arXiv subjects

Ettore Teixeira Turatti

Publications and source records attributed to Ettore Teixeira Turatti.

13 recordsLinked to original sources

arXiv2026

Quadrature rules with few nodes supported on algebraic curves

We investigate quadrature rules for measures supported on real algebraic and rational curves, focusing on the {odd-degree} case $2s-1$. Adopting an optimization viewpoint, we minimize suitable penalty functions over the space of quadrature rules of strength $2s-1$, so that optimal solutions yield rules with the minimal number of nodes. For plane algebraic curves of degree $d$, we derive explicit node bounds depending on $d$ and the number of places at infinity, improving results of Riener--Schweighofer, and Zalar. For rational curves in arbitrary dimension of degree $d$, we further refine these bounds using the geometry of the parametrization and recover the classical Gaussian quadrature bound when $d=1$. Our results reveal a direct link between the algebraic complexity of the supporting curve and the minimal size of quadrature formulas, providing a unified framework that connects real algebraic geometry, polynomial optimization, and moment theory.

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arXiv2026

On defective spans of singular vector tuples beyond the boundary format

In this paper, we study tensor spaces beyond the boundary format and analyze whether the general critical space coincides with the general span of singular vector tuples. For all tensor spaces exceeding the boundary format by one in an arbitrary number of factors, we relate the codimension of this span within the critical space to the dimension of the kernel of a map in cohomology. Furthermore, we exhibit an infinite family of order-three tensors with a defective behavior: the general span of singular vector tuples achieves the maximum possible codimension rather than the expected minimum. Finally, we conjecture a classification of the behavior of critical spaces in this regime and draw a connection to Koszul cohomology.

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arXiv2026

Weak and strong Lefschetz properties for Hartshorne-Rao modules of curves in $\mathbb P^3$

Let $C\subset \mathbb P^3$ be a curve over an algebraically closed field of characteristic zero, and let $M(C)$ denote its Hartshorne-Rao module. We study how the geometry of $C$ influences whether $M(C)$ satisfies the Weak and Strong Lefschetz Properties. We first consider unions of general skew lines and prove that multiplication by $L^i$, for a general linear form $L$, has maximal rank on $M(C)$ for $i=1,2,3$. The proof uses a specialization to zero-dimensional schemes that can be written as a union of curvilinear schemes, each of a particular type and of degree at most three, together with generic Hilbert function results for such schemes, which are of independent interest. We then examine how special geometric configurations can affect the Weak Lefschetz Property. In particular, we show that curves on a smooth quadric surface have Hartshorne-Rao modules with the Weak Lefschetz Property, and that the property persists for unions of skew lines with all but one line on a quadric. By contrast, for $r\geq 10$, we construct configurations of $r$ skew lines with all but two lines on a quadric whose Hartshorne-Rao modules fail the Weak Lefschetz Property. Finally, we study smooth irreducible curves. We prove the Weak Lefschetz Property in several low-degree cases, construct a degree 15 curve for which it fails, and show that general nondegenerate rational curves have Hartshorne-Rao modules with the Weak Lefschetz Property. These results illustrate both the strength and the limitations of geometric hypotheses in controlling Lefschetz properties of Hartshorne-Rao modules.

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arXiv2026

On Waring rank jumps via critical rank-one approximations

We investigate whether eigenvectors, also known as critical rank-one approximations, of a symmetric tensor can be used to increase or decrease its Waring rank. First, we study the variety of degree-d rank-r forms which admit an eigenvector as part of a minimal Waring decomposition. In the case of binary forms, we show that this is of codimension-one in the r-th secant variety of the rational normal curve. On the other hand, we prove that for any binary form of rank less than (d+1)/2 (subgeneric), any eigenvector increases the rank. Additionally, when the degree is odd, the same holds for generic forms of generic rank. Our approach employs the strict relation between the apolar action and the Bombieri-Weyl product.

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arXiv2025

Decomposing tensors via rank-one approximations

Matrices can be decomposed via rank-one approximations: the best rank-one approximation is a singular vector pair, and the singular value decomposition writes a matrix as a sum of singular vector pairs. The singular vector tuples of a tensor are the critical points of its best rank-one approximation problem. In this paper, we study tensors that can be decomposed via successive rank-one approximations: compute a singular vector tuple, subtract it off, compute a singular vector tuple of the new deflated tensor, and repeat. The number of terms in such a decomposition may exceed the tensor rank. Moreover, the decomposition may depend on the order in which terms are subtracted. We show that the decomposition is valid independent of order if and only if all singular vectors in the process are orthogonal in at least two factors. We study the variety of such tensors. We lower bound its dimension, showing that it is significantly larger than the variety of odeco tensors.

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arXiv2025

When does subtracting a rank-one approximation decrease tensor rank?

Subtracting a critical rank-one approximation from a matrix always results in a matrix with a lower rank. This is not true for tensors in general. Motivated by this, we ask the question: what is the closure of the set of those tensors for which subtracting some of its critical rank-one approximation from it and repeating the process we will eventually get to zero? In this article, we show how to construct this variety of tensors and we show how this is connected to the bottleneck points of the variety of rank-one tensors (and in general to the singular locus of the hyperdeterminant), and how this variety can be equal to and in some cases be more than (weakly) orthogonally decomposable tensors.

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arXiv2024

Generalized identifiability of sums of squares

Let $f$ be a homogeneous polynomial of even degree $d$. We study the decompositions $f=\sum_{i=1}^r f_i^2$ where $\mathrm{deg} f_i=d/2$. The minimal number of summands $r$ is called the $2$-rank of $f$, so that the polynomials having $2$-rank equal to $1$ are exactly the squares. Such decompositions are never unique and they are divided into $\mathrm{O}(r)$-orbits, the problem becomes counting how many different $\mathrm{O}(r)$-orbits of decomposition exist. We say that $f$ is $\mathrm{O}(r)$-identifiable if there is a unique $\mathrm{O}(r)$-orbit. We give sufficient conditions for generic and specific $\mathrm{O}(r)$-identifiability. Moreover, we show the generic $\mathrm{O}(r)$-identifiability of ternary forms.

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arXiv2024

On the degree of varieties of sum of squares

We study the problem of how many different sums of squares decompositions a general polynomial $f$ with SOS-rank $k$ admits. We show that there is a link between the variety $\mathrm{SOS}_k(f)$ of all SOS-decompositions of $f$ and the orthogonal group $\mathrm{O}(k)$. We exploit this connection to obtain the dimension of $\mathrm{SOS}_k(f)$ and show that its degree is bounded from below by the degree of $\mathrm{O}(k)$. In particular, for $k=2$ we show that $\mathrm{SOS}_2(f)$ is isomorphic to $\mathrm{O}(2)$ and hence the degree bound becomes an equality. Moreover, we compute the dimension of the space of polynomials of SOS-rank $k$ and obtain the degree in the special case $k=2$.

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arXiv2023

Geometry of first nonempty Terracini loci

After a few results on curves, we characterize the smallest nonempty Terracini loci of Veronese and Segre-Veronese varieties. For del Pezzo surfaces, we give a full description of the Terracini loci. Moreover, we present an algorithm to explicitly compute the Terracini loci of a given variety.

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arXiv2023

Characteristic polynomials and eigenvalues of tensors

We lay the geometric foundations for the study of the characteristic polynomial of tensors. For symmetric tensors of order $d \geq 3$ and dimension $2$ and symmetric tensors of order $3$ and dimension $3$, we prove that only finitely many tensors share any given characteristic polynomial, unlike the case of symmetric matrices and the case of non-symmetric tensors. We propose precise conjectures for the dimension of the variety of tensors sharing the same characteristic polynomial, in the symmetric and in the non-symmetric setting.

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arXiv2023

The span of singular tuples of a tensor beyond the boundary format

A singular $k$-tuple of a tensor $T$ of format $(n_1,\ldots,n_k)$ is essentially a complex critical point of the distance function from $T$ constrained to the cone of tensors of format $(n_1,\ldots,n_k)$ of rank at most one. A generic tensor has finitely many complex singular $k$-tuples, and their number depends only on the tensor format. Furthermore, if we fix the first $k-1$ dimensions $n_i$, then the number of singular $k$-tuples of a generic tensor becomes a monotone non-decreasing function in one integer variable $n_k$, that stabilizes when $(n_1,\ldots,n_k)$ reaches a boundary format. In this paper, we study the linear span of singular $k$-tuples of a generic tensor. Its dimension also depends only on the tensor format. In particular, we concentrate on special order three tensors and order-$k$ tensors of format $(2,\ldots,2,n)$. As a consequence, if again we fix the first $k-1$ dimensions $n_i$ and let $n_k$ increase, we show that in these special formats, the dimension of the linear span stabilizes as well, but at some concise non-sub-boundary format. We conjecture that this phenomenon holds for an arbitrary format with $k>3$. Finally, we provide equations for the linear span of singular triples of a generic order three tensor $T$ of some special non-sub-boundary format. From these equations, we conclude that $T$ belongs to the linear span of its singular triples, and we conjecture that this is the case for every tensor format.

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arXiv2021

On tensors that are determined by their singular tuples

In this paper we study the locus of singular tuples of a complex valued multisymmetric tensor. The main problem that we focus on is: given the set of singular tuples of some general tensor, which are all the tensors that admit those same singular tuples. Assume that the triangular inequality holds, that is exactly the condition such that the dual variety to the Segre-Veronese variety is an hypersurface, or equivalently, the hyperdeterminant exists. We show in such case that, when at least one component has degree odd, this tensor is projectively unique. On the other hand, if all the degrees are even, the fiber is an $1$-dimensional space.

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arXiv2020

Binary forms of suprageneric rank and the multiple root loci

We state the relation between the variety of binary forms of given rank and the dual of the multiple root loci. This is a new result for the suprageneric rank, as a continuation of the work by Buczyński, Han, Mella and Teitler. We describe the strata of these varieties and explore their singular loci.

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