Let $m$ and $n$ be positive integers. Suppose that we are given $m$ homogeneous polynomial equations (with complex coefficients) such that each of $n$ variables occurs at least once in all of the polynomials. No other variables occur in any of these polynomials – and they are not necessarily all of the same degree. If $m$ is less than $n$, will these $m$ equations always have at least one common root in the field of complex numbers that is non-trivial? By "non-trivial", I mean a root in which not all the variables are equal to zero.
Theorem: In n dimensional complex projective space over an algebraically closed field (such as the complex numbers), the intersection of m < n hypersurfaces is always nonempty.
Proof: Suppose that hypersurfaces are given by equation $f_1, \ldots f_m$. Consider these as equations cutting out points in the affine space whose lines will make up the projective space ($A^n \setminus 0 \to P^n$). Here each $X_i = V(f_i)$ contains the origin, so the intersection of the $X_i$ is nonempty. However, by dimension theory, the minimum dimension of the components of $X = \cap X_i$ is $n - m \geq 1$. Thus, $X$ is at least one dimensional and nonempty, so it contains a point not passing through the origin. This means that the corresponding projective algebraic set is non-empty.
Lemma: (From dimension theory) (Generalizing the theorem of algebra about the finiteness of roots of a polynomial) If $X$ is an algebraic variety of dimension $k$ in $A^n$, and $Y$ is a hypersurface, then the irreducible components of $X \cap Y$ have dimension $\geq k - 1$.
Proof: Relatively hard commutative algebra. (Depending on your definition of dimension.) But morally, $Y$ is given by an equation $f$. Restricting this equation to $X$ as $f|X$, the zeros of $f|_X$ correspond to points of the intersection $X \cap Y$. Now, generally speaking, looking the zeros of a polynomial equation on a variety $X$ can't cut down the dimension by more than 1. (Krull's Principal Ideal Theorem.)