Let $\mathbf{Sch}_k$ be the category of $k$-schemes, and let $K_0(\mathbf{Sch}_k)$ be the Grothendieck ring of $k$-schemes. Let $\mathbb{Z}[\mathbb{L}]$ the subring generated by the virtual Lefschetz motive $\mathbb{L} := [\mathbb{A}^1_k]$. Is every element of $\mathbb{Z}[\mathbb{L}]$ the class of a $k$-scheme (or at least a $k$-constructible set) ? It might be convenient to recall that $K_0(\mathbf{Sch}_k)$ is generated by isomorphism classes of $k$-schemes $[X]$, and that $$[X] = [X \setminus C] + [C]$$ whenever $C$ is a closed subscheme in $X$; the product is given by $$[X] \cdot [Y] = [X \times Y].$$

Let $\mathbf{Sch}_k$ be the category of $k$-schemes of finite type, and let $K_0(\mathbf{Sch}_k)$ be the Grothendieck ring of $k$-schemes. Let $\mathbb{Z}[\mathbb{L}]$ the subring generated by the virtual Lefschetz motive $\mathbb{L} := [\mathbb{A}^1_k]$. Is every element of $\mathbb{Z}[\mathbb{L}]$ the class of a $k$-scheme (or at least a $k$-constructible set) ? It might be convenient to recall that $K_0(\mathbf{Sch}_k)$ is generated by isomorphism classes of $k$-schemes $[X]$, and that $$[X] = [X \setminus C] + [C]$$ whenever $C$ is a closed subscheme in $X$; the product is given by $$[X] \cdot [Y] = [X \times Y].$$