L’algebra lineare studia in modo rigoroso le proprietà dei vettori e delle operazioni che li riguardano, definendo le strutture algebriche che consentono di formalizzare operazioni fondamentali come la somma di vettori, la moltiplicazione di un vettore per uno scalare e il prodotto scalare tra due vettori. Da un punto di vista più generale, i vettori assumono un significato molto più profondo. Nella visione classica essi sono definiti come grandezze descritte da modulo, direzione e verso; tuttavia, l’obiettivo dell’algebra lineare è proprio quello di generalizzare tale concetto ed estendere le proprietà dei vettori a oggetti appartenenti a insiemi più astratti. Una delle applicazioni più immediate dell’algebra lineare è la risoluzione dei sistemi lineari. In senso più ampio, essa trova impiego in un vasto numero di ambiti scientifici: oltre alla meccanica classica, è essenziale in fluidodinamica, meccanica quantistica, ingegneria strutturale, statistica, economia, grafica 3D, machine learning e intelligenza artificiale, nonché in biologia, dove contribuisce alla costruzione di modelli di vario tipo. Comprendere l’algebra lineare dal punto di vista teorico permette quindi di affrontare problemi complessi e di applicare i concetti appresi in contesti molto diversi, dalla matematica pura alle scienze applicate. Lo scopo principale di questo volume non è tanto quello di fornire un elenco di nozioni, quanto piuttosto di offrire una preparazione che consenta di affrontare in modo consapevole e autonomo i vari argomenti.

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 upper bounds for the border rank in general and prove that they are optimal under mild conditions. We give applications to complexity theory. Many of the results rest on the notion of an encompassing polynomial, which we introduce.

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 their rank and also to obtain decompositions whose size is as close as possible to this value. This problem is classical and these forms assume importance especially because of their invariance under the action of the special orthogonal group. We give the detailed procedure to prove that the apolar ideal of the s-th power of a quadratic form is generated by the harmonic polynomials of degree s+1. We also generalize and improve some of the results on real decompositions given by B. Reznick in his notes of 1992, focusing on possibly minimal decompositions and providing new ones, both real and complex. We investigate the rank of the second power of a non-degenerate quadratic form in n variables, which in most cases is equal to (n^2+n+2)/2, and also give some results on powers of ternary quadratic forms.

We establish an upper bound for the rank of every power of an arbitrary quadratic form. Specifically, for any s ∈ N, we prove that the s-th power of a quadratic form of rank n grows as n^s. Furthermore, we demonstrate that its rank is subgeneric for all n>(2s−1)^2.

We determine the border rank of each power of any quadratic form in three variables. Since the problem for rank 1 and rank 2 quadratic forms can be reduced to determining the rank of powers of binary forms, we primarily focus on non-degenerate quadratic forms. We begin by considering the quadratic form q_n = x_1^2 + ··· + x_n^2 in an arbitrary number n of variables. We determine the apolar ideal of any power q_n^s , proving that it corresponds to the homogeneous ideal generated by the harmonic polynomials of degree s + 1. Using this result, we select a specific ideal contained in the apolar ideal for each power of a quadratic form in three variables, which, without loss of generality, we assume to be the form q_3. After verifying certain properties, we utilize the recent technique of border apolarity to establish that the border rank of any power q_3^s is equal to the rank of its middle catalecticant matrix, namely (s + 1)(s + 2)/2.

We determine the successive pages of the Frölicher spectral sequence of the Iwasawa manifold and some of its small deformations, providing new examples and counterexamples on its properties, including the behaviour under small deformations.

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. The minimum size of such decompositions for a harmonic polynomial is called its isotropic rank. As with the Waring rank, the problem of determining the isotropic rank of a given harmonic form is very hard. We determine the isotropic rank of a general harmonic form providing a full classification of the dimensions of secant varieties of the variety of d-powers of isotropic linear forms in n+1 variables, for every n, d ∈ N, thus obtaining the analogue of the widely-celebrated Alexander-Hirschowitz theorem. Moreover, we completely solve the problem of determining the isotropic rank for the following classes of harmonic forms: ternary forms, quadrics and monomials.

We generalize Mammana's classification of limits of direct sums to more than two factors. We also extend it from polynomials to arbitrary Segre-Veronese format, generalising and unifying results of Buczyńska-Buczyński-Kleppe-Teitler, Hwang, Wang, and Wilson. Remarkably, in such much more general setup it is still possible to characterise the possible limits. Our proofs are direct and based on the theory of centroids, in particular avoiding the delicate Betti number arguments.

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 homogeneous polynomials. We present three main results. First, we reinterpret Ruppert's classical equations for reducible ternary forms in the language of representation theory and we extend them to an arbitrary number of variables. Second, we construct new determinantal equations for polynomials of small strength based on syzygies of their partial derivatives. Finally, we establish a reduction theorem for cubic forms, proving that slice rank two is determined by generic linear sections in 14 variables; this gives one of the few explicit upper bounds for defining equations for the image of a polynomial map in the framework of noetherianity for polynomial functors.

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 individual ranks. In the last decades, this conjecture has been a central topic in tensor rank theory and its implications for computational complexity. In 2019, Yaroslav Shitov disproved this conjecture in its general form by showing the existence of a counter-example using a dimension counting argument. In this paper, we provide an overview of the Strassen problem and Shitov's work and revisit his counterexample with a detailed explanation, offering an alternative proof.

A decomposition of a homogeneous polynomial is a representation of that polynomial as a sum of powers of linear forms; in particular, the minimum number of addends in this sum is said to be the rank of the polynomial. We analyze a way to determine explicit decompositions of a polynomial corresponding to a power of a non-degenerate quadratic form. The main instrument used in this context is the Apolarity Lemma, which is a classic result relating the summands of a decomposition to its apolar ideal.

We analyze the Waring decompositions of the powers of any quadratic form over the field of complex numbers. Our main objective is to provide detailed information about their rank and border rank. These forms are of significant importance because of the classical decomposition expressing the space of polynomials of a fixed degree as a direct sum of the spaces of harmonic polynomials multiplied by a power of the quadratic form. Using the fact that the spaces of harmonic polynomials are irreducible representations of the special orthogonal group over the field of complex numbers, we show that the apolar ideal of the s-th power of a non-degenerate quadratic form in n variables is generated by the set of harmonic polynomials of degree s+1. We also generalize and improve upon some of the results about real decompositions, provided by B. Reznick in his notes from 1992, focusing on possibly minimal decompositions and providing new ones, both real and complex. We investigate the rank of the second power of a non-degenerate quadratic form in n variables, which is equal to (n^2+n+2)/2 in most cases. We also study the border rank of any power of an arbitrary ternary non-degenerate quadratic form, which we determine explicitly using techniques of apolarity and a specific subscheme contained in its apolar ideal. Based on results about smoothability, we prove that the smoothable rank of the s-th power of such form corresponds exactly to its border rank and to the rank of its middle catalecticant matrix, which is equal to (s+1)(s+2)/2.