Posted by Jason Polak on 05. March 2017 · Write a comment · Categories: commutative-algebra · Tags:

I’d like to invite readers of this blog to download my latest paper, to appear in the Canadian Mathematical Bulletin:

What is this paper about? It uses the theory of separable algebras to study separable polynomials in $\Z/n[x]$, which extends the usual definition of separability for polynomials over a field.

Let $d\geq 2$. A classical theorem of Leonard Carlitz says that for a prime $p$ with $q=p^k$, the number of monic separable polynomials of degree $d$ in $\F_q[x]$ is $q^d – q^{d-1}$. One can also define separable for polynomials in $\Z/n[x]$. In this case, since a polynomial cannot always be converted to a monic one by multiplying by a unit, it makes more sense count all separable polynomials. Deriving a formula for this number is exactly what my paper does.

Read the paper to see how it’s done! Although it talks about separable algebras, you can actually read it without knowing anything about this more advanced stuff as the interface between separable algebra theory and the concrete combinatorics is pretty clean. Or, you can just look at the final answer: the number of separable polynomials in $\Z/n[x]$ of degree at most $d$ for $d\geq 1$ is given by
$$\phi(n)n^d\prod_{i=1}^m(1 + p_i^d)$$
where $n = p_1^{k_1}\cdots p_m^{k_m}$ is the prime factorization of $n$ and $\phi(n) = |(\Z/n)^\times|$ is Euler’s phi function. The formulas in the paper have been checked mutliple times with Sage.