Rigid continuation paths I. Quasilinear average complexity for solving polynomial systems

@article{Lairez2017RigidCP,
  title={Rigid continuation paths I. Quasilinear average complexity for solving polynomial systems},
  author={Pierre Lairez},
  journal={ArXiv},
  year={2017},
  volume={abs/1711.03420}
}
  • Pierre Lairez
  • Published 2017
  • Mathematics, Computer Science
  • ArXiv
  • How many operations do we need on the average to compute an approximate root of a random Gaussian polynomial system? Beyond Smale's 17th problem that asked whether a polynomial bound is possible, we prove a quasi-optimal bound $\text{(input size)}^{1+o(1)}$. This improves upon the previously known $\text{(input size)}^{\frac32 +o(1)}$ bound. The new algorithm relies on numerical continuation along \emph{rigid continuation paths}. The central idea is to consider rigid motions of the equations… CONTINUE READING
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