arXiv subjects

Tomasz Mańdziuk

Publications and source records attributed to Tomasz Mańdziuk.

9 recordsLinked to original sources

arXiv2026

Border rank lower bounds beyond weak border apolarity

We provide first examples of border rank lower bounds obtained using border apolarity beyond ``weak border apolarity''. Border apolarity introduced by Buczyńska-Buczyński characterizes tensors of border rank greater than $r$ as those whose annihilator does not contain an ideal with certain properties depending on $r$ and the ambient space of the tensor. Previous applications of border apolarity depended on the algorithm developed by Conner-Harper-Landsberg. For a tensor with a large symmetry group it lists a set of ideals that satisfy some of the properties mentioned above. We give first examples where the list is non-empty but we are able to prove that no ideal on this list satisfies all necessary conditions coming from border apolarity. The new lower bound examples include tensors of interest in complexity: the first Schönhage triple sum of matrix multiplication tensors where the border rank was unknown, and the direct sum of two copies of a tensor that could potentially be used to prove the exponent of matrix multiplication is two. A third example of interest in group theory related to work of A. Leitner is also analyzed.

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arXiv2026

Symmetrization maps and minimal border rank Comon's conjecture

One of the fundamental open problems in the field of tensors is the border Comon's conjecture: given a symmetric tensor $F\in(\mathbb{C}^n)^{\otimes d}$ for $d\geq 3$, its border and symmetric border ranks are equal. In this paper, we prove the conjecture for large classes of concise tensors in $(\mathbb{C}^n)^{\otimes d}$ of border rank $n$, i.e., tensors of minimal border rank. These families include all tame tensors and all tensors whenever $n\leq d+1$. Our technical tools are border apolarity and border varieties of sums of powers.

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arXiv2025

Limits of saturated ideals

We investigate the question whether a given homogeneous ideal is a limit of saturated ones. We provide cohomological necessary criteria for this to hold and apply them to a range of examples. Our motivation comes from the theory of border apolarity and varieties of sums of powers, where the question above is tightly connected to proving new lower bounds for border ranks of tensors.

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arXiv2024

Border apolarity and varieties of sums of powers

We study border varieties of sums of powers ($\underline{\mathrm{VSP}}$'s for short), recently introduced by Buczyńska and Buczyński, parameterizing border rank decompositions of a point (e.g. of a tensor or a homogeneous polynomial) with respect to a smooth projective toric variety and living in the Haiman-Sturmfels multigraded Hilbert scheme. Their importance stems from the role of border tensor rank in theoretical computer science, especially in the estimation of the exponent of matrix multiplication, a fundamental and still unknown quantity in the theory of computation. We compare $\underline{\mathrm{VSP}}$'s to other well-known loci in the Hilbert scheme, parameterizing scheme-theoretic versions of decompositions. The latter ones are crucial in that they naturally explain the existing severe barriers to giving good lower bounds on ranks. We introduce the notion of border identifiability and provide sufficient criteria for its appearance, which rely on the multigraded regularity of Maclagan and Smith. We link border identifiability to wildness of points. Finally, we determine $\underline{\mathrm{VSP}}$'s in several instances and regimes, in the contexts of tensors and homogeneous polynomials. These include concise $3$-tensors of minimal border rank and in particular of border rank three.

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arXiv2023

Tensoring by a plane maintains secant-regularity in degree at least two

Starting from an integral projective variety $Y$ equipped with a very ample, non-special and not-secant defective line bundle $\mathcal{L}$, the paper establishes, under certain conditions, the regularity of $(Y \times \mathbb P^2,\mathcal{L}[t])$ for $t\geq 2$. The mildness of those conditions allow to classify all secant defective cases of any product of $(\mathbb P^1)^{ j}\times (\mathbb P^2)^{k}$, $j,k \geq 0$, embedded in multidegree at least $(2, \ldots , 2)$ and $(\mathbb{P}^m\times\mathbb{P}^n\times (\mathbb{P}^2)^k, \mathcal{O}_{\mathbb{P}^m\times\mathbb{P}^n\times (\mathbb{P}^2)^k} (d,e,t_1, \ldots, t_k))$ where $d,e \geq 3$, $t_i\geq 2$, for any $n$ and $m$.

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arXiv2023

Limits of ideals of points in a toric variety

We consider the multigraded Hilbert scheme corresponding to the Hilbert function of a finite number of points in general position in a smooth projective complex toric variety. We develop several criteria for a point of that parameter space to be in the distinguished irreducible component. To obtain the criteria we study the behaviour of the locus of saturated ideals under morphisms of multigraded Hilbert schemes. We apply our results to classify the irreducible multigraded Hilbert schemes corresponding to points in general position in a product of projective spaces.

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arXiv2022

Distinguishing secant from cactus varieties

Cactus varieties are a generalization of secant varieties. They are defined using linear spans of arbitrary finite schemes of bounded length, while secant varieties use only isolated reduced points. In particular, any secant variety is always contained in the respective cactus variety, and, except in a few initial cases, the inclusion is strict. It is known that lots of natural criteria that test membership in secant varieties are actually only tests for membership in cactus varieties. In this article, we propose the first techniques to distinguish actual secant variety from the cactus variety in the case of the Veronese variety. We focus on two initial cases, $κ_{14}(ν_d(\mathbb{P}^n))$ and $κ_{8,3}(ν_d(\mathbb{P}^n))$, the simplest that exhibit the difference between cactus and secant varieties. We show that for $d\geq 5$, the component of the cactus variety $κ_{14}(ν_d(\mathbb{P}^6))$ other than the secant variety $σ_{14}(ν_d(\mathbb{P}^6))$ consists of degree $d$ polynomials divisible by a $(d-3)$-rd power of a linear form. We generalize this description to an arbitrary number of variables. We present an algorithm for deciding whether a point in the cactus variety $κ_{14}(ν_d(\mathbb{P}^n))$ belongs to the secant variety $σ_{14}(ν_d(\mathbb{P}^n))$ for $d\geq 6,$ $n \geq 6$. We obtain similar results for the Grassmann cactus variety $κ_{8,3}(ν_d(\mathbb{P}^n))$. Our intermediate results give also a partial answer to analogous problems for other cactus varieties and Grassmann cactus varieties to any Veronese variety.

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arXiv2021

Cox Rings and Algebraic Maps

Given a morphism $F : X \rightarrow Y$ from a Mori Dream Space $X$ to a smooth Mori Dream Space $Y$ and quasicoherent sheaves $\mathcal{F}$ on $X$ and $\mathcal{G}$ on $Y$ , we describe the inverse image of $\mathcal{G}$ by $F$ and the direct image of $\mathcal{F}$ by $F$ in terms of the corresponding modules over the Cox rings graded in the class groups.

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arXiv2021

Identifying limits of ideals of points in the case of projective space

We study the closure of the locus of radical ideals in the multigraded Hilbert scheme associated with a standard graded polynomial ring and the Hilbert function of a homogeneous coordinate ring of points in general position in projective space. In the case of projective plane, we give a sufficient condition for an ideal to be in the closure of the locus of radical ideals. For projective space of arbitrary dimension we present a necessary condition. The paper is motivated by the border apolarity lemma which connects such multigraded Hilbert schemes with the theory of ranks of polynomials.

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