Smale's 17th problem asks for an algorithm which finds an approximate zero of polynomial systems ... more Smale's 17th problem asks for an algorithm which finds an approximate zero of polynomial systems in average polynomial time (see [21]). The main progress on Smale's problem is [6] and [10]. In this paper we will improve on both approaches and prove an interesting intermediate result on the 1 average value of the condition number. Our main results are Theorem 1 on the complexity of a randomized algorithm which improves the result of [6], Theorem 2 on the average of the condition number of polynomial systems which improves the estimate found in [10], and Theorem 3 on the complexity of finding a single zero of polynomial systems. This last theorem is similar to the main result of [10] but relies only on homotopy methods, thus removing the need for the elimination theory methods used in [10]. We build on methods developed in [2].
Smale's 17th problem asks for an algorithm which finds an approximate zero of polynomial systems ... more Smale's 17th problem asks for an algorithm which finds an approximate zero of polynomial systems in average polynomial time (see [21]). The main progress on Smale's problem is [6] and [10]. In this paper we will improve on both approaches and prove an interesting intermediate result on the 1 average value of the condition number. Our main results are Theorem 1 on the complexity of a randomized algorithm which improves the result of [6], Theorem 2 on the average of the condition number of polynomial systems which improves the estimate found in [10], and Theorem 3 on the complexity of finding a single zero of polynomial systems. This last theorem is similar to the main result of [10] but relies only on homotopy methods, thus removing the need for the elimination theory methods used in [10]. We build on methods developed in [2].
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Papers by Felipe Cucker