TITLE: How to tell if a system of polynomial equations has no real solutions
QUESTION [6 upvotes]: I have a system of $3n + 3$ polynomial equations in $6n$ variables, where $n$ is probably going to be less than about $5$.  I can compute its Groebner basis and I see that it does not contain $\{1\}$, so I know that it has complex solutions at least.  However, I know for a fact that, by varying some of the parameters, I can cause it to have no real solutions.  Is there a way to check that the number of real solutions is nonzero? It doesn't have to be fast.  In fact, if I could turn the problem around and get a set of constraints on my parameters for which a real solution exists that would be great, but I'm not sure if that's even possible...

REPLY [2 votes]: The system of polynomial equations you have given in the comments has $3n+3$ equations in $6n$ variables. For $n=2$ it has obvious real solutions (we do not need a Gröbner basis), namely
$$
x_{11}=1, x_{21}=-1, x_{14}=x_{24}=x_{16}=x_{26}=1, 
$$
and all other equal to zero. In general it seems we can do this. For which $n$ do you think that there is no real solution ?