naming question: linear family of endoms (rank, regular)

Stream: maths

[Johan Commelin]

Suppose that we have a linear family of endomorphisms $\phi \colon L \to \text{End}(M)$, where $L,M$ are two $R$-modules. (Example: the adjoint representation of a Lie algebra $L$. In this case $M = L$.)
Then the characteristic polynomials $\chi_{\phi(x)}$ have coefficients that are homogeneous polynomials as function of $x$. (One way to make this precise: represent $x$ on some basis of $L$.)

In Lie theory, the smallest index for which this homogeneous "coefficient polynomial" is non-zero is called the rank, and elements $x$ whose $\chi_{\phi(x)}$ has a non-zero coefficient at index rank are called regular elements.

Here is my question: these definitions make sense outside Lie theory for an arbitrary $\phi$ as above. (Oliver has been very patiently pointing this out to me.)

1. Are these concepts used outside of Lie theory?
2. If yes, under what names?
3. If no, what are reasonable names for these concepts? (I currently have LinearMap.polyRank φ and LinearMap.IsRegular φ x, and I'm not very fond of those names.)

[Oliver Nash]

I only have a moment so a quick remark. Maybe LinearMap.nilpotentRank or LinearMap.nilRank for one of the names since when $L$ is one-dimensional (so that we really just have a single endomorphism) this number is the dimension of the Fitting 0 component.

[Johan Commelin]

Ok, that is a good suggestion. Should we go with something like LinearMap.IsNilRegular for the other one?

[Johan Commelin]

Ok, I pushed those names to

feat(Algebra/Lie/Rank): the rank of a Lie algebra, and regular elements #10628

[Damiano Testa]

I think that some people might call these (or very small variations) determinantal varieties.

[Antoine Chambert-Loir]

Then you can even have three linear spaces and $\phi \colon L\to \mathscr L(M;N)$.