Absolute irreducibility of a polynomial over a finite field

Question

Let $k$ be a positive integer, $\mathbb{F}_q$ be the finite field with $q$ elements. Consider the polynomial $f(X, Y) = 1 + X^k + Y^k + X^{2k} + Y^{2k}$ over $\mathbb{F}_q$. How to verify whether it is absolutely irreducible (irreducible over $\overline{\mathbb{F}_q}$) or not?

The context is the following: I am considering existence of rational points over finite fields. In that direction I wanted to use Andre Weil's result as in "Numbers of solutions of equations in finite fields". In order to use Weil's results, I need to first prove absolute irreducibility of the polynomial.

Answer 1

Your attempt is not quite right: if $f\in F[x,y]$ factors as $f=f_1f_2$, then $V(f)$ is singular along $V(f_1)\cap V(f_2)\subset\Bbb A^2$, but these subvarieties need not meet in $\Bbb A^2$ (e.g. $x(x+1)$). The fix is to consider $f^h\in F[x,y,z]$, the homogenization of $f$ with respect to $z$, and look at $V(f^h)\subset\Bbb P^2$: if $f$ factors nontrivially as $f=f_1f_2$, then $f^h=f_1^hf_2^h$, and $V(f_1^h)\cap V(f_2^h)\subset\Bbb P^2$ must be nonempty by Bezout, so if $f$ is reducible then $V(f^h)$ must be singular. We can check this by the Jacobian criterion.

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Homogenizing our $f$, we get $f^h=z^{2k}+x^kz^k+y^kz^k+x^{2k}+y^{2k}$ which has Jacobian $$\begin{pmatrix} kx^{k-1}(2x^k+z^k) & ky^{k-1}(2y^k+z^k) & kz^{k-1}(x^k+y^k+2z^k) \end{pmatrix}$$ and $v\in V(f^h)$ is singular iff all the entries of the Jacobian vanish at $v$. Now we do some casework:

• If $p\mid k$, then all entries of our Jacobian vanish identically, and indeed our original polynomial is $(1+x^{k/p}+y^{k/p}+x^{2k/p}+y^{2k/p})^p$, which isn't irreducible. From now on, assume $(k,p)=1$.
• If $p=2$, then the entries of our Jacobian vanish iff $x^{k-1}z^k$, $y^{k-1}z^k$, and $x^kz^{k-1}+y^kz^{k-1}$ do. So either $x=y=0$ or $z=0$. In the first case, $[0:0:1]$ is not on our curve, but in the second case we do have singular points: $[\zeta:1:0]$ for $\zeta$ a $2k^{th}$ root of unity. From now on, assume $p\neq 2$.
• If any two of $x,y,z$ are zero, then the third must be zero as well, which gives that our curve is nonsingular. Therefore we may assume that either one or none of $x,y,z$ are zero.
• If $z=0$ and $x,y\neq 0$, then the first two entries of our Jacobian don't vanish anywhere.
• If $x=0$ and $y,z\neq 0$ then we must have $2y^k+z^k=0$ and $y^k+2z^k=0$, which implies $(2y^k+z^k)-(y^k+2z^k)=y^k-z^k=0$. For such a point to be on our curve, it must also satisfy $z^{2k}+y^kz^k+y^{2k}=0$, which gives us that $3y^{2k}=3z^{2k}=0$. When $p\neq 3$, we get that $y=z=0$ and we have don't have singular points. When $p=3$, the singular points with $x=0$ are of the form $[0:\zeta:1]$ for $\zeta$ a $k^{th}$ root of unity. The case $y=0$ and $x,z\neq 0$ follows by symmetry.
• If none of $x,y,z$ are zero, then we must have $2x^k+z^k=2y^k+z^k=x^k+y^k+2z^k=0$. But then two times the last equation minus the first two equations implies $2z^k=0$, so there are no singular points here.

What we've shown so far is that if $p\neq 2,3$ and $(k,p)=1$, then $f$ is irreducible, while if $p\mid k$ then $f$ is a $p^{th}$ power and reducible. The cases $p=2$ and $p=3$ require more investigation - our curve has singularities there, but this does not automatically imply that $f$ is reducible (consider $y^2z=x^3+x^2z$, for instance).

When $p=2$, let $\alpha\in\overline{\Bbb F_2}$ solve $t^2+t+1$. Then we have $$(\alpha x^k+\alpha y^k+1)((\alpha+1)x^k+(\alpha+1)y^k+1)=x^{2k}+y^{2k}+x^k+y^k+1,$$ and $f$ is reducible.

When $p=3$, if $k$ is even, we may find the following factorization: $$(x^k+\sqrt{2}x^{k/2}y^{k/2}+y^k-1)(x^k-\sqrt{2}x^{k/2}y^{k/2}+y^k-1)=x^{2k}+y^{2k}+x^k+y^k+1.$$ I claim that when $k$ is odd, $f$ is irreducible. By considering $f$ over $F[x^{1/2},y^{1/2}]$, we have $f$ factors as before, so it suffices to prove that $x^{2l}\pm\sqrt{2}x^ly^l+y^{2l}-1$ is irreducible in $k[x,y]$. The Jacobian of the homogenization of these polynomials are $$\begin{pmatrix} lx^{l-1}(2x^l\pm\sqrt{2}y^l) & ly^{l-1}(2y^l\pm\sqrt{2}x^l) & -2lz^{2l-1} \end{pmatrix},$$ whose entries simultaneously vanish iff $x=y=z=0$ or $2x^l\pm\sqrt{2}y^l=2y^l\pm\sqrt{2}x^l=z=0$. The first case can't happen, while the second case gives $z=0$ and $(2\pm\sqrt{2})(x^l+y^l)=0\Rightarrow x^l+y^l=0$, which is not satisfied at any point on $V(x^{2l}\pm\sqrt{2}x^ly^l+y^{2l}-z^{2l})$.

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In summary: $f$ is absolutely irreducible iff $(k,p)=1$ and either 1) $p=3$ while $k$ is odd or 2) $p>3$.

Answer 2

Let $k$ be a positive integer, and let $F_q$ be the finite field with $q$ elements.

First the characteristic of $F_q$ is not $2$. Based on the suggestion by @Jyrki Lahtonen, it follows that if the curve defined the equation $f(X,Y) = 1 + X^k + y^k + X^{2k} + Y^{2k}$ is nonsingular it is absolutely irreducible, otherwise, the irreducible factors of the equation over the algebraic closure $\bar{F_q}$ intersect, due to Bezout's theorem, necessarily in singular points. Furthermore, if $k$ and $q-1$ are relatively prime, then homomorphism $x \mapsto x^k: F_q \to F_q$ is a surjection, so by change of variables it suffices to treat the case when $f(X,Y) = 1 + X + Y + X^2 + Y^2$. The partial derivative with respect to $X$, $Y$ respectively are is $1 + 2 X$ are $1 + 2 Y$, and they vanish at the point $(-1/2, -1/2)$, but the value of $f$ at this point is $1 + (-1/2) + (-1/2) + 1/4 + 1/4 = 1/8$ which is nonzero, because we have assumed that characteristic of $F_q$ to be not $2$ (so that the point does not lie on the curve). Since the equations $f = 0,f_x = 0 , f_y = 0$ have no common zero, it follows that the curve is nonsingular, and therefore by Weil's theorem, the number $N$ of $F_q$ points satisfies $$ | N- q | \leq (d-1)(d-2) \sqrt{q} + d $$ where $d$ is the degree of the curve, which in our case is $2$. Therefore, $$ q - 2 \leq N \leq q + 2 $$ so that if $q > 2$, we have at least one solution.

Now suppose the characteristic of $F_q$ is $2$. Then $f_x = 1$ and $f_y = 1$, so that $f(X,Y)$ is nonsingular too, and the above estimate remains valid.

Please correct me if there is a gap in the proof.