# homogeneous polynomials with suitable number of variables that has a non-trivial solution

It is an elementary fact that when the number of variables exceeds the number of linear equations then the system has a nontrivial solution.

I want to know whether there is such thing about homogeneous polynomials of degree $2$ or $3$, that is, for every natural number $n$ there exists $N \in \mathbb{N}$ s.t. for every $m>N$, the system of equations consisting $n$ homogeneous polynomials of degree $2$ in $\mathbb{C}[x_1,...,x_m]$ (or of degree $3$ in $\mathbb{R}[x_1,...,x_m]$ or $\mathbb{C}[x_1,...,x_m]$) has a non-trivial solution?

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1) It is not entirely true what you wrote about linear equations. If your equations are contradictory, then no matter how many additional variables you add, you will not find a solution. For instance $x+y=0$, $x+2y=1$, $2x+y=1$ has no solution regardless possible additional variables.
3) For homogenous systems of equations over $\mathbb C$ the same bound holds. Here is why: since it is homogenous, again, taking all variables to be zero gives a solution. Therefore the hypersurfaces determined by the equations have a common point and by the affine dimension theorem (which is essentially saying that one equation can cut down the dimension of the solution set by at most one) the solution set has positive dimension.
4) Over $\mathbb R$ you run into the problem of not having solutions no matter what. If your equations contain something like $x^2+y^2+...=0$ involving all the variables, then you are out of luck.