Familiar equations in more general settings

Question

What equations, or results about equations, generalize in interesting ways from number theory or geometry to more abstract settings? The motivating example for this question was as follows:

Pythagorean triples are one of the first mathematical concepts many people encounter. But is the equation a² + b² = c² of any relevance in more general rings? The rings of square matrices over the integers (or rationals), for example? Rings of polynomials? The ring of integers of a number field - or just the number field itself? How about more abstract settings?

Other equations spring to mind, however. To take a related example, what rings is Fermat's last theorem thought (or known) to generalize to?

What about the equation, x² = x + 1, defining the Golden ratio? Is that of any interest? Or the iterated mapping which generates the Mandelbrot set (say over C^n×n)?

What use are things like exponentiation, trigonometric functions and π when studying, say, matrices over a complete field? Are they only used for solving differential equations, or are there more number-theory-like applications (as in the relationship between π and the primes, for example)?

Apologies if all this is a bit naive; likewise if it's too much for one question - I'm sure FLT alone would lead to interesting answers. Also, feel free to make this community wiki if that's appropriate: I'm still relatively new to this site, and unsure of etiquette.

Answer 1

This is a really really broad question, but I'll mention something about FLT: it's known to be true for $\mathbb{C}[x]$ via the Mason-Stothers theorem, whose analogue over $\mathbb{Z}$ is the abc conjecture, which is known to imply FLT for sufficiently large exponents. This is a good example of how statements which are true about polynomial rings can inform conjectures about $\mathbb{Z}$.

Answer 2

Your question reminds me of the following quote from my advisor, which I can't resist posting here:

"Finally I want to remark that the treatment of big Cohen-Macaulay modules here serves as a reminder that algebra, after all, has to do with solving equations. Abstract algebra is the daughter of the theory of equations, (in the broadest sense), and perhaps its best theorems (like M. Artin's approximation theorem) still deal with that subject. In any case, we shall see that even in abstruse homological matters it is best not to forget this fact."

(in case you wonder what equations he was talking about, I found some wiki link)

Answer 3

Since you are looking for generalizations of the classical theory of algebraic equations to more general rings, maybe this can be of some interests for you.

It is possible to study finite extensions $R[\alpha]$, where $R$ is an arbitrary UFD. This means that $\alpha$ is the root of an irreducible polynomial in $R[z]$:

$f(z)=z^n+a_{n-1}z^{n-1}+ \cdots + a_1z+a_0, \quad a_i \in R$.

Reductions similar to Tschirnhaus method apply also in this setting, for instance Jerrard proved that the second, third and fourth terms after the leading term can be removed, using transformations involving only square ad cube roots. Moreover, Vasconcelos and others have given algorithms to effectively compute the integral closure of $R[\alpha]$ in some cases.

In Algebraic Geometry, this can be applied to the study of finite covers of factorial varieties, in particular in the non-Galois case (which is a rather difficult problem already in degree 3).

The paper of Tan and Zhang "The determination of integral closures and geometric applications" is a good introduction to this circle of ideas.

Answer 4

The equation for the variance of a sum of correlated random variables looks like the law of cosines. More on that here. Both the variance formula and the law of cosines are consequences of general properties of Hilbert space.