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.

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.

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.