Skip to Main content Skip to Navigation
Preprints, Working Papers, ...

Condition Numbers for the Cube. I: Univariate Polynomials and Hypersurfaces

Abstract : The condition-based complexity analysis framework is one of the gems of modern numerical algebraic geometry and theoretical computer science. One of the challenges that it poses is to expand the currently limited range of random polynomials that we can handle. Despite important recent progress, the available tools cannot handle random sparse polynomials and Gaussian polynomials, that is polynomials whose coefficients are i.i.d. Gaussian random variables. We initiate a condition-based complexity framework based on the norm of the cube that is a step in this direction. We present this framework for real hypersurfaces and univariate polynomials. We demonstrate its capabilities in two problems, under very mild probabilistic assumptions. On the one hand, we show that the average run-time of the Plantinga-Vegter algorithm is polynomial in the degree for random sparse (alas a restricted sparseness structure) polynomials and random Gaussian polynomials. On the other hand, we study the size of the subdivision tree for Descartes' solver and run-time of the solver by Jindal and Sagraloff (2017). In both cases, we provide a bound that is polynomial in the size of the input (size of the support plus logarithm of the degree) for not only on the average, but all higher moments. [This is the journal version of the conference paper with the same title.]
Complete list of metadatas

https://hal.inria.fr/hal-03086875
Contributor : Josué Tonelli-Cueto <>
Submitted on : Tuesday, December 22, 2020 - 11:02:40 PM
Last modification on : Thursday, December 24, 2020 - 3:32:32 AM

File

Condition_Numbers_for_the_Cube...
Files produced by the author(s)

Identifiers

  • HAL Id : hal-03086875, version 1
  • ARXIV : 2006.04423

Citation

Josué Tonelli-Cueto, Elias Tsigaridas. Condition Numbers for the Cube. I: Univariate Polynomials and Hypersurfaces. 2020. ⟨hal-03086875⟩

Share

Metrics

Record views

19

Files downloads

129