A Lower Bound on Determinantal Complexity

Mrinal Kumar, Ben Lee Volk*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


The determinantal complexity of a polynomial P∈ F[x1, … , xn] over a field F is the dimension of the smallest matrix M whose entries are affine functions in F[x1, … , xn] such that P= Det (M). We prove that the determinantal complexity of the polynomial ∑i=1nxin is at least 1.5 n- 3.For every n-variate polynomial of degree d, the determinantal complexity is trivially at least d, and it is a long-standing open problem to prove a lower bound which is super linear in max { n, d}. Our result is the first lower bound for any explicit polynomial which is bigger by a constant factor than max { n, d} , and improves upon the prior best bound of n+ 1 , proved by Alper et al. (2017) for the same polynomial.

Original languageEnglish
Article number12
JournalComputational Complexity
Issue number2
StatePublished - Dec 2022
Externally publishedYes


  • 68Q06
  • 68Q15
  • 68Q17
  • Algebraic Circuits
  • Algebraic Complexity Theory
  • Determinantal Complexity
  • Lower Bounds


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