Starting from a problem in spectral graph theory, I got dragged into a problem in combinatorial matrix theory about constructing $n\times n$ real orthogonal matrices with a specified pattern of zero/non-zero entries. (I can expand on this if required.)


Anyway, I have a construction which appears to work (which I have tested in Maple for some small values of $n$), but to prove it in full generality would require showing that a certain system of simultaneous real quadratic equations has a real solution.


I know very little about algebraic geometry and, well, nothing at all about real algebraic geometry, but I was wondering if there were any existing results in the literature which may help with this---as it stands I'm not even sure where to start looking!


[edited after Alexandre's answer]


The aim (in the first instance) is to construct an orthogonal $n\times n$ matrix with zero diagonal entries and non-zero off-diagonal entries. The idea is to use the following matrix: $$ A=\left[ \begin{array}{c|ccc} 0 & 1 & \cdots & 1 \\ \hline 1 & & & \\ \vdots && B & \\ 1 &&& \end{array} \right] $$
where the rows of $B$ are formed of the $n-1$ cyclic permutations of $v=(0,x\_1,\ldots,x\_{n-2})$. For this to be orthogonal (up to scaling by $\sqrt{n-1}$), the inner product of two distinct rows must be $0$, and the norm of each row must be $\sqrt{n-1}$. This gives the equations $\sum\_{i=1}^{n-2} x\_i = 0$, $\sum\_{i=1}^{n-2} x\_i^2 = n-2$, and $\lceil (n-2)/2\rceil$ other equations arising from inner products of rows containing cyclic permutations of $v$ (from adjacent rows, rows 2 apart, 3 apart, etc.). Also, we have the constraint that any solution for $x\_1,\ldots.x\_n$ may not contain a zero.


For example, when $n=5$, Maple finds the solution $x\_2=1$, $x\_1,x\_3 = \frac{-1\pm \sqrt{3}}{2}$.



Systems of simultaneous real quadratic equations
Existence of real solutions of systems of algebraic equation is a difficult problem
and there is no general theory. You can get an impression of the modern state of it
from the nice book:


MR2830310 
Sottile, Frank,
Real solutions to equations from geometry. 
American Mathematical Society, Providence, RI, 2011. 


Also this one:
MR1925796 Sturmfels, Bernd, 
Solving systems of polynomial equations. American Mathematical Society, Providence, RI, 2002.


Only some very special systems can be presently studied. That all equations are of degree 2
does not help much. But if you write your system, perhaps it is of the special kind about which
something is known.


