Number of Irreducible Polynomial Factors of a Polynomial in $\mathbb{F}_p[X]$

This question raises from the fact that over the integers, the prime omega $$\Omega(N)$$ function tells us the total number of prime factors of a number $$N$$. This is because the number $$N$$ factors as $$N=\prod_{i=1}^{r}p_i^{e_i}$$. Then $$\Omega(N)= \sum_{i=1}^r e_i$$.

Right now, my research conduces me to estimate the (average) number of irreducible factors of a monic polynomial over $$\mathbb{F}_p$$.

This is, take a polynomial $$f(X) \in \mathbb{F}_p[X]$$. Factor it over $$\mathbb{F}_p$$ as $$f(X) = \prod_{i=1}^r f_i(X)^{e_i}$$, where $$f_i(X)$$ is irreducible over $$\mathbb{F}_p$$. And finally obtain $$\Omega(f):=\sum_{i=1}^re_i$$.

My question is how I can estimate the number of irreducible factors of any polynomial $$f(X)$$ over $$\mathbb{F}_p$$.

• 1) For integers, $\omega(N)$ usually denotes the number $r$, and $\sum_{i=1}^r e_i$ is denoted by $\Omega(N)$. Aug 9 '20 at 23:17
• 2) The leading coefficient of a polynomial doesn't play any significant role in its factorization, so it sounds reasonable to fix it (to $1$), i.e. to consider only monic polynomials, right? Aug 9 '20 at 23:18
• 3) How exactly do you take the average? A possible way is to do so over polynomials of a fixed degree $d$. For the $\omega$ analogue, the computation is quite hard (I haven't reached the result yet, though I'm close); for $\Omega$, it is much easier: say, the average appears to tend to $H_d=\sum_{k=1}^d 1/k$ as $p\to\infty$. Aug 9 '20 at 23:33
• And 0) I don't see any relationship to the integer versions (besides the definitions)... Aug 9 '20 at 23:34
• I'm interested on $\Omega(N)$ then. I want to know how many total irreducible factor does $p(X)$ have in $F_p[X]$. Moreover, $p(X)$ is monic. This is because my algorithm is bounded by a partition length which is always equal to the nº of total factors of $p(X)$. Gonna change my description to fit these requirements. Aug 9 '20 at 23:48

A possible approach is to analyze the number $$N_{k,d}$$ (throughout the answer, $$p$$ is considered fixed, and its primality is not used, so it can be a power of a prime as well) of monic polynomials $$f$$ over $$\mathbb{F}_p$$ of degree $$d$$ with $$\Omega(f)=k$$. We assume $$\deg 1=\Omega(1)=0$$ for the constant polynomial $$1$$, so that $$N_{k,d}$$ is defined for $$k,d\geqslant 0$$. Then, $$N_{1,d}$$ is the number of irreducible monic polynomials of degree $$d$$, and $$\sum_{k=0}^d N_{k,d}=p^d$$.

This is a road to various averages expressible. Say, $$\Omega(f)$$ averaged over monic $$f$$ with $$\deg f=d$$ is $$\overline\Omega(d):=p^{-d}\sum_{k=1}^d kN_{k,d}.$$ We denote by $$M$$ the set of all monic polynomials (over $$\mathbb{F}_p$$), and by $$I$$ the subset of $$M$$ consisting of all irreducible polynomials. Under our agreements above, we have $$1\in M$$ and $$1\notin I$$.

The analysis is done using generating functions: $$G_\Omega(s,t)=\sum_{k,d\geqslant 0}N_{k,d}s^k t^d=\sum_{f\in M}s^{\Omega(f)}t^{\deg f},\\G_I(t)=\sum_{d\geqslant 1}N_{1,d}t^d=\sum_{f\in I}t^{\deg f}=\frac{\partial}{\partial s}G_\Omega(s,t)\Bigg|_{s=0},\\G_{\overline\Omega}(t)=\sum_{d\geqslant 1}\overline\Omega(d)t^d=\frac{\partial}{\partial s}G_\Omega\left(s,\frac{t}{p}\right)\Bigg|_{s=1}.$$

The crucial fact (the unlabelled multiset construction of Flajolet-Sedgewick, applied) is $$G_\Omega(s,t)=\exp\left\{\sum_{n\geqslant 1}\frac{s^n}{n}G_I(t^n)\right\}$$ (TODO: write an elaborated appendix if needed). It also allows to find $$G_I$$, since $$\sum_{k=0}^d N_{k,d}=p^d$$ implies $$G_\Omega(1,t)=\sum_{d\geqslant 0}(pt)^d=(1-pt)^{-1}$$. So, with $$\ell(z):=-\log(1-z)$$, we obtain $$\sum_{n\geqslant 1}\frac1n G_I(t^n)=\ell(pt)\implies G_I(t)=\sum_{n\geqslant 1}\frac{\mu(n)}{n}\ell(pt^n)$$ by a variant of the Möbius inversion. Since we don't really need $$G_\Omega$$, let's compute directly $$G_{\overline\Omega}(pt)=\exp\big(\ell(pt)\big)\sum_{n\geqslant 1}G_I(t^n)=(1-pt)^{-1}\sum_{n\geqslant 1}\ell(pt^n)\sum_{m\,\mid\,n}\frac{\mu(m)}{m};$$ since $$\sum_{m\mid n}\big(\mu(m)/m\big)=\varphi(n)/n$$ using Euler's totient function, we find finally $$\overline\Omega(d)=\sum_{n=1}^d\frac{1}{np^n}\sum_{m\,\mid\,n}\varphi(m)p^{n/m}=\sum_{m=1}^d\frac{\varphi(m)}{m}\sum_{n=1}^{\lfloor d/m\rfloor}\frac{p^{(1-m)n}}{n}.$$

For large $$p$$, this is $$\overline\Omega(d)=H_d+\dfrac{1}{2p}+\mathcal{O}(p^{-2})$$ where $$H_d=\displaystyle\sum_{n=1}^d\frac1n=\log d+\gamma+\ldots$$

With $$\omega$$ in place of $$\Omega$$, things would go harder. We would use the powerset construction (instead of the multiset construction), but with another generating function in place of $$G_I$$: $$G_\omega(s,t)=\exp\left\{\sum_{n\geqslant 1}(-1)^{n-1}\frac{s^n}{n}\sum_{k\geqslant 1}G_I(t^{nk})\right\},$$ with somewhat laborous number-theoretic computations. We find $$G_{\overline\omega}(pt)=A(t)B(t)$$, where \begin{align*} A(t)&=\exp\sum_{n\geqslant 1}a_n\ell(pt^n),&B(t)&=\sum_{n\geqslant 1}b_n\ell(pt^n),\\a_n&=\sum_{\substack{a,b,c\geqslant 1\\abc=n}}\frac{(-1)^{a-1}}{a}\frac{\mu(c)}{c},&b_n&=\sum_{\substack{a,b,c\geqslant 1\\abc=n}}(-1)^{a-1}\frac{\mu(c)}{c}. \end{align*}

Now both $$a_n$$ and $$b_n$$ can be found using Dirichlet series. For $$a_n$$, the result is simple: $$\sum_{n\geqslant 1}\frac{a_n}{n^s}=\sum_{a,b,c\geqslant 1}\frac{(-1)^{a-1}}{a^{1+s}}\frac{1}{b^s}\frac{\mu(c)}{c^{1+s}}=(1-2^{-s})\zeta(s)\implies a_n=\begin{cases}1,&n\text{ is odd}\\0,&n\text{ is even}\end{cases};\\\sum_{n\geqslant 1}\frac{b_n}{n^s}=(1-2^{1-s})\frac{\zeta^2(s)}{\zeta(1+s)}=\frac{(1-2^{1-s})(1-2^{-1-s})}{(1-2^{-s})^2}\prod_{p\neq 2}\frac{1-p^{-1-s}}{(1-p^{-s})^2},$$ which gives, for odd $$n=p_1^{r_1}\cdots p_k^{r_k}$$ and $$r>0$$, $$b_n=\prod_{j=1}^k\left[1+\left(1-\frac{1}{p_j}\right)r_j\right],\quad b_{2^r n}=-\frac{r}{2}b_n.$$

Thus, $$G_{\overline\omega}(pt)=\left(\sum_{n\geqslant 1}b_n\ell(pt^n)\right)\prod_{n\geqslant 1}(1-pt^{2n-1})^{-1}$$. This is where I got to at the moment.

• I find your answer great and helpful. Right now, I focus on the first sum you gave on $\overline{\Omega}(d)$ as In my work I've to analyse the case where $p$ is small i.e $p=2,d=128$ and $\overline{\Omega}(d) \approx 6.1227$. Have to say that I couldn't get there as I cannot derive these expressions from generating functions by myself. Have you published these results somewhere else? I'd like to include it as a reference. Aug 10 '20 at 15:45
• No, the above is done from scratch. I've just found the $\Omega$ part in TAOCP vol. $2$, as an exercise to section $4.6.2$ (the formula for $\overline\Omega$ is not stated there, but it's just one step away). Aug 10 '20 at 15:59