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For m nonlinear polynomial equations with n variables and the highest degree 3, how is the current ability to solve such equations? In the webpage of IBM cplex, it says that:

IBM ILOG CPLEX Optimizer has solved problems with millions of constraints and variables.

Does it represent the highest level to solve such equations? If not, what is the biggest m and n that can be solved?

I'm sorry that I'm asking about the systems of equations, not optimization.

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  • $\begingroup$ it seems you can solve anything you like, given infinite time and able hardware... $\endgroup$
    – gt6989b
    Jun 7, 2016 at 15:03
  • $\begingroup$ Are you asking about solving systems of equations or optimization? Cplex solves linear or quadratic optimization problems, not systems of polynomial equations. $\endgroup$ Jun 7, 2016 at 15:06
  • $\begingroup$ Thx. I mean systems of equations... $\endgroup$
    – haik
    Jun 7, 2016 at 15:08
  • $\begingroup$ You can typically get numerical solutions using Newton-type methods, especially if you have a good idea of where to look for a solution. Some problems are harder than others, though. The number of variables is not necessarily correlated to the difficulty, as long as an $n \times n$ matrix will fit in your computer's memory. $\endgroup$ Jun 7, 2016 at 15:15
  • $\begingroup$ In other words, if I design an encryption system, I'd like to know the size of m and n to ensure the security of the system. Therefore, it relates to the current computing power of today's computer, right? $\endgroup$
    – haik
    Jun 7, 2016 at 15:27

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