Appearance of Formal Derivative in Algebra

When studying polynomials, I know it is useful to introduce the concept of a formal derivative. For example, over a field, a polynomial has no repeated roots iff it and its formal derivative are coprime. My question is, should we be surprised to see the formal derivative here?

Is there some way we can make sense of the appearance of the derivative (which to me is an analytic object) in algebra? I suspect it might have something to do with the fact that the derivative is linear and satisfies the product rule, which makes it a useful object to consider. It would also be interesting to hear an explanation which explains this in the context of algebraic geometry.

Thanks!

Answer

Rather than being a deep surprise, the formal derivative is somewhat closer to being a triviality that is easy to overlook!

For example, the MacLaurin series for a polynomial:

f(x)=i=0fixi

is simply writing out f(x) in the usual fashion as the sum of its terms. It’s also clear that every polynomial has a finite Taylor series around any point:

f(x)=i=0ci(xa)i

and we can consider all points at once by letting a be a variable. Then the coefficients are now functions of a:

f(x)=i=0ci(a)(xa)i

in fact, it’s easy to work out that they are polynomials in a. Going back to the original example, it’s now clear that those a such that f(x) has a double root at a are precisely the roots of gcd.

Of course, we know that c_0(a) = f(a) and c_1(a) = f'(a) and so forth; the derivatives of f can effectively by defined by this equation. In fact, a common algebraic definition of the formal derivative is that it is the unique polynomial such that

f(x + \epsilon) = f(x) + \epsilon f'(x) \pmod{\epsilon^2}

where we consider \epsilon a variable. The ring F[\epsilon] / \epsilon^2, incidentally, is called the ring of dual numbers.

This should look an awful lot like asymptotic notation for the first-order behavior of f at 0….

A posteriori we know that this point of view turns out to be extremely fruitful (not just in analysis, but even purely algebraically), and pushed to even greater extremes — e.g. formal power series rings. And it generalizes to rings other than F[x] where we consider things modulo powers of an ideal, which in turn leads to things like the p-adic numbers.

But without this knowledge, one might look at all I’ve said above, and think that all I’ve done is take a simple thing and make things complicated.

One might even be so bold to argue that the derivative is actually more of an algebraic idea that has been fruitfully generalized to the study of analytic objects rather than the other way around.

Attribution
Source : Link , Question Author : Dylan Yott , Answer Author : Community

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