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Ronald van Luijk

Publications and source records attributed to Ronald van Luijk.

At least 19 recordsLinked to original sources
arXiv2025

IMProofBench: Benchmarking AI on Research-Level Mathematical Proof Generation

As the mathematical capabilities of large language models (LLMs) improve, it becomes increasingly important to evaluate their performance on research-level tasks at the frontier of mathematical knowledge. However, existing benchmarks are limited, as they focus solely on final-answer questions or high-school competition problems. To address this gap, we introduce IMProofBench, a private benchmark consisting of 77 peer-reviewed problems developed by expert mathematicians. Each problem requires a detailed proof and is paired with subproblems that have final answers, supporting both an evaluation by human experts and a large-scale quantitative analysis through automated grading. Furthermore, unlike prior benchmarks, the evaluation setup simulates a realistic research environment: models operate in an agentic framework with tools like web search for literature review and mathematical software such as SageMath. Our results show that current LLMs can already solve a significant percentage of research-level questions. IMProofBench will continue to evolve as a dynamic benchmark in collaboration with the mathematical community, ensuring its relevance for evaluating the next generation of LLMs.

cs.CL↗
arXiv2023

On the Galois-invariant part of the Weyl group of the Picard lattice of a K3 surface

Let $X$ denote a K3 surface over an arbitrary field $k$. Let $k^\text{s}$ denote a separable closure of $k$ and let $X^\text{s}$ denote the base change of $X$ to $k^\text{s}$. The action of the absolute Galois group Gal($k^\text{s}/k$) of $k$ on Pic $X^\text{s}$ respects the intersection pairing, which gives Pic $X^\text{s}$ the structure of a lattice. Let O(Pic $X$) and O(Pic $X^\text{s}$) denote the group of isometries of Pic $X$ and Pic $X^\text{s}$, respectively. Let $R_X$ denote the Galois invariant part of the Weyl group of O(Pic $X^\text{s}$). One can show that each element in $R_X$ can be restricted to an element of O(Pic $X$). The following question arises: Is the image of the restriction map $R_X \to $O(Pic $X$) a normal subgroup of O(Pic $X$) for every K3 surface $X$? We show that the answer is negative by giving counterexamples over $k=\mathbb{Q}$.

math.AG↗
arXiv2022

Concurrent lines on del Pezzo surfaces of degree one

Let $X$ be a del Pezzo surface of degree one over an algebraically closed field $k$, and let $K_X$ be its canonical divisor. The morphism $φ$ induced by the linear system $|-2K_X|$ realizes $X$ as a double cover of a cone in $\mathbb{P}^3$ that is ramified over a smooth curve of degree 6. The surface $X$ contains 240 curves with negative self-intersection, called exceptional curves. We prove that for a point~$P$ on the ramification curve of $φ$, at most sixteen exceptional curves go through~$P$ in characteristic $2$, and at most ten in all other characteristics. Moreover, we prove that for a point $Q$ outside the ramification curve of $φ$, at most twelve exceptional curves go through $Q$ in characteristic $3$, and at most ten in all other characteristics. We show that these upper bounds are sharp in all cases except possibly in characteristic 5 outside the ramification curve.

math.AG↗
arXiv2021

The action of the Weyl group on the $E_8$ root system

Let $Γ$ be the graph on the roots of the $E_8$ root system, where any two distinct vertices $e$ and $f$ are connected by an edge with color equal to the inner product of $e$ and $f$. For any set $c$ of colors, let $Γ_c$ be the subgraph of $Γ$ consisting of all the $240$ vertices, and all the edges whose color lies in $c$. We consider cliques, i.e., complete subgraphs, of $Γ$ that are either monochromatic, or of size at most $3$, or a maximal clique in $Γ_c$ for some color set $c$, or whose vertices are the vertices of a face of the $E_8$ root polytope. We prove that, apart from two exceptions, two such cliques are conjugate under the automorphism group of $Γ$ if and only if they are isomorphic as colored graphs. Moreover, for an isomorphism $f$ from one such clique $K$ to another, we give necessary and sufficient conditions for $f$ to extend to an automorphism of $Γ$, in terms of the restrictions of $f$ to certain special subgraphs of $K$ of size at most 7.

math.CO↗
arXiv2019

Finiteness theorems for K3 surfaces over arbitrary fields

Over an algebraically closed field, various finiteness results are known regarding the automorphism group of a K3 surface and the action of the automorphisms on the Picard lattice. We formulate and prove versions of these results over arbitrary base fields, and give examples illustrating how behaviour can differ from the algebraically closed case.

math.AG↗
arXiv2015

Unirationality of del Pezzo surfaces of degree two over finite fields

We prove that every del Pezzo surface of degree two over a finite field is unirational, building on the work of Manin and an extension by Salgado, Testa, and Várilly-Alvarado, who had proved this for all but three surfaces. Over general fields of characteristic not equal to two, we state sufficient conditions for a del Pezzo surface of degree two to be unirational.

math.AG↗
arXiv2014

Computing Néron-Severi groups and cycle class groups

Assuming the Tate conjecture and the computability of étale cohomology with finite coefficients, we give an algorithm that computes the Néron-Severi group of any smooth projective geometrically integral variety, and also the rank of the group of numerical equivalence classes of codimension p cycles for any p.

math.AG↗
arXiv2014

Density of rational points on del Pezzo surfaces of degree one

We state conditions under which the set S(k) of k-rational points on a del Pezzo surface S of degree 1 over an infinite field k of characteristic not equal to 2 or 3 is Zariski dense. For example, it suffices to require that the elliptic fibration over the projective line induced by the anticanonical map has a nodal fiber over a k-rational point. It also suffices to require the existence of a point in S(k) that does not lie on six exceptional curves of S and that has order 3 on its fiber of the elliptic fibration. This allows us to show that within a parameter space for del Pezzo surfaces of degree 1 over the field of real numbers, the set of surfaces S defined over the field Q of rational numbers for which the set S(Q) is Zariski dense, is dense with respect to the real analytic topology. We also include conditions that may be satisfied for every del Pezzo surface S and that can be verified with a finite computation for any del Pezzo surface S that does satisfy them.

math.AG↗
arXiv2013

Explicit Selmer groups for cyclic covers of P^1

For any abelian variety J over a global field k and an isogeny phi: J -> J, the Selmer group Sel^phi(J,k) is a subgroup of the Galois cohomology group H^1(Gal(ksep/k), J[phi]), defined in terms of local data. When J is the Jacobian of a cyclic cover of P^1 of prime degree p, the Selmer group has a quotient by a subgroup of order at most p that is isomorphic to the `fake Selmer group', whose definition is more amenable to explicit computations. In this paper we define in the same setting the `explicit Selmer group', which is isomorphic to the Selmer group itself and just as amenable to explicit computations as the fake Selmer group. This is useful for describing the associated covering spaces explicitly and may thus help in developing methods for second descents on the Jacobians considered.

math.AG↗
arXiv2012

The Cayley-Oguiso automorphism of positive entropy on a K3 surface

Recently Oguiso showed the existence of K3 surfaces that admit a fixed point free automorphism of positive entropy. The K3 surfaces used by Oguiso have a particular rank two Picard lattice. We show, using results of Beauville, that these surfaces are therefore determinantal quartic surfaces. Long ago, Cayley constructed an automorphism of such determinantal surfaces. We show that Cayley's automorphism coincides with Oguiso's free automorphism. We also exhibit an explicit example of a determinantal quartic whose Picard lattice has exactly rank two and for which we thus have an explicit description of the automorphism.

math.AG↗
arXiv2010

Density of rational points on elliptic surfaces

Suppose V is a surface over a number field k that admits two elliptic fibrations. We show that for each integer d there exists an explicitly computable closed subset Z of V, not equal to V, such that for each field extension K of k of degree at most d over the field of rational numbers, the set V(K) is Zariski dense as soon as it contains any point outside Z. We also present a version of this statement that is universal over certain twists of V and over all extensions of k. This generalizes a result of Swinnerton-Dyer, as well as previous work of Logan, McKinnon, and the author.

math.AG↗
arXiv2009

Lines on Fermat surfaces

We prove that the Neron-Severi groups of several complex Fermat surfaces are generated by lines. Specifically, we obtain these new results for all degrees up to 100 that are relatively prime to 6. The proof uses reduction modulo a supersingular prime. The techniques are developed in detail. They can be applied to other surfaces and varieties as well.

math.AG↗
arXiv2009

Density of rational points on diagonal quartic surfaces

Let a,b,c,d be nonzero rational numbers whose product is a square, and let V be the diagonal quartic surface in PP^3 defined by ax^4+by^4+cz^4+dw^4=0. We prove that if V contains a rational point that does not lie on any of the 48 lines on V or on any of the coordinate planes, then the set of rational points on V is dense in both the Zariski topology and the real analytic topology.

math.AG↗
arXiv2007

Cubic points on cubic curves and the Brauer-Manin obstruction on K3 surfaces

We show that if over some number field there exists a certain diagonal plane cubic curve that is locally solvable everywhere, but that does not have points over any cubic galois extension of the number field, then the algebraic part of the Brauer-Manin obstruction is not the only obstruction to the Hasse principle for K3 surfaces.

math.NT↗
arXiv2007

Nontrivial elements of Sha explained through K3 surfaces

In this paper we present a new method to show that a principal homogeneous space of the Jacobian of a curve of genus two is nontrivial. The idea is to exhibit a Brauer-Manin obstruction to the existence of rational points on a quotient of this principal homogeneous space. In an explicit example we apply the method to show that a specific curve has infinitely many quadratic twists whose Jacobians have nontrivial Tate-Shafarevich group.

math.AG↗
arXiv2006

Non-Euclidean Pythagorean triples, a problem of Euler, and rational points on K3 surfaces

We discover suprising connections between three seemingly different problems: finding right triangles with rational sides in a non-Euclidean geometry, finding three integers such that the difference of the squares of any two is a square, and the problem of finding rational points on an algebraic surface in algebraic geometry. We will also reinterpret Euler's work on the second problem with a modern point of view.

math.NT↗