If I have system of n equations (any equations in general) with n variables can I always solve it?

Question

If I have system of n equations (any equations in general) with n variables can I always find all the solutions or prove that there are no such a solutions ? If it's true is there such theorem ?

Answer 1

Of course, if the OP presents his question like here, he is exposed to be downvoted...

A good question would be (for example).

Let $(S)$ be a system of $n$ algebraic equations in $n$ complex unknowns. Can we know if $(S)$ admits at least one solution.

Using any Grobner basis software, the answer is yes if $n$ and the degrees of the equations are small. More precisely, if the Grobner basis is not reduced to $[1]$, then there is at least one complex solution of $(S)$; the problem is to find such a basis although it is very complicated.

For example, assume that $n=4$ and the $4$ equations are of degree $4$ with respect to the $4$ variables; we obtain a result in $\approx 20"$. In the generic case, according to Bezout, there are $4^4=256$ solutions.

When the underlying field is $\mathbb{R}$, the problem is much more complicated.