arXiv subjects

Arthur Bik

Publications and source records attributed to Arthur Bik.

At least 19 recordsLinked to original sources

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↗

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↗

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

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

Higher-degree symmetric rank-metric codes

Over fields of characteristic unequal to $2$, we can identify symmetric matrices with homogeneous polynomials of degree $2$. This allows us to view symmetric rank-metric codes as living inside the space of such polynomials. In this paper, we generalize the construction of symmetric Gabidulin codes to polynomials of degree $d>2$ over field of characteristic $0$ or $>d$. To do so, we equip the space of homogeneous polynomials of degree $d\geq 2$ with the metric induced by the essential rank, which is the minimal number of linear forms needed to express a polynomial. We provide bounds on the minimal distance and dimension of the essential-rank metric codes we construct and provide an efficient decoding algorithm. Finally, we show how essential-rank metric codes can be seen as special instances of rank-metric codes and compare our construction to known rank-metric codes with the same parameters.

cs.IT↗

arXiv2022

Isogeny classes of cubic spaces

A cubic space is a vector space equipped with a symmetric trilinear form. Two cubic spaces are isogeneous if each embeds into the other. A cubic space is non-degenerate if its form cannot be expressed as a finite sum of products of linear and quadratic forms. We classify non-degenerate cubic spaces of countable dimension up to isogeny: the isogeny classes are completely determined by an invariant we call the residual rank, which takes values in $\mathbf{N} \cup \{\infty\}$. In particular, the set of classes is discrete and (under the partial order of embedability) satisfies the descending chain condition.

math.RT↗

arXiv2022

Classifying one-dimensional discrete models with maximum likelihood degree one

We propose a classification of all one-dimensional discrete statistical models with maximum likelihood degree one based on their rational parametrization. We show how all such models can be constructed from members of a smaller class of 'fundamental models' using a finite number of simple operations. We introduce 'chipsplitting games', a class of combinatorial games on a grid which we use to represent fundamental models. This combinatorial perspective enables us to show that there are only finitely many fundamental models in the probability simplex $\Delta_n$ for $n\leq 4$.

math.CO↗

arXiv2022

The Geometries of Jordan nets and Jordan webs

A Jordan net (resp. web) is an embedding of a unital Jordan algebra of dimension $3$ (resp. $4$) into the space $\mathbb{S}^n$ of symmetric $n\times n$ matrices. We study the geometries of Jordan nets and webs: we classify the congruence-orbits of Jordan nets (resp. webs) in $\mathbb{S}^n$ for $n\leq 7$ (resp. $n\leq 5$), we find degenerations between these orbits and list obstructions to the existence of such degenerations. For Jordan nets in $\mathbb{S}^n$ for $n\leq5$, these obstructions show that our list of degenerations is complete. For $n=6$, the existence of one degeneration is still undetermined. To explore further, we used an algorithm that indicates numerically whether a degeneration between two orbits exists. We verified this algorithm using all known degenerations and obstructions, and then used it to compute the degenerations between Jordan nets in $\mathbb{S}^7$ and Jordan webs in $\mathbb{S}^n$ for $n=4,5$.

math.AG↗

arXiv2021

The geometry of polynomial representations

We define a GL-variety to be a (typically infinite dimensional) algebraic variety equipped with an action of the infinite general linear group under which the coordinate ring forms a polynomial representation. Such varieties have been used to study asymptotic properties of invariants like strength and tensor rank, and played a key role in two recent proofs of Stillman's conjecture. We initiate a systematic study of GL-varieties, and establish a number of foundational results about them. For example, we prove a version of Chevalley's theorem on constructible sets in this setting.

math.AG↗

arXiv2021

Universality of high-strength tensors

A theorem due to Kazhdan and Ziegler implies that, by substituting linear forms for its variables, a homogeneous polynomial of sufficiently high strength specialises to any given polynomial of the same degree in a bounded number of variables. Using entirely different techniques, we extend this theorem to arbitrary polynomial functors. As a corollary of our work, we show that specialisation induces a quasi-order on elements in polynomial functors, and that among the elements with a dense orbit there are unique smallest and largest equivalence classes in this quasi-order.

math.AG↗

arXiv2021

Strength and slice rank of forms are generically equal

We prove that strength and slice rank of homogeneous polynomials of degree $d \geq 5$ over an algebraically closed field of characteristic zero coincide generically. To show this, we establish a conjecture of Catalisano, Geramita, Gimigliano, Harbourne, Migliore, Nagel and Shin concerning dimensions of secant varieties of the varieties of reducible homogeneous polynomials. These statements were already known in degrees $2\leq d\leq 7$ and $d=9$.

math.AG↗

arXiv2020

The set of forms with bounded strength is not closed

The strength of a homogeneous polynomial (or form) is the smallest length of an additive decomposition expressing it whose summands are reducible forms. Using polynomial functors, we show that the set of forms with bounded strength is not always Zariski-closed. More specifically, if the ground field is algebraically closed, we prove that the set of quartics with strength $\leq3$ is not Zariski-closed for a large number of variables.

math.AG↗

arXiv2020

Topological Noetherianity of polynomial functors II: base rings with Noetherian spectrum

In a previous paper, the third author proved that finite-degree polynomial functors over infinite fields are topologically Noetherian. In this paper, we prove that the same holds for polynomial functors from free $R$-modules to finitely generated $R$-modules, for any commutative ring $R$ whose spectrum is Noetherian. As Erman-Sam-Snowden pointed out, when applying this with $R = \mathbb{Z}$ to direct sums of symmetric powers, one of their proofs of a conjecture by Stillman becomes characteristic-independent. Our paper advertises and further develops the beautiful but not so well-known machinery of polynomial laws. In particular, to any finitely generated R-module M we associate a topological space, which we show is Noetherian when $\operatorname{Spec}(R)$ is; this is the degree-zero case of our result on polynomial functors.

math.AC↗

arXiv2020

Jordan Algebras of Symmetric Matrices

We study linear spaces of symmetric matrices whose reciprocal is also a linear space. These are Jordan algebras. We classify such algebras in low dimensions, and we study the associated Jordan loci in the Grassmannian.

math.RA↗

arXiv2020

On the strength of general polynomials

A slice decomposition is an expression of a homogeneous polynomial as a sum of forms with a linear factor. A strength decomposition is an expression of a homogeneous polynomial as a sum of reducible forms. The slice rank and strength of a polynomial are the minimal lengths of such decompositions, respectively. The slice rank is an upper bound for the strength and the gap between these two values can be arbitrary large. However, in line with a conjecture by Catalisano et al. on the dimensions of secant varieties of the varieties of reducible forms, we conjecture that equality holds for general forms. By using a weaker version of Fr\"oberg's Conjecture on the Hilbert series of ideals generated by general forms, we show that our conjecture holds up to degree $7$ and in degree $9$.

math.AG↗

arXiv2019

Semi-algebraic properties of Minkowski sums of a twisted cubic segment

We find a semi-algebraic description of the Minkowski sum $\mathcal{A}_{3,n}$ of $n$ copies of the bounded twisted cubic $\{(t,t^2,t^3)\mid -1\leq t\leq 1\}$ for each integer $n\geq3$. These descriptions provide efficient membership tests for the sets $\mathcal{A}_{3,n}$. These membership tests in turn can be used to resolve some instances of the underdetermined matrix moment problem, which was formulated by Michael Rubinstein and Peter Sarnak in order to study problems related to $L$-functions and their zeros.

math.AG↗

arXiv2019

The monic rank

We introduce the monic rank of a vector relative to an affine-hyperplane section of an irreducible Zariski-closed affine cone $X$. We show that the monic rank is finite and greater than or equal to the usual $X$-rank. We describe an algorithmic technique based on classical invariant theory to determine, in concrete situations, the maximal monic rank. Using this technique, we establish three new instances of a conjecture due to B. Shapiro which states that a binary form of degree $d\cdot e$ is the sum of $d$ $d$-th powers of forms of degree $e$. Furthermore, in the case where $X$ is the cone of highest weight vectors in an irreducible representation---this includes the well-known cases of tensor rank and symmetric rank---we raise the question whether the maximal rank equals the maximal monic rank. We answer this question affirmatively in several instances.

math.AG↗