $f$ constrained to the sphere and the affine cone over $X$. The number of $X$-eigenpoints of a generic polynomial $f$ is the Rayleigh-Ritz degree of $X$. This invariant is a version of the Euclidean distance degree of a Veronese embedding of $X$. We provide concrete formulas in various scenarios, including those involving varieties of rank-one tensors.

math.AG↗

arXiv2025

The canonical form, scissors congruence and adjoint degrees of polytopes

We study the canonical form $\Omega$ as a valuation in the context of scissors congruence for polytopes. We identify the degree of its numerator - the adjoint polynomial $\operatorname{adj}_P$ - as an important invariant in this context. More precisely, for a polytope $P$ we define the degree drop that measures how much smaller than expected the degree of the adjoint polynomial of $P$ is. We show that this drop behaves well under various operations, such as decompositions, restrictions to faces, projections, products and Minkowski sums. Next we define the reduced canonical form $\Omega_0$ and show that it is a translation-invariant 1-homogeneous valuation on polytopes that vanishes if and only if $P$ has positive degree drop. Using it we can prove that zonotopes can be characterized as the $d$-polytopes that have maximal possible degree drop $d-1$. We obtain a decomposition formula for $\Omega_0$ that expresses it as a sum of edge-local quantities of $P$. Finally, we discuss valuations $\Omega_s$ that can distinguish higher values of the degree drop.

math.CO↗

arXiv2025

Complete monotonicity of log-functions

In this article we investigate the property of complete monotonicity within a special family $\mathcal{F}_s$ of functions in $s$ variables involving logarithms. The main result of this work provides a linear isomorphism between $\mathcal{F}_s$ and the space of real multivariate polynomials. This isomorphism identifies the cone of completely monotone functions with the cone of non-negative polynomials. We conclude that the cone of completely monotone functions in $\mathcal{F}_s$ is semi-algebraic. This gives a finite time algorithm to decide whether a function in $\mathcal{F}_s$ is completely monotone

math.CA↗

arXiv2025

Mixed Eulerian numbers and beyond

We derive explicit formulas for the matroidal mixed Eulerian numbers. We resolve a question posed by Berget, Spink, and Tseng, demonstrating that the invariant defined by matroidal mixed Eulerian numbers is precisely equivalent to Derksen's $\mathcal{G}$-invariant. As an application, we provide the first explicit, non-recursive formula for mixed Eulerian numbers. Our combinatorial approach draws inspiration from the classical work of Schubert and incorporates the cutting-edge contributions of Huh.

math.AG↗

arXiv2025

White's conjecture for matroids and inner projections

White's conjecture predicts quadratic generators for the ideal of any matroid base polytope. We prove that White's conjecture for any matroid $M$ implies it also for any matroid $M'$, where $M$ and $M'$ differ by one basis. Our study is motivated by inner projections of algebraic varieties.

math.CO↗

arXiv2023

Equivariant Tutte Polynomial

We use the equivariant cohomology ring of the permutohedral variety to study matroids and their invariants. Investigating the pushforward of matroid Chern classes defined by A. Berget, C. Eur, H. Spink and D. Tseng to the product space $\mathbb{P}^n \times \mathbb{P}^n$, we establish an equivariant generalization of the Tutte polynomial of a matroid. This was suggested in a survey paper by M.Micha{\l}ek. We discuss how this polynomial encodes properties of the matroid by looking at special evaluations. We further introduce an equivariant generalization of the reduced characteristic polynomial of a matroid.

math.AG↗