Suppose that we have a parametrization via polynomials as follows:

$$t\longrightarrow (f_1(t),\ldots,f_n(t)),$$

where $t$ is a vector in $\mathbb{C}^r$ and $f_i$ are polynomials of arbitrary degree.

Can we always find equations such that the image is an affine algebraic variety?

The question is motivated by Exercise 1.11 in Hartshorne:

Let $Y\subseteq A^3$ be the curve given parametrically by $x = t^3, y= t^4, z = t^5$. Show that $I(Y)$ is a prime ideal of height 2 in $k[x,y,z]$ which cannot be generated by 2 elements.

I am not interested in the exercise in particular. Finding the variety is easy sometimes, for instance $t\rightarrow (t^2,t^3)$ is given by $I(x^3-y^2)$.

I am looking for a result which says that the image is always an affine algebraic variety AND a procedure to find the ideal.