# On Waring's problem for several algebraic forms

### Enrico Carlini

Monash University, Clayton, Australia### Jaydeep V. Chipalkatti

University of British Columbia, Vancouver, Canada

## 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.

## Cite this article

Enrico Carlini, Jaydeep V. Chipalkatti, On Waring's problem for several algebraic forms. Comment. Math. Helv. 78 (2003), no. 3, pp. 494–517

DOI 10.1007/S00014-003-0769-6