Binary forms with three different relative ranks

Bruce Reznick, Neriman Tokcan

Research output: Contribution to journalArticlepeer-review


Suppose f(x, y) is a binary form of degree d with coefficients in a field K ⊆ C. The K-rank of f is the smallest number of d-th powers of linear forms over K of which f is a K-linear combination. We prove that for d ≥ 5, there always exists a form of degree d with at least three different ranks over various fields. The K-rank of a form f (such as x3 y2) may depend on whether -1 is a sum of two squares in K.

Original languageEnglish (US)
Pages (from-to)5169-5177
Number of pages9
JournalProceedings of the American Mathematical Society
Issue number12
StatePublished - Dec 2017


  • Binary forms
  • Complex rank
  • Real rank
  • Stufe
  • Sums of powers
  • Sylvester
  • Tensor decompositions

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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