# Level sets of the polynomial function $𝑥^2+𝑦^2−𝑥+𝑦−𝑎𝑥𝑦$ over $\mathbb{F}_𝑝$

Let 𝑝 be an odd prime and assume $$𝑥^2+ax+1$$ is irreducible over the field $$\mathbb{F}_p$$. The polynomial function

$$\Psi:\mathbb{F}_p^2⟶\mathbb{F}_p,\quad (x,y)\mapsto 𝑥^2+𝑦^2−x+y-axy$$

is surjective, as proved here: Image of a polynomial function $x^2+y^2-x+y-axy$ over $\mathbb{F}_p$.

I would like to compute a set of representatives of the classes of the kernel of $$\Psi$$ (i.e., the relation $$\ker(\Psi)=\{(x,y,t,w)\in \mathbb{F}_p^4, \, \Psi(𝑥,𝑦)=\Psi(t,w)\})$$. So basically I would like to have an explicit set of $$p-1$$ elements of $$\mathbb{F}_p^2$$ that take on all the nonzero values in $$\mathbb{F}_p$$ when you apply $$\Psi$$. (Finding a solution to $$\Psi(x,y)=0$$ is obvious.)

Is there a way to do this in general, regardless of the value of $$p$$?

Here is a solution for $$\frac{p-1}{2}$$ of the level sets. Let $$\varphi=x^2+y^2-x+y-axy$$, and assume that $$a\not=\pm 2$$.

Following the suggestion in https://mathoverflow.net/q/356936 we first apply the change of variables given by $$x= z+(a/2)y$$ to obtain $$\varphi=z^2-z-b(y^2+(1+a/2)^{-1}y),$$ where $$b=a^2/4-1$$. Then letting $$u=z-1/2$$, $$v=y+(2+a)^{-1}$$, and $$c=\frac{-1}{4}\left(\frac{1-a/2}{1+a/2}\right)$$ we have $$\varphi=u^2-bv^2+c.$$

To compute the level sets you need to compute the fibres $$\varphi^{-1}(d)$$ for an image point $$d$$. If $$d-c$$ is a quadratic residue, then $$d-c=e^2$$ and so letting $$s=u/e$$ and $$t=v/e$$ we obtain: $$s^2-bt^2=1.$$

Having all solutions $$(s,t)$$ determines the solutions $$(u,v)$$, in turn the solutions $$(y,z)$$, and in turn the solutions $$(x,y)$$.

So it suffices to find the solutions to $$s^2-bt^2=1.$$ But such solutions are explicitly determined in:

Tekcan, Ahmet, The number of rational points on conics $$C_{p,k}:x^2−ky^2=1$$ over finite fields $$\mathbb{F}_p$$. Int. J. Math. Sci. 1 (2007), no. 2, 150–153.

MathSciNet Summary:

"Let $$p$$ be a prime number, $$\mathbb{F}_p$$ be a finite field, and let $$k\in \mathbb{F}_p^*$$. In this paper, we consider the number of rational points on conics $$C_{p,k}: x^2−ky^2=1$$ over $$\mathbb{F}_p$$. We prove that the order of $$C_{p,k}$$ over $$\mathbb{F}_p$$ is $$p−1$$ if $$k$$ is a quadratic residue mod $$p$$ and is $$p+1$$ if $$k$$ is not a quadratic residue mod $$p$$....''

I am not sure what happens when $$d-c$$ is a non-residue (I haven't tried), but since $$\varphi$$ is onto and $$(p-1)/2$$ of the non-zero values are quadratic residues, this gets you roughly half way.

• I am surprised that Tekcan could publish this result, which is a standard exercise in number theory (for any diagonal conic over $\mathbb{F}_p$, in any number of variables). This was also discussed multiple times on this site, see e.g. my response for mathoverflow.net/questions/65183/… – GH from MO Sep 2 '20 at 21:58