Mathematics Problem Archive
Showing 1-8 of 8 problems
Smale's 4th Problem: Integer Zeros of Polynomials
Find efficient algorithms for deciding whether a polynomial with integer coefficients has an integer root....
Smale's 5th Problem: Height Bounds for Diophantine Curves
Find effective uniform bounds for the heights of rational points on algebraic curves....
Smale's 6th Problem: Finiteness of Central Configurations
For the Newtonian $n$-body problem with positive masses, are there only finitely many central configurations (relative equilibria) for each $n$?...
Smale's 7th Problem: Distribution of Points on the 2-Sphere
What is the optimal arrangement of $n$ points on the 2-sphere to minimize energy for various potential functions?...
Smale's 9th Problem: Linear Programming in Polynomial Time
Find a strongly polynomial algorithm for linear programming....
Smale's 10th Problem: The Pugh Closing Lemma
Is the $C^r$ closing lemma true for dynamical systems?...
The Jacobian Conjecture
If $F: \mathbb{C}^n \to \mathbb{C}^n$ is a polynomial map with constant non-zero Jacobian determinant, then $F$ is invertible....
Smale's 12th Problem: Centralizers of Diffeomorphisms
Determine the structure of centralizers of generic diffeomorphisms....