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Commentarii Mathematici Helvetici

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On Waring's problem for several algebraic forms

Enrico Carlini (1) and Jaydeep V. Chipalkatti (2)

(1) Dipartimento di Matematica, Università di Pavia, Via Ferrata 1, I-27100, PAVIA, ITALY
(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