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? 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 generally a proper subvariety of. In particular, can anything be said about the dimension or irreducibility of $X(d)$ from that of $X$? If $X$ is irreducible, is $X(d)$ eventually irreducible for large enough $d$?
Question: "Has this construction been studied?"
Answer: If $X:=V(f(t)) \subseteq \mathbb{A}^1_k$ with $k:=\mathbb{C}$ and you consider maps $\phi: A:=k[t]/(f(t)) \rightarrow k[T]$ of $k$-algebras , it follows the set of such maps $\phi$ are in 1-1 correspondence with polynomials $u(T) \in k[T]$ with $f(u(T))=0$. You get an equality
$$ Hom_{Sch/k}(Spec(k[T]) , X) \cong Hom_{k-alg}(A, k[T]):=X(k[T]).$$
Let $X:=Spec(A)$. If you truncate and instead consider $B:=k[T]/(T^{l+1})$ you get
$$Hom_{Sch/k}(Spec(B), X) :=X(k[T]/(T^{l+1}))$$
and this is the "$l$'th arc space" of $X$. If you google scholar "arc space" you find many articles on this issue.