Let $L|K$ be a finite Galois extension of degree $d$ and $X$ be a variety over $K$. Is there a simple criterion, similar to Galois descent, allowing to determine whether $X$ is the Weil restriction (restriction of scalars) of a variety $Y$ over $L$?
Since Weil restriction is adjoint to extension of scalars, I assume a condition is that for every $K$-scheme $S$, $X(S)$ depends only on $S \times_{\mathrm{Spec}\,K} \mathrm Spec\,L$, i.e. the functor of point factorizes as the composition of $\bullet \times_{\mathrm{Spec}\,K} \mathrm Spec\,L$ and some presheaf on the category of $L$-schemes, which should be represented by the variety we are looking for. However, such a criterion is a little impractical.
I am looking more for something like the Cauchy-Riemann equations. In terms of coordinates, I think I can write what I am trying to do: choose a $K$-basis $(\beta_1, \ldots, \beta_d)$ of $L$ and write $B_1, \ldots, B_d \in \mathrm{GL}_d(K)$ the matrices corresponding to multiplication by $\beta_1, \ldots, \beta_d$. Let's do the case of a single polynomial. If $P_1, \ldots, P_d \in K[X_1, \ldots, X_d]$ comes from a polynomial $Q \in L[X]$, I know that: $$ \sum_{i=1}^d \beta_i\frac{\partial P_i(X_1,\ldots,X_d)}{\partial X_j} = \frac{\partial Q(\beta_1 X_1 + \ldots + \beta_d X_d)}{\partial X_j} = \beta_j Q'(\beta_1 X_1 + \ldots + \beta_d X_d) $$ and in particular: $$ \left(\frac{\partial P_i(X_1,\ldots,X_d)}{\partial X_j} \right)_{1 \leq i \leq d} = B_j \left( \frac{\partial P_i(X_1,\ldots,X_d)}{\partial X_1} \right)_{1 \leq i \leq d} $$ and conversely these equality for all $2 \leq j \leq d$ certainly imply that $(P_1, \ldots, P_d)$ come from a polynomial $Q \in L[X]$. However, this is very unsatisfying as I had to choose a basis and this is way too coordinatey. I think I would love to have a condition that is instead saying something like "differentials have some sort of equivariance under the action of $\mathrm{Gal}(L|K)$", ideally in a scheme-theoretic "abstract" way.
Has such a condition ever been described?
Edit: I did not mention it, but I am also interested (actually, rather) in a criterion for Galois ascent of morphisms and not just of varieties.