I've come across an interesting connection between irreducible polynomials and polynomial constraints. For example, consider the basic quadratic:
$$af^2 + bf + c = 0$$
If we're working in a ring, and $f$ is assumed to be irreducible, then $c$ must be some multiple of $f$:
$$c = mf$$
Furthermore, substituting this back into the original equation, an additional constraint is discovered for $b$:
$$b + m = qf$$
Substituting this back into the quadratic equation, we find that $a=-q$.
So we now have a system of equations:
$$af^2 + bf +c = 0$$ $$c - mf = 0$$ $$b + m + af = 0$$
and I'm starting to wonder if I can translate my assumption of irreducibility into polynomial constraints that can be handled by a Groebner basis calculation.
For example, if the ring is $\mathbb{Z}$, and $f^2-6f-7=0$ is solved by 7 (irreducible), then $c$ is a multiple of 7, its multiple is -1, and $b$ (-6) plus that multiple is also a multiple of 7 (again -1), and $a$ is the negative of that multiple.
I'm working in a more complex system where (for example) $dx(nd)$ has to be a multiple of some irreducible $f$ with a zero $x$-derivative, so I'm led to conclude that $nd$ must have the form $fq+c$ where $c$ has a zero $x$-derivative. Now I'm thinking of adding $nd=fq+c$ and $c_x=0$ to my system and doing a differential Groebner basis calculation (maybe).
Can I find an exhaustive list of polynomial constraints triggered by the assumption of irreducibility? Exhaustive in the sense of a finite basis for an ideal.
Has anyone seen anything like this before? Any references?
