Hi,


I am interested in the following question. Let $F(x\_1, x\_2, x\_3, x\_4)$ be a quadratic form in four variables with integer coefficients. Let $B > 0$ be a parameter. Define $N\_1(F,B)$ to be the number of rational solutions to the equation $F(x\_1, x\_2, x\_3, x\_4) = 0$, such that $|x\_i| \leq B$ for $i = 1,2,3,4$. I am looking for an upper bound of the form
$$\displaystyle N\_1(F,B) \ll\_\epsilon B^{\theta + \epsilon},$$
where the constant $\theta < 2$. Is there any such results known? One can assume that $F$ is absolutely irreducible and if necessary, assume that $F$ is non-singular.



Rational roots to quadratic forms in 4 variables
I'm afraid that, even if we assume (as suggested in A.Meyerowitz's comment) that the intention is *integer* solutions of $F(x\_1,x\_2,x\_3,x\_4) = 0$, the number of solutions with $\max\_i |x\_i| \leq B$ will grow at least as $B^2$ as long as there's any nonzero solution, and sometimes the growth will even be a bit faster.


The usual heuristics suggest that if there's no local obstruction (such as there is for W.Zudilim's example of $X\_1^2 + X\_2^2 + X\_3^2 + X\_4^2$, with no nontrivial zeros even over $\bf R$) then the number solutions up to height $B$ should grow as $B^2$: there are about $B^4$ candidates, and for each one of them $F(x\_1,x\_2,x\_3,x\_4)$ has size at most $B^2$, so if we imagine they're randomly distributed then about $B^4 / B^2 = B^2$ values should be zero.


It is not hard to prove that for some choices of quadratic form $F$, even nonsingular ones, the count is $\gg B^2$. Namely suppose $F(x\_1,x\_2,x\_3,x\_4)$ has the form $Q(x\_1,x\_2) - Q(x\_3,x\_4)$ for some quadratic $Q$. Then the line $(x\_1,x\_2,x\_1,x\_2)$ already gives $B^2$ zeros. Geometrically, the two rulings of the quadric $Q=0$ in ${\bf P}^3$ are defined over ${\bf Q}$, and any line — not just the trivial $\{(x\_1:x\_2:x\_1:x\_2)\}$ — will give some positive multiple of $B^2$. Using all the lines in the ruling one finds that in fact the correct growth rate is $B^2 \log B$.


If I remember right it is known that for an unobstructed smooth quadric in ${\bf P}^3$ whose rulings are not rational but Galois conjugate over ${\bf Q}$, the counting function $N\_1(F,B)$ is asymptotically proportional to $B^2$ as the heuristics suggested.


This is all consistent with Manin's conjecture: the quadrics with rational rulings are precisely those for which the rational subgroup of the Néron-Severi group has rank 2 rather than 1, which accounts for the extra factor of $\log B$.


