In what follows, I will work over $\mathbb{C}$.

Let $d_1,\ldots,d_n$ be positive integers. Consider generic homogeneous polynomials $F_1(x_1,\ldots,x_n),\ldots,F_n(x_1,\ldots,x_n)$ of degrees $d_1,\ldots,d_n$ respectively. The resultant of format $(d_1,\ldots,d_n)$ is the unique polynomial in variables given by the coefficents of the polynomials $F_1,\ldots,F_n$ which satisfies:

1. It vanishes iff $\exists x\in\mathbb{C}^n\backslash\{0\}$, $\forall i, 1\le i\le n, F_i(x)=0$.

2. It takes the value 1 when $\forall i, F_i(x)=x_{i}^{d_i}$.

3. It is irreducible.

The determinant is just the resultant of format $(1,1,\ldots,1)$.