Assumme $f_0, f_1 ... f_n$ is n+1 homogenous polynomials with fixed degree $d_0,d_1...d_n$ in variable $x_0...x_n$. So the number of polynomials is equal to the number of variables.
Then does there exists a polynomial D in coefficients of $f_i$'s s.t. D=$0$ iff $f_0=f_1=...=f_n=0$ has a nontrivial solution. For example, when all the $d_i$=1, the determinant of the corresponding matrix is the D.
But in general, how can I prove such polynomial D exists?
Such a polynomial exists and it's called the resultant.
To prove that it exists, use the linked definition (the product of the differences of roots in an algebraic closure) and use the theory of symmetric polynomials (and/or a bit of Galois theory) to show that the coefficients lie in the base field.