Estimates for exponential sums with a large automorphism group

We prove some improvements of the classical Weil bound for one variable additive and multiplicative character sums associated to a polynomial over a finite field $k=\text{Fq}$ for two classes of polynomials which are invariant under a large abelian group of automorphisms of the affine line ${\text{AAA}}_k^1$: those invariant under translation by elements of $k$ and those invariant under homotheties with ratios in a large subgroup of the multiplicative group of $k$. In both cases, we are able to improve the bound by a factor of $\sqrt{q}$ over an extension of $k$ of cardinality sufficiently large compared to the degree of $f$.