On Smale's 17th Problem: A Probabilistic Positive Solution
نویسندگان
چکیده
Smale’s 17th Problem asks “Can a zero of n complex polynomial equations in n unknowns be found approximately, on the average [for a suitable probability measure on the space of inputs], in polynomial time with a uniform algorithm?” We present a uniform probabilistic algorithm for this problem and prove that its complexity is polynomial. We thus obtain a partial positive solution to Smale 17th Problem.
منابع مشابه
A Facility Location Problem with Tchebychev Distance in the Presence of a Probabilistic Line Barrier
This paper considers the Tchebychev distance for a facility location problem with a probabilistic line barrier in the plane. In particular, we develop a mixed-integer nonlinear programming (MINLP) model for this problem that minimizes the total Tchebychev distance between a new facility and the existing facilities. A numerical example is solved to show the validity of the developed model. Becau...
متن کاملA Multi-Period 1-Center Location Problem in the Presence of a Probabilistic Line Barrier
This paper investigates a multi-period rectilinear distance 1-center location problem considering a line-shaped barrier, in which the starting point of the barrier follows the uniform distribution function. In addition, the existing points are sensitive to demands and locations. The purpose of the presented model is to minimize the maximum barrier distance from the new facility to the existing ...
متن کاملThe Reliable Hierarchical Location-allocation Model under Heterogeneous Probabilistic Disruptions
This paper presents a novel reliable hierarchical location-allocation model where facilities are subject to the risk of disruptions. Based on the relationship between various levels of system, a multi-level multi-flow hierarchy is considered. The heterogeneous probabilistic disruptions are investigated in which the constructed facilities have different site-dependent and independent failure rat...
متن کاملUniversality of Polynomial Positivity and a Variant of Hilbert’s 17th Problem
We observe that the decision problem for the ∃ theory of real closed fields (RCF) is simply reducible to the decision problem for RCF over a connective-free ∀ language in which the only relation symbol is a strict inequality. In particular, every ∃ RCF sentence φ can be settled by deciding a proposition of the form “polynomial p (which is a sum of squares) takes on strictly positive values over...
متن کاملOn the Complexity of Non Universal Polynomial Equation Solving: old and new results
These pages summarize some results on the efficiency of polynomial equation solving. We focus on semantic algorithms, i.e., algorithms whose running time depends on some intrinsic/semantic invariant associated with the input data. Both computer algebra and numerical analysis algorithms are discussed. We show a probabilistic and positive answer to Smale’s 17th problem. Estimates of the probabili...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Foundations of Computational Mathematics
دوره 8 شماره
صفحات -
تاریخ انتشار 2008