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The product of \(n\) real homogeneous linear forms. (English) Zbl 0034.31601

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Number theory
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[1] Minkowski proved inGeometrie der Zahlen, (Leipzig, 1910), § 40 that one can find integersu 1, ...u n , not all zero, satisfying |x 1 |+...+|x n |n!) 1/n and (1) follows from this by the inequality of the arithmetic and geometric means.
[2] H. F. Blichfeldt,Monatshefte für Math. und Phys., 43 (1936), 410–414. · Zbl 0013.34503 · doi:10.1007/BF01707621
[3] R. A. Rankin,Proc. Kon. Ned. Akad. v. Wet., Amsterdam, 51 (1948), 846–853 (848).
[4] C. A. Rogers,Journal London Math. Soc., 24 (1949), 31–39. · Zbl 0032.01504 · doi:10.1112/jlms/s1-24.1.31
[5] H. F. Blichfeldt,Monatshefte für Math. und Phys., 48 (1939), 531–533. Blichfeldt considers the linear forms $$w_{k1} x_1 + ... + w_{kn} x_n ,{\(\backslash\)text{ }}k{\(\backslash\)text{ = 1,}}...{\(\backslash\)text{, }}n{\(\backslash\)text{ ,}}$$ of determinant \(\Delta\), wherew 11,...,w 1n is a basis of a totally real algebraic field of discriminant \(\Delta\) andw 1k1 , ...,w kn ,k=2, ..., n are conugate bases of the conjugate fields. But Blichfeldt proves, without use of his assumption concerning the nature ofw 11, ...,w nn , that for any integerm>1 there are integersu 1, ...,u n not all zero such that $$\(\backslash\)prod _{st} = \(\backslash\)mathop \(\backslash\)prod \(\backslash\)limits_{k = 1} (w_{k1} u_1 + ... + w_{kn} u_n )\^2 \(\backslash\)leqslant \(\backslash\)Delta \^2 \(\backslash\)left[ {\(\backslash\)frac{1}{{\(\backslash\)pi n(m - 1)}}} \(\backslash\)right]\^n (m - 1)\^2 [1 \(\backslash\)cdot 2\^2 \(\backslash\)cdot 3\^3 \(\backslash\)cdot {\(\backslash\)text{ }} \(\backslash\)cdot \(\backslash\)cdot \(\backslash\)cdot {\(\backslash\)text{ }} \(\backslash\)cdot m\^{\(\backslash\)mathop m\(\backslash\)limits_ \(\backslash\)cdot } ]$$ Takingm=[n logn] and lettingn tend to infinity one obtains (7). · Zbl 0021.38805 · doi:10.1007/BF01696206
[6] H. Minkowski,Geometrie der Zahlen, (1910), § 42. · JFM 41.0948.05
[7] H. F. Blichfeldt, (1939),—-loc. cit. 43, 410–414. · Zbl 0013.34503 · doi:10.1007/BF01707621
[8] This result should be compared with the result obtained by C.L. Siegel (Annals of Math., 46 (1945), 302–312) for the case when 0<z 1 <z 2 <...<z m . The proofs are quite different.
[9] We work throughout with the principal values of our integrals; we useg 2 (x) to denote (g(x))2 andg’(x) to denote the derivative ofg(x).
[10] See Hardy, Littlewood and Pólya,Inequalities, (Cambridge 1934), § 10.12, page 276.
[11] H. F. Blichfeldt,Trans. American Math. Soc., 40 (1914), 227–256. We restate Blichfeldt’s Theorem 1 (page 228), in the form he considers in § 7 (page 230). · JFM 45.0314.01 · doi:10.1090/S0002-9947-1914-1500976-6
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