# Commentarii Mathematici Helvetici

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**Volume 78, Issue 3, 2003, pp. 494–517**

**DOI: 10.1007/s00014-003-0769-6**

On Waring's problem for several algebraic forms

Enrico Carlini^{[1]}and Jaydeep V. Chipalkatti

^{[2]}(1) School of Mathematical Sciences, Monash University, 1 Wellington Rd, 3800, CLAYTON, VIC, AUSTRALIA

(2) Department of Mathematics, University of British Columbia, BC V6T 1Z2, VANCOUVER, CANADA

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.

*Keywords: *Waring's problem for forms, apolarity

Carlini Enrico, Chipalkatti Jaydeep: On Waring's problem for several algebraic forms. *Comment. Math. Helv.* 78 (2003), 494-517. doi: 10.1007/s00014-003-0769-6