Steven Smale, one of the most recognized mathematicians of the XXI century, left 18 problems known as Smale's list, which his creator hismself considered as the biggest problems of this century. And one of those problems was solved by a spanish mathematician called Carlos Beltrán.
Carlos Beltrán received the Jose Luis Rubio de Francia for resolving the 17th problem of the list. Despite his thesis director affirmed that Beltrán has enormous skills in a lot of math areas, his pupil stated that the problem was less harder than other problems from the list than aren't unresolved yet. But even that, the truth is that this problem never had a progress aside from Smale's contribution.
The problem number 17 consist of solving polynomial equations in polynomal time in the average case. But we're not talking about a second degree ecuation, we're talking about this kind of ecuation:
Normally, this kind of ecuation always has many ways to be solved, and has lots of solutions, but the problem is that solving this must be efficient (that's what ti means to solve the problem "in polynomal time"). Resolving this has caused quite a change in Maths, particulary in programming, because with this resolution, now someone who solves an ecuation on a math program knows if the program is going to take a long time or not, or even success on all the possible solutions. There are different ways on solving an ecuation, but with Beltran's method, a uniform probabilistic algorithm called Las Vegas algorirthm, will give always the correct results and will only gamble on the resources used for the computation. And it's no secret that on Maths, making sure the result is correct has to be checked with other ways.
"There are lots of ways to caculate things, but it's no easy to prove that the way you're doing if the right one" Carlos Beltrán.
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