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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