Hi All,


Let $F$ be a finite field, $\lambda\in F$, and $$p\_\lambda (x,y,z)=\left|\begin{array}{ccc}x & y & z \\ y & z & x +\lambda z \\ z& x+\lambda z & y+\lambda x+\lambda ^2z \end{array}\right|.$$
Can one choose $\lambda$ so that $(0,0,0)$ is the only zero in $F^3$ for $p\_\lambda (x,y,z)$?
Also, I'm wondering if the matrix that defines $p\_\lambda$ looks familiar to anyone. Thanks!



zeros of a homogeneous polynomial
Let $t$ be a root of $t^3 + \lambda t + 1 = 0$. Suppose $\lambda$ can be chosen so this polynomial is irreducible. Then the cubic decomposes into three lines, one of which is given by $x = ty + z/t$ (and the other two are the Galois conjugates). Now it's easy to see this line has no $\mathbb{F}\_p$-rational points (else $t$ would satisfy an equation of degree less than $3$). Similarly the others don't have rational points either. So all you have to do is ensure that this poly is irreducible, and also that $4\lambda^3 + 27$ is nonzero, so you don't have a singular point over the base field.


