Poké-Nomials

We are all somewhat familiar with polynomials, and they may even appear basic on the surface, but there are still many things to be discovered about these objects. For example, consider polynomials in three variables x, y, z and define this function φ that translates each variable by a new variable θ like so:

x ↦ x + θ

y ↦ y + θ

z ↦ z + θ

Now the important question is: which polynomials return to their original self after applying φ? One such example is x - y:

φ(x - y) = x + θ - (y + θ) = x + θ - y - θ = x - y.

For fun, we will call these "invariant" polynomials Poke-nomials and our first task will be to Catch 'Em All by classifying them! After classifying them, we will look at "types" of Poke-nomials whose sets of solutions have special geometric properties.

For example, one "type" of Poke-nomial we want to catch, are those that cannot be factored as products of other Poke-nomials, and that also are as small as possible in the sense that we cannot find another Poke-nomial that uses only a subset of its monomials.

Finding the Poke-nomials of that particular "type" gets surprisingly complicated very quickly, but there is actually a connection between these polynomials and geometry, and changing to this more visual perspective opens up many opportunities for discovery. The question transforms into a spectacularly visual balancing act.