I have the system of multi-variable polynomial (quadratic) equations with real coefficients. The number of equations is given scales as $K$ and the number of unknowns goes as $K^2$. So for for large $K$, this is an underdetermined system.


Can I conclude that I can always find $K$ large enough so that this system has at least one real solution or where should I look for counterexamples?



Real solutions to underdetermined system of polynomial equations
However large $n$ may be, the equation


$x\_1^2 + \ldots + x\_n^2 = -1$ has no real solution.


(Similarly, the homogeneous equation $x\_1^2 + \ldots + x\_n^2 = 0$ has no nontrivial solution.)


If you want to go there, the theorem which gives you a necessary and sufficient condition for a system of real polynomial equations to have a real solution is the **Real Nullstellensatz**. See for instance Section 3.9` of


[http://alpha.math.uga.edu/~pete/modeltheory2010Chapter3.pdf](http://alpha.math.uga.edu/%7Epete/modeltheory2010Chapter3.pdf)


(*Very* roughly, this theorem says that the above is essentially the only way that a system of $n$ real polynomial equations in more than $n$ variables can fail to have a real solution.)


