I am trying to construct a counterexample in $\mathbb{R}^4$. There are 10 relations, each of which is given as the vanishing set of a determinant. There are 14 variables in total. Is there a way to find a solution easily, possibly with the help of some computer program?

1$\begingroup$ If I understand you correctly, you have 10 equations of degree 4 in 14 variables. This looks too complicated for a Computer approach using Gröbner bases or Macualy determinants, unless there are some unexpected simplifications. If your problem has some structure, unexpected things are more likely to happen, so you should try e.g. math.uiuc.edu/Macaulay2 . $\endgroup$– JanChristoph SchlagePuchtaJan 27 '17 at 9:33

5$\begingroup$ Yes, this sounds underdetermined perhaps you could post your system?! $\endgroup$– Moritz FirschingJan 27 '17 at 9:33

$\begingroup$ you can try minimising a nice function, say, linear, on your algebraic set. Then the Langrange multipliers method will give you more equations (that only hold where an optimum is reached). $\endgroup$– Dima PasechnikJan 27 '17 at 10:46

1$\begingroup$ I second the suggestion that you should post your actual set of equations (in a form that can be cut and pasted into a computer programme; do not LaTeX them). If they are very big and you do not have a convenient way to put them on the web then you can email them to me (N.P.Strickland@sheffield.ac.uk) $\endgroup$– Neil StricklandJan 27 '17 at 17:45

2$\begingroup$ bertini.nd.edu $\endgroup$– Steve HuntsmanJan 28 '17 at 1:03
If the system is too complicated for symbolic solution, you might try some numerical experimentation.
Since there are more variables than equations, the system is likely to be underdetermined as Moritz Firsching commented. In this case, if you choose values for $4$ of the variables (hopefully the nature of the problem gives you some idea of likely orders of magnitude for them), you may be able to numerically solve the $10$ equations for the remaining $10$ variables, using something like Maple's fsolve, and have a good chance of success. It may even be possible to prove that a true solution exists near that numerical solution.