arXiv subjects

Matěj Doležálek

Publications and source records attributed to Matěj Doležálek.

11 recordsLinked to original sources

arXiv2026

Geometric and homological interpretations of tensor flattenings

We relate several nonlinear flattenings of structure tensors of finite algebras and modules to their algebraic and homological invariants. Since smoothability of algebras is equivalent to their structure tensors having minimal border rank, this shows equivalence of several tensor-theoretic nonsmoothability certificates to known classical ones. Particularly, we prove two conjectures of Doležálek and Michałek relating the tangency flattening and the $3\times3$ minor flattening to dimensions of tangent spaces to the Hilbert scheme of points. In the process, we generalize the tangency flattening to a mixed regime with two tensor inputs, relating this to dimensions of $\operatorname{Ext}$ modules, and introduce the dual notion of the cotangency flattening, related to dimensions of $\operatorname{Tor}$ modules. In the case of algebras, we also realize the kernel of the dualized tangency flattening within the setting of Harrison cohomology, making the connection to the tangent space of the Hilbert scheme explicit.

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arXiv2026

On degenerate geometric rank

A tensor of degenerate geometric rank corresponds to a linear space of matrices in which some locus of matrices of rank at most $r$ has unexpectedly large dimension. We prove a sufficient criterion for when such a tensor cannot be lifted to a larger tensor and characterize those tensors for which the degeneracy is caused by the rank $r=1$ locus. We also provide new examples of tensors with degenerate geometric rank stemming from nonlinear loci.

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arXiv2026

Representing rational integers by generalized quadratic forms over quadratic fields

We investigate generalized quadratic forms with values in the set of rational integers over quadratic fields. We characterize the real quadratic fields which admit a positive definite binary generalized form of this type representing every positive integer. We also show that there are only finitely many such fields where a ternary generalized form with these properties exists.

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arXiv2026

Nonlinear methods for tensors: determinantal equations for secant varieties beyond cactus

We present a family of flattening methods of tensors which we call Kronecker-Koszul flattenings, generalizing the famous Koszul flattenings and further equations of secant varieties studied among others by Landsberg, Manivel, Ottaviani and Strassen. We establish new border rank criteria given by vanishing of minors of Kronecker-Koszul flattenings. We obtain the first explicit polynomial equations -- tangency flattenings -- vanishing on secant varieties of Segre variety, but not vanishing on cactus varieties. Additionally, our polynomials have simple determinantal expressions. As another application, we provide a new, computer-free proof that the border rank of the $2\times2$ matrix multiplication tensor is $7$.

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arXiv2026

Secant varieties of Segre-Veronese varieties $\mathbb{P}^m\times\mathbb{P}^n$ embedded by $\mathcal{O}(1,2)$ are non-defective for $n\gg m^3$, $m\geq3$

We prove that for any $m\geq3$, $n\gg m^3$, all secant varieties of the Segre-Veronese variety $\mathbb{P}^m\times\mathbb{P}^n$ have the expected dimension. This was already proved by Abo and Brambilla in the subabundant case, hence we focus on the superabundant case. We generalize an approach due to Brambilla and Ottaviani into a construction we call the inductant. With this, the proof of non-defectivity reduces to checking a finite collection of base cases, which we verify using a computer-assisted proof.

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arXiv2025

Generalizing Hurwitz's quaternionic proof of Lagrange's and Jacobi's four-square theorems

A proof of Lagrange's and Jacobi's four-square theorem due to Hurwitz utilizes orders in a quaternion algebra over the rationals. Seeking a generalization of this technique to orders over number fields, we identify two key components: an order with a good factorization theory and the condition that all orbits under the action of the group of elements of norm $1$ acting by multiplication intersect the suborder corresponding to the quadratic form to be studied. We use recent results on class numbers of quaternion orders and then find all suborders satisfying the orbit condition. Subsequently, we obtain universality and formulas for the number of representations by the corresponding quadratic forms. We also present a quaternionic proof of Götzky's four-square theorem.

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arXiv2025

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.Michałek. 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.

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arXiv2025

Extending bounds on minimal ranks of universal quadratic lattices to larger number fields

There exist numerous results in the literature proving that within certain families of totally real number fields, the minimal rank of a universal quadratic lattice over such a field can be arbitrarily large. Kala introduced a technique of extending such results to larger fields -- e.g. from quadratic fields to fields of arbitrary even degree -- under some conditions. We present improvements to this technique by investigating the structure of subfields within composita of number fields, using basic Galois theory to translate this into a group-theoretic problem. In particular, we show that if totally real number fields with minimal rank of a universal lattice $\geq r$ exist in degree $d$, then they also exist in degree $kd$ for all $k\geq3$.

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arXiv2024

No proper generalized quadratic forms are universal over quadratic fields

We consider generalized quadratic forms over real quadratic number fields and prove, under a natural positive-definiteness condition, that a generalized quadratic form can only be universal if it contains a quadratic subform that is universal. We also construct an example illustrating that the positive-definiteness condition is necessary.

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arXiv2023

The Six-Vertex Model with a Non-Standard Boundary Condition

We consider the enumeration of states in the Brubaker-Bump-Friedberg six-vertex model, whose boundary conditions are determined by an integer partition. In general, we find the number of states is a polynomial in the largest part of the partition. By explicating this technique, we also enumerate the states completely for hook shapes and staircases.

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arXiv2020

Quaternions and universal quadratic forms over number fields

We study quadratic forms over totally real number fields by using an associated ring of quaternions. We examine some properties of residue class rings of these quaternions and use geometry of numbers to prove that certain ideals of the ring of quaternions contain elements of a small norm. We prove that $x^2+y^2+z^2+w^2+xy+xz+xw$ is universal over $\mathbb{Q}(ζ_7+ζ_7^{-1})$ and that $x^2+xy+y^2+z^2+zw+w^2$ represents all totally positive multiples of certain special elements.

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