What object's symmetry does the group of polynomials of degree at most $n$ preserve?

Question

So the set of polynomial of degree at most $n$ form a commutative group under addition. And groups describe the symmetry of some object. So, I'm wondering what object does the group the polynomials of degree at most $n$ describes?

I think the set of polynomials of degree at most $n$ is isomorphic to $\mathcal{R}^{n+1}$. So then the question becomes what symmetry does $\mathcal{R}^{n+1}$ preserve. So then that would be the space of $\mathcal{R}^{n+1}$ itself?

Answer 1

Your idea that

And groups describe the symmetry of some object.

was an important source of early examples of groups. For example, in the 19th century as Galois theory. The Wikipedia article states:

The central idea of Galois' theory is to consider permutations (or rearrangements) of the roots such that any algebraic equation satisfied by the roots is still satisfied after the roots have been permuted.

Another example is Klein's Erlangen program. The Wikipedia article states:

With every geometry, Klein associated an underlying group of symmetries. The hierarchy of geometries is thus mathematically represented as a hierarchy of these groups, and hierarchy of their invariants.

However, later in the 19th century Group theory was developed. The Wikipedia article states:

Most groups considered in the first stage of the development of group theory were "concrete", having been realized through numbers, permutations, or matrices. It was not until the late nineteenth century that the idea of an abstract group as a set with operations satisfying a certain system of axioms began to take hold.

The Wikipedia article on Groups states:

Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; Poincaré groups, which are also Lie groups, can express the physical symmetry underlying special relativity; and point groups are used to help understand symmetry phenomena in molecular chemistry.

So you are not completely wrong about the importance of symmetry as a source of groups, but these are only examples of the more general and abstract modern concept of a group. So the answer to your question

what object does the group the polynomials of degree at most n describes?

is that since polynomials form an vector space, any vector in the spaced acts on the space itself via translation. This is a special case of a Group action The Wikipedia article states

In every group $G$, left multiplication is an action of $G$ on $G: g⋅x = g\,x$ for all $g, x$ in $G$. This action forms the basis of a rapid proof of Cayley's theorem - that every group is isomorphic to a subgroup of the symmetric group of permutations of the set $G$.

Thus, in some sense every group is a group of symmetries of objects, and in more than one way.

Answer 2

Sure, that's one possibility. The set $\Bbb R^{n+1}$ (or say metric / topological space, to have something more concrete to work with) has translation symmetry, and the group of translation symmetries is the group $\Bbb R^{n+1}$ with standard (vector) addition.