We study the problem of determining, for a polynomial function$f$on a vector space$V$, the linear transformations$g$of$V$such that$f\circ g=f$. When$f$is invariant under a simple algebraic group$G$acting irreducibly on$V$, we note that the subgroup of$\text{GL}(V)$stabilizing$f$often has identity component$G$, and we give applications realizing various groups, including the largest exceptional group$E_{8}$, as automorphism groups of polynomials and algebras. We show that, starting with a simple group$G$and an irreducible representation$V$, one can almost always find an$f$whose stabilizer has identity component$G$, and that no such$f$exists in the short list of excluded cases. This relies on our core technical result, the enumeration of inclusions$G<H\leqslant \text{SL}(V)$such that$V/H$has the same dimension as$V/G$. The main results of this paper are new even in the special case where$k$is the complex numbers.