Ontology of polynomials

When considering polynomials over a commutative ring, it is often useful to limit yourself to considering special cases. Even though the most interesting polynomials are multivariable, we tend to limit ourselves to univariate polynomials or even more then that to monic polynomials such as in the definition of integral extensions. The whole field of linear algebra basically built around the simple idea of a linear polynomial. Clearly, a classification system for polynomials is necessary. Here are some classes to start with:

Polynomials are relatively simple data structures, which make them a solid part of any computer algebra system. They can simply be represented by a collection of individual monomial terms, which only need to contain a coefficient and some variables. With such a simple representation as a sum of terms, each of these different classes of polynomials can easily be turned into computable predicates on polynomials. This is then a hierarchy of computable predicates, which is generally the nicest kind of ontology.

Homogeneous polynomials: it is trivial to check if a given polynomial has only terms of the same degree, degree in this case means the sum of all exponents of variables in each term. $x^2 + y^2 + xy$ for example is a homogeneous polynomial and a binary quadratic form while $ax^3 + bx^2y + cxy^2 + dy^3$ is a binary cubic form. Homogeneous polynomials are fundamental in algebraic geometry because roots are invariant under scaling, which produces a natural link to projective geometry. Applying a little creativity, we could define anti-homogeneous polynomials to be ones with different degrees for each term.

Max power one polynomials: these are polynomials in which the exponent of each variable in each term is never greater then one. For example $xy + yz + xz$ is a max power one quadratic form but $x^2 + 2x + 1$ is not because of the exponent of two in one of the variables. Although not as familiar, these might appear from semirings in which multiplication is idempotent.

Diagonal polynomials: the dual of a polynomial in which each variable is different, is one in which each variable is the same. We can call these diagonal polynomials and diagonal forms are special cases that are also homogeneous. For example, $x^3 + y^3 + z^3$ is a diagonal form.

Additional classes: this is a very limited upper ontology, which is applicable to general rings. We could classify polynomials themselves based upon the ring that they emerge from, so for example we could consider real polynomials, complex polynomials, etc but these wouldn't fit into an upper ontology. As these classes are based upon sum representation, factorisation based considerations like separable and irreducible polynomials are not included. In algebraically closed fields, irreducible polynomials are simply linear univariate ones, so over some rings it is not necessary to consider irreducible polynomials as a separate class, so they must dealt with separately.