Smale’s Fundamental Theorem of Algebra Reconsidered

In his 1981 Fundamental Theorem of Algebra paper Steve Smale initiated the complexity theory of finding a solution of polynomial equations of one complex variable by a variant of Newton’s method. In this paper we reconsider his algorithm in the light of work done in the intervening years. Smale’s up...

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Veröffentlicht in:Foundations of computational mathematics 2014-02, Vol.14 (1), p.85-114
Hauptverfasser: Armentano, Diego, Shub, Michael
Format: Artikel
Sprache:eng
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Zusammenfassung:In his 1981 Fundamental Theorem of Algebra paper Steve Smale initiated the complexity theory of finding a solution of polynomial equations of one complex variable by a variant of Newton’s method. In this paper we reconsider his algorithm in the light of work done in the intervening years. Smale’s upper bound estimate was infinite average cost. Ours is polynomial in the Bézout number and the dimension of the input. Hence it is polynomial for any range of dimensions where the Bézout number is polynomial in the input size. In particular it is not just for the case that Smale considered but for a range of dimensions as considered by Bürgisser–Cucker, where the max of the degrees is greater than or equal to n 1+ ϵ for some fixed ϵ . It is possible that Smale’s algorithm is polynomial cost in all dimensions and our main theorem raises some problems that might lead to a proof of such a theorem.
ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-013-9155-y