I Analogy question for algebraists

[Stephen Tashi]

TL;DR Summary

Polynomials with one variable and coefficients in the field K are to finite dimensional K vector spaces as polynomials in several variables over the field K are to ....?

An "analogy question":

Polynomials with one variable and coefficients in the field K are to finite dimensional K vector spaces as polynomials in several variables over the field K are to ....?

As a teenager, I recall taking tests that had "analogy questions" on them. The format was: Thing A is to thing B as Thing C is to ...what? They had objectively correct answers - at least in the mind of the people that posed them.

So I'm wondering if the above question has an answer that most algebraists would agree with. Thanks to Axler's book Linear Algebra Done Right, I can see that iterating a linear transformation $T$ applied to a particular vector $v$ leads to the existence of a (finite degree) polynomial in powers of $Tv$ that must equal to zero. That heads toward the topic of eigenvalues and eigenvectors. I don't know if there is some analogy to that situation that involves polynomials in several variables.

 

[fresh_42]

... to the tensor product of as many vector spaces as there are variables.