JournalscmhVol. 78 , No. 3DOI 10.1007/s00014-003-0769-6

On Waring's problem for several algebraic forms

  • Enrico Carlini

    Monash University, Clayton, Australia
  • Jaydeep V. Chipalkatti

    University of British Columbia, Vancouver, Canada
On Waring's problem for several algebraic forms cover


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.