Factorization type probabilities of polynomials with prescribed coefficients over a finite field

TL;DR

This paper extends previous results on the factorization probabilities of polynomials over finite fields, providing criteria for irreducibility properties when the field and degree are not coprime.

Contribution

It offers a new criterion for the irreducibility probability of shifted polynomials over finite fields, even when the field size and degree are not coprime.

Findings

Probability of irreducibility is approximately 1/d for most shifts

Extension of criteria to cases where (q,2d)>1

Quantitative bounds on the deviation from the expected probability

Abstract

Let $f(T)$ be a monic polynomial of degree $d$ with coefficients in a finite field $\mathbb{F}_q$. Extending earlier results in the literature, but now allowing $(q,2d)>1$, we give a criterion for $f$ to satisfy the following property: for all but $d^2-d-1$ values of $s$ in $\mathbb{F}_q$, the probability that $f(T)+sT+b$ is irreducible over $\mathbb{F}_q$ (as $b\in\mathbb{F}_q$ is chosen uniformly at random) is $1/d+O(q^{-1/2})$.

Equations

$$I(f,m)=\{f(T)+a_m{T}^m+\cdots +a_{1}T+a_0|a_0,...,a_m\in {\mathbb{F}}_q\}.$$

$$D^{j}f=\sum \left(\genfrac{}{}{0ex}{}{i}{j}\right)a_i{T}^{i-j},$$

$$f(x)-f(y)=(x-y)\tilde{f}(x,y).$$

$$B_1=\{-f^{\prime }(\alpha )|D^{2}f(\alpha )=0\};$$

$$X_1:=V(\tilde{f}(x,y)-f^{\prime }(x))\subset {\mathbb{A}}^2\text{and}{X}_2:=V(\widetilde{f^{\prime }}(x,y))\subset {\mathbb{A}}^2.$$

$$B_2=\{-f^{\prime }(\alpha )|(\alpha ,\beta )\in (X_1\cap X_2)(\overline{{\mathbb{F}}_q})\text{for some}\beta \in \overline{{\mathbb{F}}_q},\beta \ne \alpha \};$$

$${\Omega }_s:=\{(t,b)\in {\mathbb{A}}^2|f(t)+st+b=0\}.$$

$${\Delta }^c:=\{(x,y)\in {\mathbb{A}}^2|\tilde{f}(x,y)+s=0,x\ne y\}.$$

$$\Omega :=\{(t,s,b)\in {\mathbb{A}}^3|f(t)+st+b=0\}$$

Full text

1 Introduction

Fix a positive integer $d$. Gauss proved that the probability for a monic polynomial of degree $d$ with coefficients in a finite field $\mathbb{F}_{q}$ to be irreducible is $1/d+O_{d}(q^{-1/2})$. In fact, for any partition $\lambda=(\lambda_{i})_{i=1}^{k}$ of $d$, the probability for a random monic $\mathbb{F}_{q}$-polynomial of degree $d$ to have exactly $k$ irreducible factors over $\mathbb{F}_{q}$ of degrees $\lambda_{1},...,\lambda_{k}$ (i.e., to have factorization type $\lambda$) is $p_{\lambda}+O_{d}(q^{-1/2}),$ where $p_{\lambda}$ is the probability that a permutation in $S_{d}$ has cycle structure $\lambda$.

The setting in which some coefficients of the polynomial are fixed and the remaining ones vary in $\mathbb{F}_{q}$ has been studied extensively. For a monic polynomial $f(T)\in\mathbb{F}_{q}[T]$ of degree $d$ and an integer $m$ with $0\leq m<d,$ it is conventional to define the $m$-th “short interval” in $\mathbb{F}_{q}[T]$ around $f$ to be

[TABLE]

We are particularly interested in the small cases $m=0,1$. One naturally asks for assumptions on $f$ under which the following expected statement holds true:

(*)

For any partition $\lambda$ of $d$, the probability for an element of $I(f,m)$ to have factorization type $\lambda$ is $p_{\lambda}+O_{d}(q^{-1/2})$.

While a “sufficiently general” polynomial $f\in\mathbb{F}_{q}[T]$ will satisfy () with $m=1$ or even with $m=0$, one is interested in an explicit criterion that can be used to check that a specific $f$ satisfies (). Along these lines, Bank, Bary-Soroker, and Rosenzweig ([3]) prove the following

Theorem 1**.**

Let $f(T)\in\mathbb{F}_{q}[T]$ be a monic polynomial of degree $d$. Suppose $(q,d(d-1))=1$. Then $f$ satisfies () with $m=1$.*

A monic polynomial $f(T)\in\mathbb{F}_{q}[T]$ of degree $d$ is called a Morse polynomial if the equation $f^{\prime}(T)=0$ has exactly $d-1$ distinct roots over $\overline{\mathbb{F}_{q}}$, and the values of $f$ at them are all distinct. For a Morse polynomial $f(T)$, (*) holds with $m=0$ (see [5] or [6]). For $j\geq 0,$ the $j$-th Hasse derivative of a polynomial $f=\sum a_{i}T^{i}$ is defined as

[TABLE]

so $f$ has a zero of order at least $k$ at $\alpha$ precisely when $D^{j}f(\alpha)=0$ for all $j=0,...,k-1$.

The proposition below weakens the assumption $(q,d(d-1))=1$; it is stated as Proposition 7 in [6] and attributed to Jarden and Razon (Proposition 4.3 in [5]).

Proposition 2**.**

Let $f(T)\in\mathbb{F}_{q}[T]$ be a monic polynomial of degree $d$. Suppose $f^{\prime\prime}\neq 0$ and $(q,2d)=1$. Then for all but $O_{d}(1)$ values of $s\in\mathbb{F}_{q}$, the polynomial $f(T)+sT$ is a Morse polynomial, and hence satisfies () with $m=0$. In particular, $f$ satisfies () with $m=1$.

Remark 3*.*

The assumption $(q,2d)=1$ is essential in Proposition 2. Indeed, if $\operatorname{char}\mathbb{F}_{q}\mid d$, a polynomial of degree $d$ is never a Morse polynomial. Also, even if the condition $f^{\prime\prime}\neq 0$ is replaced by the weaker $D^{2}f\neq 0$ (see the paragraph preceding Proposition 7 in [6]), Proposition 2 still does not hold in characteristic $2$. For example, $T^{7}+sT$ is never a Morse polynomial when $q$ is a power of $2$, and in fact $f(T)=T^{7}$ fails to satisfy (*) with $m=1$.

The goal of this note is to give a criterion for a polynomial to satisfy (*), but allowing $(q,2d)>1$.

For a field $k$ and a polynomial $f(T)\in k[T]$, let $\widetilde{f}(x,y)$ denote the polynomial in $k[x,y]$ defined by

[TABLE]

We now state our main result.

Theorem 4**.**

Let $f(T)\in\mathbb{F}_{q}[T]$ be a monic polynomial of degree $d$. Suppose $D^{2}f\neq 0$, $\deg f^{\prime}\geq 1$, and the polynomials $\widetilde{f}(x,y)-f^{\prime}(x)$ and $\widetilde{f^{\prime}}(x,y)$ have no common factors besides possibly a power of $x-y$. Then for all but $d^{2}-d-1$ values of $s\in\mathbb{F}_{q}$, the polynomial $f(T)+sT$ satisfies () with $m=0.$*

Corollary 5**.**

Let $f(T)\in\mathbb{F}_{q}[T]$ be a polynomial as in Theorem 4. Then $f$ satisfies () with $m=1$.*

When $q$ is odd, Corollary 5 also follows from Corollary 1.4 in [4].

Example 6*.*

Theorem 4 and Corollary 5 apply to $f(T)=T^{12}+T^{3}\in\mathbb{F}_{q}[T]$ with $q$ a power of $2$; the gcd of $\widetilde{f}(x,y)-f^{\prime}(x)$ and $\widetilde{f^{\prime}}(x,y)$ is $x-y$.

Remark 7*.*

The statements of Theorem 4 and Corollary 5 would be false if one drops the gcd assumption. A counterexample is $f(T)=T^{7}$ in characteristic $2$. Thus Theorem 4 here corrects the false Theorem 1.3 in our previous version [9] of this paper.

To apply Theorem 4 to a specific polynomial, one has to compute the greatest common divisor of the two polynomials that appear in the statement; this task is computationally easy. In fact, based on modest numerical evidence, we state the following

Conjecture 8**.**

Let $k$ be a field and let $f\in k[T]$. Suppose $f^{\prime\prime}\neq 0$. Then the polynomials $\widetilde{f}(x,y)-f^{\prime}(x)$ and $\widetilde{f^{\prime}}(x,y)$ in $k[x,y]$ have no common factors.

In other words, we conjecture that the assumptions in Theorem 4 not only cover further examples when $q$ is a power of $2$ or $\operatorname{char}{\mathbb{F}_{q}}\mid d$ but are actually strictly weaker than the assumptions in Proposition 2.

The proof of Theorem 4 is based on the technique employed by Entin in a variety of problems solved in [4], with an extra ingredient (Lemma 9 below) developed by the author in an earlier work, concerning the irreducibility of the perturbations of a certain curve. Namely, for $s\in\mathbb{F}_{q},$ we set up a generically étale map $\varphi_{s}\colon\mathbb{A}^{1}\to\mathbb{A}^{1}$ of degree $d$ such that for any $b\in\mathbb{A}^{1}(\mathbb{F}_{q})$ with $d$ preimages over $\overline{\mathbb{F}_{q}}$, the conjugacy class in $S_{d}$ that the action of the Frobenius $\text{Fr}_{q}$ on $\varphi_{s}^{-1}(b)$ gives rise to has cycle structure corresponding to the factorization type of the polynomial $f(T)+sT+b$ in $\mathbb{F}_{q}[T]$. The statement will then follow by the Chebotarev density theorem for function fields, once we show that the monodromy group of $\varphi_{s}$ is the full symmetric group $S_{d}$. To this end, we check the criterion proven in [2].

2 The proof

We say that a polynomial $f(T)\in\overline{\mathbb{F}_{q}}[T]$ is “affine linearized” if it has the form $f(T)=\sum a_{i}T^{p^{i}}+f(0)$, where $p=\text{char}\,\mathbb{F}_{q}$.

Lemma 9**.**

Let $f(T)\in\mathbb{F}_{q}[T]$ be a polynomial of degree $d$, which is not affine linearized. For all but at most $d-1$ values of $s\in\overline{\mathbb{F}_{q}}$, the polynomial $\widetilde{f}(x,y)+s$ is geometrically irreducible.

Proof.

The author has proven this as Lemma 19 in [8]. We sketch the proof here as well. First, an elementary undetermined coefficients argument shows that if $f$ is not affine linearized, the polynomial $\widetilde{f}(x,y)$ cannot be written as $Q(h(x,y))$ for a polynomial $Q$ with $\deg Q>1$. Then we apply the main result of [7]. ∎

Lemma 10**.**

Let $f\in\mathbb{F}_{q}[T]$ be a polynomial that satisfies the hypotheses of Theorem 4. Then for all but $d^{2}-2d$ values of $s\in\overline{\mathbb{F}_{q}}$, there exists a $b\in\overline{\mathbb{F}_{q}}$ such that the polynomial $f(T)+sT+b$ has a unique root of multiplicity $2$ and $d-2$ simple roots over $\overline{\mathbb{F}_{q}}$.

Proof.

Let

[TABLE]

then $|B_{1}|\leq d-2$ and for any $s\notin B_{1}$ and $b\in\overline{\mathbb{F}_{q}}$, the polynomial $f(T)+sT+b$ has no roots over $\overline{\mathbb{F}_{q}}$ of multiplicity $3$ or more.

Define

[TABLE]

By Bézout’s theorem, there are at most $(d-1)(d-2)$ pairs $(\alpha,\beta)\in(X_{1}\cap X_{2})(\overline{\mathbb{F}_{q}})$ with $\alpha\neq\beta$. Let

[TABLE]

then $|B_{2}|\leq(d-1)(d-2).$ Suppose $\alpha\neq\beta$ in $\overline{\mathbb{F}_{q}}$ are both roots of multiplicity at least $2$ of some polynomial $f(T)+sT+b$ with $s,b\in\overline{\mathbb{F}_{q}}$. Explicitly, $f(\alpha)+s\alpha+b=f(\beta)+s\beta+b=0$ and $f^{\prime}(\alpha)+s=f^{\prime}(\beta)+s=0$. These imply $(\alpha,\beta)\in(X_{1}\cap X_{2})(\overline{\mathbb{F}_{q}})$, hence $s\in B_{2}$.

The set $B:=B_{1}\cup B_{2}$ satisfies $|B|\leq d^{2}-2d$. Let $s\notin B$. Choose $\alpha\in\overline{\mathbb{F}_{q}}$ such that $f^{\prime}(\alpha)+s=0$ and set $b:=-f(\alpha)-s\alpha$. The polynomial $f(T)+sT+b$ satisfies the requirement. ∎

Proof of Theorem 4..

For any $s\in\mathbb{F}_{q}$, define

[TABLE]

The projection $\Omega_{s}\to\mathbb{A}^{1}_{t}$, $(t,b)\mapsto t$ is an isomorphism and the map $\varphi_{s}\colon\Omega_{s}\to\mathbb{A}^{1}_{b}$, $(t,b)\mapsto b$ is a generically étale morphism of degree $d$ between geometrically irreducible $\mathbb{F}_{q}$-varieties.

The polynomial $f(T)$ is not affine linearized, since $\deg f^{\prime}\geq 1$. Combining Lemma 9 and Lemma 10, there exists a set $B\subset\mathbb{F}_{q}$ of cardinality at most $d^{2}-d-1$ such that for any $s\in\mathbb{F}_{q}-B$, the following hold:

• (i)

the polynomial $\widetilde{f}(x,y)+s$ is geometrically irreducible, and

• (ii)

there exists a $b\in\overline{\mathbb{F}_{q}}$ such that the polynomial $f(T)+sT+b$ has a unique root of multiplicity $2$ and $d-2$ simple roots over $\overline{\mathbb{F}_{q}}$.

Let $s\in\mathbb{F}_{q}-B$. By (ii), the fiber of $\varphi_{s}$ over some $b\in\overline{\mathbb{F}_{q}}$ consists of $d-1$ points over $\overline{\mathbb{F}_{q}}$, with $\varphi_{s}$ being étale at $d-2$ of them. Thus the assumption of Proposition 3 in [2] is satisfied. Moreover, the complement of the diagonal in $\Omega_{s}\times_{\mathbb{A}^{1},\varphi_{s}}\Omega_{s}$ is isomorphic to

[TABLE]

It is nonempty because we can pick a $\beta\in\overline{\mathbb{F}_{q}}$ such that $f^{\prime}(\beta)+s\neq 0$, set $b:=-f(\beta)-s\beta$ (so $\beta$ is a simple root of $f(T)+sT+b$), let $\gamma$ be any other root of $f(T)+sT+b$ (note: $\deg f\geq 2$, since $\deg f^{\prime}\geq 1$) and observe that $(\beta,\gamma)\in\Delta^{c}$. Thus $\Delta^{c}$ is a nonempty open subset of $V(\widetilde{f}(x,y)+s)$ and by (i) is geometrically irreducible. Therefore the assumption of Proposition 2 in [2] is satisfied as well. We conclude that the geometric monodromy group of the map $\varphi_{s}$ is the full $S_{d}$.

Let $U$ be a dense open subset of $\mathbb{A}^{1}_{b}$ such that $\varphi_{s}|_{\varphi_{s}^{-1}(U)}\colon\varphi_{s}^{-1}(U)\to U$ is finite and étale. The statement now follows from Theorem 3 in [4], which is a version of the Chebotarev density theorem for function fields. ∎

Remark 11*.*

We can also deduce Corollary 5 directly from the criterion in [2], without going through Theorem 4. Namely, consider

[TABLE]

and $\varphi\colon\Omega\to\mathbb{A}^{2}_{s,b}.$ If $\Delta$ and $\Delta^{\prime}$ denote the diagonals of $\Omega\times_{\mathbb{A}^{2}_{s,b}}\Omega$ and $\mathbb{A}^{1}_{t}\times\mathbb{A}^{1}_{t}$ respectively, then the map $\Omega\times_{\mathbb{A}^{2}_{s,b}}\Omega-\Delta\to\mathbb{A}^{1}_{t}\times\mathbb{A}^{1}_{t}-\Delta^{\prime}$ is an isomorphism, so the source is geometrically irreducible. The existence of $(s,b)\in\mathbb{A}^{2}(\overline{\mathbb{F}_{q}})$ such that $f(T)+sT+b$ has a unique root of multiplicity $2$ and $d-2$ simple roots over $\overline{\mathbb{F}_{q}}$ follows from Lemma 10.

Acknowledgments

I thank Bjorn Poonen and Alexei Entin for comments.