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New conjectures on multiplicities of tensor eigenvalues
Stefano Canino, Cosimo Flavi, Francesco Galuppi +1
We work on two conjectures on tensor eigenvalue multiplicities. By using the language of algebraic geometry we give stronger, more refined versions of the conjectures and we prove…
Polynomials of minimal border rank
Cosimo Flavi, Weronika Obcowska, Tim Seynnaeve
We use the correspondence between iterated multiplication tensors of Gorenstein algebras and homogeneous polynomials of minimal smoothable rank to classify polynomials of minimal b…
Isotropic Rank of Harmonic Polynomials
S. Canino, C. Flavi
Any homogeneous harmonic polynomial can be decomposed as a sum of powers of isotropic linear forms, that is, linear forms whose coefficients are the coordinates of isotropic points…
Polynomials of small slice rank and strength
Cosimo Flavi, Fulvio Gesmundo, Alessandro Oneto +1
This paper investigates defining equations for secant varieties of the variety of reducible polynomials, which geometrically encode the notions of strength and slice rank of homoge…
On the construction of a counterexample to Strassen's rank additivity conjecture
Viktoriia Borovik, Cosimo Flavi, Paweł Pielasa +2
The rank additivity conjecture, first formulated by Volker Strassen in 1973, states that the rank of the direct sum of two independent tensors is equal to the sum of their individu…
Decompositions of powers of quadrics
Cosimo Flavi
We analyze the problem of determining Waring decompositions of the powers of any quadratic form over the field of complex numbers. Our main goal is to provide information about the…
Symmetric powers: structure, smoothability, and applications
Cosimo Flavi, Joachim Jelisiejew, Mateusz Michałek
We investigate border ranks of twisted powers of polynomials and smoothability of symmetric powers of algebras. We prove that the latter are smoothable. For the former, we obtain u…
Upper bounds for the rank of powers of quadrics
Cosimo Flavi
We establish an upper bound for the rank of every power of an arbitrary quadratic form. Specifically, for any , we prove that the -th power of a quadratic form o…
Border rank of powers of ternary quadratic forms
Cosimo Flavi
We determine the border rank of each power of any quadratic form in three variables. Since the problem for rank and rank quadratic forms can be reduced to determining the r…