  We construct invariant polynomials on truncated multicurrent algebras, which
are Lie algebras of the form $\mathfrak{g} \otimes_\mathbb{F}
\mathbb{F}[t_1,\dotsc,t_\ell]/I$, where $\mathfrak{g}$ is a finite-dimensional
Lie algebra over a field $\mathbb{F}$ of characteristic zero, and $I$ is a
finite-codimensional ideal of $\mathbb{F}[t_1,\dotsc,t_\ell]$ generated by
monomials. In particular, when $\mathfrak{g}$ is semisimple and $\mathbb{F}$ is
algebraically closed, we construct a set of algebraically independent
generators for the algebra of invariant polynomials. In addition, we describe a
transversal slice to the space of regular orbits in $\mathfrak{g}
\otimes_\mathbb{F} \mathbb{F}[t_1,\dotsc,t_\ell]/I$. As an application of our
main result, we show that the center of the universal enveloping algebra of
$\mathfrak{g} \otimes_\mathbb{F} \mathbb{F}[t_1,\dotsc,t_\ell]/I$ acts
trivially on all irreducible finite-dimensional representations provided $I$
has codimension at least two.
