## The number of minimum points of a positive quadratic form.(English)Zbl 0215.34901

Let $$f_n$$ be a positive-definite $$n$$-ary quadratic form, with real coefficients, and for sirnplicity let its minimum be 1. Denote by $$s(f_n)$$ the number of pairs $$(\pm x)$$ of integer points $$(x =(x_1, \ldots, x_n)$$) at which $$f_n(x) = 1$$. I prove that
$s(f_n) \le 3, 6, 12, 20, 36, 63, 120, 136\quad\text{for }n=2,\ldots, 9\text{ respectively}. \tag{1}$
There exist forms $$E_2,\ldots, E_8$$ such that equality holds in (1) if and only if $$f_n$$ is equivalent to $$E_n$$. $$E_2,\ldots, E_8$$ are absolutely extreme.
When the case of equality is excluded, we can in some cases do much better than subtract 1 from the bound in (1): more precisely, for $$f_n$$ not equivalent to $$E_n$$
$s(f_n) \le 10, 16, 30, 46, 75\quad\text{for }n = 4,\ldots, 8. \tag{2}$
(2) is also best possible, with equality for imperfect $$f_n$$ when $$n = 5, 7$$ or $$8$$. If the case of equality in (2) is excluded we have $$s(f_n) \le 28, 42$$ for $$n = 6, 7$$; and probably something similar for $$n = 8$$.
The argument is elementary but somewhat laborious. Suppose it established, for some $$k <n$$, that either $$s(f_n)$$ satisfies the inequality to be proved or $$f_n$$ has a $$k$$-ary section $$f_k$$ with $$s_k$$ pairs of minimum points, $$s_k$$ as large as possible. Then I border $$f_k$$ so as to obtain $$f_n$$, if $$n =k + 1$$, or a $$(k + 1)$$-ary section $$f_{k+1}$$ with as many minimum points as possible if $$n\ge k + 2$$.
The main tool needed for the inductive argument outlined above is (the case $$n = k + 1$$ of) a bound for the determinant $$\Delta$$ of $$n$$ minimum points of $$f_n$$. The inequality I use is
$\Delta\le 1, 2, 4, 8\quad\text{for }n\le 4\text{ only when }f_4\text{ is equivalent to }E_4. \tag{3}$
Unfortunately, after writing the present paper, I found a further improvement on (3) which would have shortened the argument considerably. There is a perfect form $$B_6$$, with $$s(B_6) = 30$$, such that equality in (3), for $$n =6$$, implies $$f_6$$ equivalent to $$B_6$$. Further, for $$n=7$$, (3) improves to $$\Delta \le 4$$ unless $$f_7$$ is equivalent to $$E_7$$, with $$s(E_7) =63$$. These results are proved in [Mathematika 18, 60–70 (1971; Zbl 0219.10032)].
I have used them, in a paper [Proc. Lond. Math. Soc. (3) 24, 625–646 (1972; Zbl 0233.10012)], to obtain bounds for $$s(f_n)$$, smaller than those given above, when $$f_n$$ is restricted to have no three minimum points with sum $$(0,\ldots,0)$$. As a corollary of the work needed for $$n\le 9$$ show that $$s(f_n) <2^{n-2} + 8$$ for $$n\ge 10$$. This result, though hard to improve, is probably very weak even for $$n = 10$$, and certainly very weak indeed for large $$n$$.
Reviewer: G. L. Watson

### MSC:

 11H50 Minima of forms

### Citations:

Zbl 0219.10032; Zbl 0233.10012