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If I have two linear equations, $ax + by = 0$ and $cx + dy = 0$, and I wanted to find out if they had any non-trivial solutions, I would simply check if $(a,b)$ and $(c,d)$ are linearly dependent.

Now suppose I set two quadratic forms equal to zero, that is, $xAx^T=0$ and $xBx^T=0$. This is no longer a linear situation, but are there any similar criteria for checking whether or not there exists a non-trivial solution satisfying both forms, if I let $x$ be a vector over $\mathbb{C}$?

I'm not too fussed about what the solutions are, as long as I can be sure that at least one exists.

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The Weak Nullstellensatz is the following:

Let $k$ be an algebraically closed field and let $I$ be an ideal of $k[x_1,\ldots,x_n]$. If $I$ is a proper ideal, then the variety $V(I)$ is nonempty.

See page 170 of "Ideals, Varieties, and Algorithms" by Cox, Little, and O'Shea.

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I have found the criterion for which I was looking. If I have a system of $r$ quadratic forms, then a non-trivial common zero exists as long as my system contains at least $r+1$ variables.

The result is mentioned in this paper, and I believe it follows from Hilbert's Nullstellensatz.

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