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I understand that the values of the Hermite constant for $1 \leq n \leq 8$ and $n=24$ have been determined exactly. For example, Lagrange proved for $n=2$ the value of the Hermite constant is $\gamma_n = \sqrt{\frac{4}{3}}$, and this value is achieved with the unique extremal form

$$ q(x,y) = x^2 + xy + y^2. $$

However, I can't find proof in any literature for any values of the Hermite constant. Can anybody direct me towards some proof of any values of the Hermite constant?

Edit: For context, let $f: \mathbb{R}^n \to \mathbb{R}$ be a quadratic form, i.e. for $\mathbf{x} = (x_1, \cdots, x_n) \in \mathbb{R}^n$ then $f(\mathbf{x}) = \sum_{ij} f_{ij} x_i x_j$. Then we define the Hermite variable in $n$ dimensions:

$$ \gamma_n(f) = \frac{\inf_{\mathbf{x}}\{f(\mathbf{x}): \mathbf{x} \in \mathbb{Z}^n - \{\mathbf{0}\}\}}{disc(f)^{1/n}}. $$

Then the Hermite constant in $n$ dimensions is the maximal value of this variable over all possible quadratic forms, i.e.

$$ \gamma_n = \sup_{f}\{\gamma_n(f)\}. $$

Chris
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  • Please add some context what the Hermite constant is. – Peter Mar 06 '18 at 14:13
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    @Peter Let $f$ be a quadratic form of $n^2$ variables, i.e. $f(x) = \sum_{ij} f_{ij} x_i x_j$ for a real $n$ dimensional vector $x$. Define the variable

    $$ \gamma_n (f) = \frac{\inf_{x}{f(x): x \in \mathbb{Z}^n - {\mathbf{0}}}}{disc(f)^{1/n}}. $$

    Then the Hermite constant is the maximum value over all possible quadratic forms, i.e.

    $$ \gamma_n = \sup_{f} {\gamma_n (f)} $$

     – Chris Mar 06 '18 at 14:17
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    @Peter it certainly is - but finding the Hermite constant for a dimension $n$ corresponds to finding the maximum lattice packing density for a hypersphere packing, so it's a very important constant! – Chris Mar 06 '18 at 14:34
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    Interesting! (+1) – Peter Mar 06 '18 at 14:36

1 Answers1

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Blichfeldt H. F. The minimum values of positive quadratic forms in six, seven and eight variables. — Math. Z., 1934—1935, 39, S. 1—15.

Watson G. L. On the minimum of a positive quadratic form in n (< 8) variables (verification of Blichfeldt's calculations). — Proc. Cambridge Phil. Soc, 1966, 62, p 719.

Maxim Lyalin
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