Foundations of Computational Mathematics

, Volume 16, Issue 6, pp 1401–1422 | Cite as

Condition Length and Complexity for the Solution of Polynomial Systems

  • Diego Armentano
  • Carlos Beltrán
  • Peter Bürgisser
  • Felipe Cucker
  • Michael Shub
Article
First Online:
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Revised:
Accepted:

Abstract

Smale’s 17th problem asks for an algorithm which finds an approximate zero of polynomial systems in average polynomial time (see Smale in Mathematical problems for the next century, American Mathematical Society, Providence, 2000). The main progress on Smale’s problem is Beltrán and Pardo (Found Comput Math 11(1):95–129, 2011) and Bürgisser and Cucker (Ann Math 174(3):1785–1836, 2011). In this paper, we will improve on both approaches and prove an interesting intermediate result on the average value of the condition number. Our main results are Theorem 1 on the complexity of a randomized algorithm which improves the result of Beltrán and Pardo (2011), Theorem 2 on the average of the condition number of polynomial systems which improves the estimate found in Bürgisser and Cucker (2011), and Theorem 3 on the complexity of finding a single zero of polynomial systems. This last theorem is similar to the main result of Bürgisser and Cucker (2011) but relies only on homotopy methods, thus removing the need for the elimination theory methods used in Bürgisser and Cucker (2011). We build on methods developed in Armentano et al. (2014).

Keywords

Polynomial systems Homotopy methods Complexity estimates 

Mathematics Subject Classification

Primary 65H10 65H20 Secondary 58C35 
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Copyright information

© SFoCM 2016

Authors and Affiliations

  • Diego Armentano
    • 1
  • Carlos Beltrán
    • 2
  • Peter Bürgisser
    • 3
  • Felipe Cucker
    • 4
  • Michael Shub
    • 5
  1. 1.Universidad de La RepúblicaMontevideoUruguay
  2. 2.Universidad de CantabriaSantanderSpain
  3. 3.Technische Universität BerlinBerlinGermany
  4. 4.City University of Hong KongKowloon TongHong Kong
  5. 5.City University of New YorkNew YorkUSA

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