On Waring?s problem for several algebraic forms

Abstract

Abstract.We reconsider the classical problem of representing a finite number of forms of degree d in the polynomial ring over n + 1 variables as scalar combinations of powers of linear forms. We define a geometric construct called a 'grove', which, in a number of cases, allows us to determine the dimension of the space of forms which can be so represented for a fixed number of summands. We also present two new examples, where this dimension turns out to be less than what a naïve parameter count would predict. Mathematics Subject Classification (2000). 14N15, 51N35.

Block-Decoupling Multivariate Polynomials Using the Tensor Block-Term Decomposition

“…1 is a schematical representation of the proposed structure. The case in which all g i (x i ) are univariate functions is related to the Waring decomposition [1,9] and is discussed in [5]. The current paper considers the case of block-decoupling with the internal functions g i (x i ) being multivariate vector-valued functions.…”

“…The current paper considers the case of block-decoupling with the internal functions g i (x i ) being multivariate vector-valued functions. It is assumed that the decomposition (1) exists (in the exact sense). …”

“…We assume that f (u) can be written as in (1). Although we will describe the method for the case that f (u) is polynomial, the method can easily be generalized to the non-polynomial case, and is applicable as long as the derivatives of f (u) can be obtained.…”

“…The task at hand is to decompose f (u) into blocks of multivariate functions as in (1). The method generalizes the result of [5] and proceeds by collecting first-order information of f (u) in a set of sampling points u (k) .…”

On Gk-1,k-defectivity of smooth surfaces and threefolds

“…For example, varieties with G h,k -defect have a strange behaviour under projections. Waring's problem for forms (see [2,6,10]) gives us another extrinsic reason for studying defective varieties. This problem is in connection with the G h,k -behaviour of Veronese embeddings of projective spaces.…”