Hermite versus Minkowski Jacques Martinet Universit´ e de Bordeaux, IMB/A2X Luminy, November–December, 2009 Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 1 / 12
The notation In this talk, we use the language of (Euclidean) lattices, more suitable here than that of (positive, definite) quadratic forms. E stands for an n -dimensional Euclidean space The “ norm ” of x ∈ E is N ( x ) = x · x ( = � x � 2 ), the Gram matrix of a set of vectors x 1 , . . . , x n is Gram ( x i ) = ( x i · x j ) . A lattice is a discrete subgroup Λ of E of maximal rank. The determinant of Λ is det (Λ) = det ( Gram ( B )) where B is any Z -basis for Λ . Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 2 / 12
The notation In this talk, we use the language of (Euclidean) lattices, more suitable here than that of (positive, definite) quadratic forms. E stands for an n -dimensional Euclidean space The “ norm ” of x ∈ E is N ( x ) = x · x ( = � x � 2 ), the Gram matrix of a set of vectors x 1 , . . . , x n is Gram ( x i ) = ( x i · x j ) . A lattice is a discrete subgroup Λ of E of maximal rank. The determinant of Λ is det (Λ) = det ( Gram ( B )) where B is any Z -basis for Λ . Since Λ is discrete in E , we may define the minimum m (Λ) = m of Λ min Λ = x ∈ Λ � { 0 } N ( x ) min and then its Hermite invariant m (Λ) γ (Λ) = det (Λ) 1 / n Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 2 / 12
The notation In this talk, we use the language of (Euclidean) lattices, more suitable here than that of (positive, definite) quadratic forms. E stands for an n -dimensional Euclidean space The “ norm ” of x ∈ E is N ( x ) = x · x ( = � x � 2 ), the Gram matrix of a set of vectors x 1 , . . . , x n is Gram ( x i ) = ( x i · x j ) . A lattice is a discrete subgroup Λ of E of maximal rank. The determinant of Λ is det (Λ) = det ( Gram ( B )) where B is any Z -basis for Λ . Since Λ is discrete in E , we may define the minimum m (Λ) = m of Λ min Λ = x ∈ Λ � { 0 } N ( x ) min and then its Hermite invariant m (Λ) γ (Λ) = det (Λ) 1 / n Note that γ (Λ) only depends on the similarity class (“the shape”) of Λ . Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 2 / 12
An inequality of Hermite For any basis B of Λ , let � 1 / n � N ( e 1 ) . . . N ( e n ) H B (Λ) = and H (Λ) = min H B (Λ) ; det (Λ) B Again, H (Λ) only depends on the similarity class of Λ . Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 3 / 12
An inequality of Hermite For any basis B of Λ , let � 1 / n � N ( e 1 ) . . . N ( e n ) H B (Λ) = and H (Λ) = min H B (Λ) ; det (Λ) B Again, H (Λ) only depends on the similarity class of Λ . The following theorem was proved by Hermite around 1850: � ( n − 1 ) / 2 � 4 Theorem. We have H (Λ) ≤ . 3 Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 3 / 12
An inequality of Hermite For any basis B of Λ , let � 1 / n � N ( e 1 ) . . . N ( e n ) H B (Λ) = and H (Λ) = min H B (Λ) ; det (Λ) B Again, H (Λ) only depends on the similarity class of Λ . The following theorem was proved by Hermite around 1850: � ( n − 1 ) / 2 � 4 Theorem. We have H (Λ) ≤ . 3 � ( n − 1 ) / 2 � 4 Corollary. (H ERMITE , August 6th, 1845). We have γ (Λ) ≤ . 3 Set ( γ n ) = sup dim Λ= n γ (Λ) . The values of ( γ n ) are known for n ≤ 8 and n = 24. Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 3 / 12
An inequality of Minkowski (successive minima) For any system B of independent vectors e 1 , . . . , e n of Λ , let � 1 / n � N ( e 1 ) . . . N ( e n ) M B (Λ) = and M (Λ) = min M B (Λ) ; det (Λ) B thus we consider vectors of Λ which constitute a basis for E , but not necessarily for Λ . Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 4 / 12
An inequality of Minkowski (successive minima) For any system B of independent vectors e 1 , . . . , e n of Λ , let � 1 / n � N ( e 1 ) . . . N ( e n ) M B (Λ) = and M (Λ) = min M B (Λ) ; det (Λ) B thus we consider vectors of Λ which constitute a basis for E , but not necessarily for Λ . In his 1896 book Geometrie der Zahlen , Minkowski proved: Theorem. We have M (Λ) ≤ γ n . Also, using his argument consisting in bounding by 1 the density of any sphere packing, Minkowski gave a linear bound for γ n (whereas Hermite’s is exponential), which yields pretty good bounds for H . Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 4 / 12
An inequality of Minkowski (successive minima) For any system B of independent vectors e 1 , . . . , e n of Λ , let � 1 / n � N ( e 1 ) . . . N ( e n ) M B (Λ) = and M (Λ) = min M B (Λ) ; det (Λ) B thus we consider vectors of Λ which constitute a basis for E , but not necessarily for Λ . In his 1896 book Geometrie der Zahlen , Minkowski proved: Theorem. We have M (Λ) ≤ γ n . Also, using his argument consisting in bounding by 1 the density of any sphere packing, Minkowski gave a linear bound for γ n (whereas Hermite’s is exponential), which yields pretty good bounds for H . From dimension 5 onwards, the index [Λ : Λ ′ ] may be > 1 and H (Λ ) may indeed be strictly larger than M (Λ ). Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 4 / 12
Bounds for H It is easy to prove that up to dimension 4, representatives (suitably chosen if n = 4) of the successive minima constitute a basis for Λ , but discrepancy between H and M systematically occurs beyond dimension 4. In his paper Die Reduktionstheorie der positiven quadratischen Formen , Acta Math. 96 (1956), 265–309, Van des Waerden proves for H an inductive � 5 � n − 4 ; URMANN ) to H formula, which actually boils down (Achill S CH ¨ M ≤ 4 Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 5 / 12
Bounds for H It is easy to prove that up to dimension 4, representatives (suitably chosen if n = 4) of the successive minima constitute a basis for Λ , but discrepancy between H and M systematically occurs beyond dimension 4. In his paper Die Reduktionstheorie der positiven quadratischen Formen , Acta Math. 96 (1956), 265–309, Van des Waerden proves for H an inductive � 5 � n − 4 ; URMANN ) to H formula, which actually boils down (Achill S CH ¨ M ≤ 4 and he made the remark that up to n = 8, the bound might well be n 4 , thus 3 2 = 1 . 5 instead of 25 16 = 1 . 625 for n = 6, 7 4 = 1 . 755 instead of 1 . 95 . . . for n = 7, and 2 instead of 2 . 44 . . . for n = 8. Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 5 / 12
Bounds for H It is easy to prove that up to dimension 4, representatives (suitably chosen if n = 4) of the successive minima constitute a basis for Λ , but discrepancy between H and M systematically occurs beyond dimension 4. In his paper Die Reduktionstheorie der positiven quadratischen Formen , Acta Math. 96 (1956), 265–309, Van des Waerden proves for H an inductive � 5 � n − 4 ; URMANN ) to H formula, which actually boils down (Achill S CH ¨ M ≤ 4 and he made the remark that up to n = 8, the bound might well be n 4 , thus 3 2 = 1 . 5 instead of 25 16 = 1 . 625 for n = 6, 7 4 = 1 . 755 instead of 1 . 95 . . . for n = 7, and 2 instead of 2 . 44 . . . for n = 8. This we shall prove, in the following precise form: Theorem For 4 ≤ n ≤ 8 , we have H (Λ) M (Λ) ≤ n 4 , and equality holds if and only if Λ is similar � = Z n ∪ ( e 1 + ··· + e n to � Z n , e 1 + ··· + e n ) + Z n . 2 2 Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 5 / 12
And beyond ? Whenever the Minkowskian sublattice has index 2 in Λ , the bound n 4 is valid. In case of index 4 with elementary quotient, the same argument works using two codewords instead of the only word O ( 1 , 1 , . . . , 1 ) , In dimension n = 9, there is a binary, 2-dimensional code with weight system ( 6 3 ) , namely � 1 1 1 1 1 1 0 0 0 � , 0 0 0 1 1 1 1 1 1 � 6 � 2 = n for which the bound is 4 . 4 Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 6 / 12
And beyond ? Whenever the Minkowskian sublattice has index 2 in Λ , the bound n 4 is valid. In case of index 4 with elementary quotient, the same argument works using two codewords instead of the only word O ( 1 , 1 , . . . , 1 ) , In dimension n = 9, there is a binary, 2-dimensional code with weight system ( 6 3 ) , namely � 1 1 1 1 1 1 0 0 0 � , 0 0 0 1 1 1 1 1 1 � 6 � 2 = n for which the bound is 4 . 4 In all dimensions n ≥ 10, one can use binary codes to construct lattices with M > n 4 . The inequality M ≤ n 4 , with two cases of equality, might well be true in dimension n = 9. Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 6 / 12
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