arXiv subjects

Jan Draisma

Publications and source records attributed to Jan Draisma.

At least 19 recordsLinked to original sources

arXiv2026

On the dimensions of correlated equilibrium polytopes of generic games

In this paper, we study the dimension of the correlated equilibrium polytope of finite games. Under the oriented-matroid notion of genericity, we prove that if a generic game is not full-dimensional, then there exists a subgame whose correlated equilibrium polytope is affinely isomorphic to that of the original game. This settles and generalizes an earlier conjecture of Brandenburg, Hollering, and Portakal (2024). Moreover, we show that the existence of a correlated equilibrium whose slices are all non-zero implies that the correlated equilibrium polytope is either full-dimensional or a singleton.

math.CO↗

arXiv2026

The space of preorders on a commutative monoid

For a finitely generated commutative monoid $\Pi$, we present a constructive description of all (total) preorders on $\Pi$ that are compatible with the monoid structure. Equipped with a natural topology, these preorders form an irreducible spectral space, which we show can be covered by a countable union of admissible sets: subsets of $\mathbb{R}^N$ of the form $A \setminus H$ where $A$ is semialgebraic and $H$ is a countable union of hyperplanes, both defined over the rational numbers. As a consequence of this description, we show that the universal theory of commutative monoids with a total order is decidable. Our proofs use a divide-and-conquer technique that requires establishing all of our results in the greater generality of sets on which $\Pi$ acts with finitely many orbits. As a by-product, we find a new description of all monomoial orders on free modules over a polynomial ring.

math.AC↗

arXiv2026

Computing the continuous symmetries of a parametrized variety

We prove that the symmetry Lie algebra of a parametrized variety can be determined directly from the parametrization, without computing the vanishing ideal of the variety. We derive a practical polynomial-time Monte Carlo algorithm for computing the symmetry Lie algebra of a parametrized variety. We discuss applications to testing the binomiality of the ideal of a parametrized variety after changing coordinates, and test this property on varieties arising from staged tree models and colored Gaussian graphical models. Finally, we discuss symmetries and binomiality after changing coordinates for rational curves and give a characterization of the symmetries of many secant varieties.

math.AG↗

arXiv2026

Improved unirationality for GL-varieties

A $\mathbf{GL}$-variety is a typically infinite dimensional variety equipped with a suitable action of the infinite general linear group $\mathbf{GL}$. In earlier work, we established the unirationality theorem: an irreducible $\mathbf{GL}$-variety admits a dominant map from a particularly simple $\mathbf{GL}$-variety, namely, the product of an irreducible finite-dimensional variety with trivial $\mathbf{GL}$-action and an infinite-dimensional affine space on which $\mathbf{GL}$ acts linearly. The main result of this paper states that this map can in fact be constructed to be surjective rather than merely dominant. An immediate application is that secant varieties to varieties of tensors, which are typically constructed as image closures of certain $\mathbf{GL}$-equivariant maps, are in fact also images of (more complicated) $\mathbf{GL}$-equivariant maps. We derive several consequences of this improved unirationality theorem.

math.AG↗

arXiv2026

Symmetric subrank and its border analogue

The symmetric subrank of homogeneous polynomial is the largest number of terms in a diagonal form to which it can be specialized by a (typically non-invertible) linear variable substitution. Building on earlier work by Derksen-Makam-Zuiddam and Biaggi-Chang-Draisma-Rupniewski for ordinary tensors, we determine the asymptotic behavior of symmetric subrank and symmetric border subrank of degree-d forms as the number of variables tends to infinity. Furthermore, by using results from geometric invariant theory we show that for cubic (resp. quartic) forms the symmetric subrank and symmetric border subrank coincide if the latter is at most three (resp. two).

math.AG↗

arXiv2025

Binomiality of colored Gaussian models

Following earlier work by Coons-Maraj-Misra-Sorea and Misra-Sullivant, we study colored, undirected Gaussian graphical models, and present a necessary and sufficient condition for such a model to have binomial vanishing ideal. These conditions involve Jordan schemes, a variant of association schemes, well-known structures in algebraic combinatorics. Using association schemes without transitive group action, we refute the conjecture by Coons-Maraj-Misra-Sorea that binomiality implies that the color classes must be orbits under the automorphism group of the colored graph.

math.CO↗

arXiv2025

Real subrank of order-three tensors

We study the subrank of real order-three tensors and give a lower bound to the subrank of a real tensor given its complex subrank. Using similar arguments to those used by Bernardi-Blekherman-Ottaviani, we show that all subranks between the minimal typical subrank and the maximal typical subrank, which equals the generic subrank, are also typical. We then study small tensor formats with more than one typical subrank. In particular, we construct a $3 \times 3 \times 5$-tensor with subrank $2$ and show that the subrank of the $4 \times 4 \times 4$-quaternion multiplication tensor is $2$. Finally, we consider the tensor associated to componentwise complex multiplication in $\mathbb{C}^n$ and show that this tensor has real subrank $n$---informally, {\em no more than $n$ real scalar multiplications can be carried out using a device that does $n$ complex scalar multiplications}. We also prove a version of this result for other real division algebras.

math.AG↗

arXiv2025

A Criterion for the Algebraic Density Property of Affine $SL_2$-Manifolds

Let $B$ be an affine $k$-domain which admits a nontrivial fundamental pair $(D,U)$ of locally nilpotent derivations, i.e., if $E=[D,U]$ then $(D,U,E)$ is an $\mathfrak{sl}_2$-triple. We prove an algebraic criterion, characterizing under which conditions the fundamental pair $(D,U)$ resp. the triple $(D,U,E)$ is compatible in a technical sense that allows us to construct many vector fields on the spectrum of $B$ from the complete ones. This criterion enables us to prove the algebraic density property for the following widely studied classes of $\mathrm{SL}_2$-varieties arising in physics: Classical Calogero--Moser spaces, Calogero--Moser spaces with "inner degrees of freedom'' and a smooth cyclic quiver variety.

math.AC↗

arXiv2025

Strength and partition rank under limits and field extensions

The strength of a multivariate homogeneous polynomial is the minimal number of terms in an expression as a sum of products of lower-degree homogeneous polynomials. Partition rank is the analogue for multilinear forms. Both ranks can drop under field extensions, and both can jump in a limit. We show that, for fixed degree and under mild conditions on the characteristic of the ground field, the strength is at most a polynomial in the border strength. We also establish an analogous result for partition rank. Our results control both the jump under limits and the drop under field extensions.

math.AG↗

arXiv2025

Singular loci in varieties of tensors

A Vec-variety is a suitable functor from finite-dimensional vector spaces to finite-dimensional varieties. Most varieties in the geometry of tensors, e.g. the variety of d-way tensors of slice rank at most r, are of this form. We prove that the singular locus of a Vec-variety is a proper closed Vec-subvariety, analogously to the situation for ordinary finite-dimensional varieties. Via earlier work of the third author, this implies that these singular loci admit a description by finitely many polynomial equations. A natural follow-up question to our main result is whether a Vec-variety also admits a suitably functorial resolution of singularities. We establish some preliminary results in this direction in the regime where the dimension of evaluations of a Vec-variety grows linearly with that of the input vector space.

math.AG↗

arXiv2024

The geometry of polynomial representations in positive characteristic

A $\mathbf{GL}$-variety is a (typically infinite dimensional) variety modeled on the polynomial representation theory of the general linear group. In previous work, we studied these varieties in characteristic 0. In this paper, we obtain results in positive characteristic: for example, we prove a version of Chevalley's theorem on constructible sets. We give an application of our theory to strength of polynomials.

math.AG↗

arXiv2024

Topological Noetherianity of the infinite half-spin representations

We prove that the infinite half-spin representations are topologically Noetherian with respect to the infinite spin group. As a consequence we obtain that half-spin varieties, which we introduce, are defined by the pullback of equations at a finite level. The main example for such varieties is the infinite isotropic Grassmannian in its spinor embedding, for which we explicitly determine its defining equations.

math.AG↗

arXiv2024

Border subrank via a generalised Hilbert-Mumford criterion

We show that the border subrank of a sufficiently general tensor in $(\mathbb{C}^n)^{\otimes d}$ is $\mathcal{O}(n^{1/(d-1)})$ for $n \to \infty$. Since this matches the growth rate $\Theta(n^{1/(d-1)})$ for the generic (non-border) subrank recently established by Derksen-Makam-Zuiddam, we find that the generic border subrank has the same growth rate. In our proof, we use a generalisation of the Hilbert-Mumford criterion that we believe will be of independent interest.

math.AG↗

arXiv2024

Two improvements in Brauer's theorem on forms

Let $k$ be a Brauer field, that is, a field over which every diagonal form in sufficiently many variables has a nonzero solution; for instance, $k$ could be an imaginary quadratic number field. Brauer proved that if $f_1, \ldots, f_r$ are homogeneous polynomials on a $k$-vector space $V$ of degrees $d_1, \ldots, d_r$, then the variety $Z$ defined by the $f_i$'s has a non-trivial $k$-point, provided that $\dim{V}$ is sufficiently large compared to the $d_i$'s and $k$. We offer two improvements to this theorem, assuming $k$ is infinite. First, we show that the Zariski closure of the set $Z(k)$ of $k$-points has codimension $<C$, where $C$ is a constant depending only on the $d_i$'s and $k$. And second, we show that if the strength of the $f_i$'s is sufficiently large in terms of the $d_i$'s and $k$, then $Z(k)$ is actually Zariski dense in $Z$. The proofs rely on recent work of Ananyan and Hochster on high strength polynomials.

math.NT↗

arXiv2023

Uniformity for limits of tensors

There are many notions of rank in multilinear algebra: tensor rank, partition rank, slice rank, and strength (or Schmidt rank) are a few examples. Typically the rank $\le r$ locus is not Zariski closed, and understanding the closure (the locus with "border rank" $\le r$) is an important problem. We make two contributions in this direction: we prove a de-bordering result, which bounds border rank as a function of rank; and we show that the limits required to realize a point of border rank $\le r$ do not become increasingly complicated as the dimension of the vector space increases. We prove both results for a fairly general class of ranks. We deduce our theorems on ranks from foundational results on $\mathbf{GL}$-varieties, which are infinite dimensional algebraic varieties on which the infinite general linear group acts. For example, an important result concerns the existence of curves on $\mathbf{GL}$-varieties.

math.AG↗

arXiv2023

The amoeba dimension of a linear space

Given a complex vector subspace $V$ of $\mathbb{C}^n$, the dimension of the amoeba of $V \cap (\mathbb{C}^*)^n$ depends only on the matroid that $V$ defines on the ground set $\{1,\ldots,n\}$. Here we prove that this dimension is given by the minimum of a certain function over all partitions of the ground set, as previously conjectured by Rau. We also prove that this formula can be evaluated in polynomial time.

math.CO↗

arXiv2023

On subtensors of high partition rank

We prove that for every positive integer $d \ge 2$ there exist polynomial functions $F_d, G_d: \mathbb{N} \to \mathbb{N}$ such that for each positive integer $r$, every order-$d$ tensor $T$ over an arbitrary field and with partition rank at least $G_d(r)$ contains a $F_d(r) \times \cdots \times F_d(r)$ subtensor with partition rank at least $r$. We then deduce analogous results on the Schmidt rank of polynomials in zero or high characteristic.

math.CO↗

arXiv2023

3D genome reconstruction from partially phased Hi-C data

The 3-dimensional (3D) structure of the genome is of significant importance for many cellular processes. In this paper, we study the problem of reconstructing the 3D structure of chromosomes from Hi-C data of diploid organisms, which poses additional challenges compared to the better-studied haploid setting. With the help of techniques from algebraic geometry, we prove that a small amount of phased data is sufficient to ensure finite identifiability, both for noiseless and noisy data. In the light of these results, we propose a new 3D reconstruction method based on semidefinite programming, paired with numerical algebraic geometry and local optimization. The performance of this method is tested on several simulated datasets under different noise levels and with different amounts of phased data. We also apply it to a real dataset from mouse X chromosomes, and we are then able to recover previously known structural features.

q-bio.GN↗