(This was posted to https://math.stackexchange.com/q/4244260/799193 where it did not recieve an answer.)

Let $X \subseteq \mathbb{C}^n$ be an affine variety defined by $f_i(x_1, \ldots, x_n)=0, 1 \le i \le m$. I am interested in the points $(g_1, \ldots, g_n) \in \mathbb{C}[t]^n$ where $\deg(g_i) < d$ and $f_j(g_1, \ldots, g_n) = 0$ for all $i,j$. This defines a variety $X(d) \subset \mathbb{C}^{dn}$ in the following way: for each equation $f_i$ we get $\deg(f_i)(d-1)+1$ new equations in the variables $y_{j,k}$ for $1 \le j \le n$, $1 \le k \le d$, given by the coefficients of $t$ in $f_i(y_{1,1}+y_{1,2}t + \cdots + y_{1,d}t^{d-1}, \ldots, y_{n,1}+y_{n,2}t + \cdots + y_{n,d}t^{d-1})$. Note that $X = X(1)$.

Has this construction been studied before? It seems sort of similar to the space of $d$-jets of $X$, which I gather is the space of $\mathbb{C}[t]/t^d$-points on $X$, which $X(d)$ is a subvariety of. In particular, can anything be said about the dimension or irreducibility of $X(d)$? For example, if $X$ is irreducible, will $X(d)$ be irreducible for large enough $d$?