Hermite versus Minkowski Jacques Martinet Universit e de Bordeaux, - - PowerPoint PPT Presentation

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 (= x2), the Gram matrix of a set of vectors x1, . . . , xn is Gram(xi) = (xi · xj). 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 (= x2), the Gram matrix of a set of vectors x1, . . . , xn is Gram(xi) = (xi · xj). 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 = min

x∈Λ{0} N(x)

and then its Hermite invariant γ(Λ) = m(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 (= x2), the Gram matrix of a set of vectors x1, . . . , xn is Gram(xi) = (xi · xj). 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 = min

x∈Λ{0} N(x)

and then its Hermite invariant γ(Λ) = m(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 HB(Λ) = N(e1) . . . N(en) det(Λ) 1/n and H(Λ) = min

B

HB(Λ) ; 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 HB(Λ) = N(e1) . . . N(en) det(Λ) 1/n and H(Λ) = min

B

HB(Λ) ; Again, H(Λ) only depends on the similarity class of Λ. The following theorem was proved by Hermite around 1850:

• Theorem. We have H(Λ) ≤
• 4

3

(n−1)/2 .

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 HB(Λ) = N(e1) . . . N(en) det(Λ) 1/n and H(Λ) = min

B

HB(Λ) ; Again, H(Λ) only depends on the similarity class of Λ. The following theorem was proved by Hermite around 1850:

• Theorem. We have H(Λ) ≤
• 4

3

(n−1)/2 .

• Corollary. (HERMITE, August 6th, 1845). We have γ(Λ) ≤
• 4

3

(n−1)/2 . Set (γn) = supdim Λ=n γ(Λ). The values of (γn) 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 e1, . . . , en of Λ, let MB(Λ) = N(e1) . . . N(en) det(Λ) 1/n and M(Λ) = min

B

MB(Λ) ; 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 e1, . . . , en of Λ, let MB(Λ) = N(e1) . . . N(en) det(Λ) 1/n and M(Λ) = min

B

MB(Λ) ; 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) ≤ γn .

Also, using his argument consisting in bounding by 1 the density of any sphere packing, Minkowski gave a linear bound for γn (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 e1, . . . , en of Λ, let MB(Λ) = N(e1) . . . N(en) det(Λ) 1/n and M(Λ) = min

B

MB(Λ) ; 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) ≤ γn .

Also, using his argument consisting in bounding by 1 the density of any sphere packing, Minkowski gave a linear bound for γn (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

formula, which actually boils down (Achill SCH ¨

URMANN) to H

M ≤

5

4

n−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

formula, which actually boils down (Achill SCH ¨

URMANN) to H

M ≤

5

4

n−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

formula, which actually boils down (Achill SCH ¨

URMANN) to H

M ≤

5

4

n−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

to Zn, e1+···+en

2

= Zn ∪ ( e1+···+en

2

) + Zn.

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 (63), namely 1 1 1 1 1 1 0 0 0

0 0 0 1 1 1 1 1 1

• ,

for which the bound is 6

4

2 = n

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 (63), namely 1 1 1 1 1 1 0 0 0

0 0 0 1 1 1 1 1 1

• ,

for which the bound is 6

4

2 = n

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

How to prove Minkowski’s bound for M ?

A non-classical proof consists in using a deformation argument to show that when ordering the ei by increasing norms, if strict inequality holds somewhere, then we may increase M(Λ). This will show that the local maxima

• f M are attained on lattices having n independent minimal vectors

(well-rounded lattices). In this case, M = γ, whence M(Λ) ≤ γn.

Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 7 / 12

How to prove Minkowski’s bound for M ?

A non-classical proof consists in using a deformation argument to show that when ordering the ei by increasing norms, if strict inequality holds somewhere, then we may increase M(Λ). This will show that the local maxima

• f M are attained on lattices having n independent minimal vectors

(well-rounded lattices). In this case, M = γ, whence M(Λ) ≤ γn. Explicitly, if N(em) < N(em+1), we can consider the span F of e1, . . . , em, its

• rthogonal complement F ⊥, and the transformations uλ (λ < 1, close to 1)

equal to the identity on F and to the homothetic transformation x → λx on F ⊥.

Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 7 / 12

How to prove Minkowski’s bound for M ?

A non-classical proof consists in using a deformation argument to show that when ordering the ei by increasing norms, if strict inequality holds somewhere, then we may increase M(Λ). This will show that the local maxima

• f M are attained on lattices having n independent minimal vectors

(well-rounded lattices). In this case, M = γ, whence M(Λ) ≤ γn. Explicitly, if N(em) < N(em+1), we can consider the span F of e1, . . . , em, its

• rthogonal complement F ⊥, and the transformations uλ (λ < 1, close to 1)

equal to the identity on F and to the homothetic transformation x → λx on F ⊥. In a deformation as above the index in Λ of the span Λ′ of the successive minima e1, . . . , en (a Minkowskian sublattice of Λ) is preserved. Applying to well-rounded lattices the Hadamard inequality together with the definition of the Hermite invariant, we obtain the bound [Λ : Λ′] ≤ γn/2

n

.

Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 7 / 12

Back to H

The deformations used in the previous frame to increase M also increase H. A consequence is that local maxima of H are attained on well-rounded

• lattices. From now on we restrict ourselves to well-rounded lattices.

Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 8 / 12

Back to H

The deformations used in the previous frame to increase M also increase H. A consequence is that local maxima of H are attained on well-rounded

• lattices. From now on we restrict ourselves to well-rounded lattices.

A strategy for bounding H(Λ) consists in analyzing closely the possible structures of quotients Λ/Λ′. Thanks to the bound [Λ : Λ′] ≤ γn/2

n

, the annihilator d of Λ/Λ′ is also bounded. Choose n-independent minimal vectors and consider their span Λ′. The sets (a1, . . . , an) such that e = a1e1+···+anen

d

belongs to Λ are the words of a Z/dZ-code of length n. The basic tool is to classify Z/dZ-codes which arise, and then for each code, to bound H on the corresponding lattices.

Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 8 / 12

Back to H

The deformations used in the previous frame to increase M also increase H. A consequence is that local maxima of H are attained on well-rounded

• lattices. From now on we restrict ourselves to well-rounded lattices.

A strategy for bounding H(Λ) consists in analyzing closely the possible structures of quotients Λ/Λ′. Thanks to the bound [Λ : Λ′] ≤ γn/2

n

, the annihilator d of Λ/Λ′ is also bounded. Choose n-independent minimal vectors and consider their span Λ′. The sets (a1, . . . , an) such that e = a1e1+···+anen

d

belongs to Λ are the words of a Z/dZ-code of length n. The basic tool is to classify Z/dZ-codes which arise, and then for each code, to bound H on the corresponding lattices. The classification of such codes for dimensions n ≤ 8 can be read in [J. M.], Sur l’indice d’un sous-r´ eseau, L ’Ens. Math., monographie 35 (Gen` eve, 2001), a paper which extends previous work by WATSON, RYSHKOV, and ZAHAREVA; this classification has been recently extended to dimension 9 (W. KELLER – J. M. – A. SCH ¨

URMANN, preprint).

Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 8 / 12

2-elementary quotients

This is the case when Λ is a union of Λ′ and of cosets e + Λ′ with e of the form ei1 + · · · + eim 2 . Suffices to consider index 2. Let Λ = Λ′ ∪ (e + Λ′) with e = e1 + · · · + em 2 , say, m = n. One shows that the smallest possible norm for e′ ∈ e + Λ′ is attained on e′ = ±e1 ± · · · ± en 2 for some convenient choice of signs, then that the largest possible norm of such an e′ occurs for pairwise othogonal ei, i.e., for Λ′ ∼ Zn.

Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 9 / 12

Explicit bounds for the index (1)

The bound ı ≤ ⌊γn/2

n

⌋ for the maximal possible value of [Λ : Λ′] happens to be

• ptimal for n ≤ 8, and reads

ı ≤ 1 for n = 1, 2, 3, ı ≤ 2 for n = 4, 5, ı ≤ 4, 8, 16 for n = 6, 7, 8, respecvtively.

Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 10 / 12

Explicit bounds for the index (1)

The bound ı ≤ ⌊γn/2

n

⌋ for the maximal possible value of [Λ : Λ′] happens to be

• ptimal for n ≤ 8, and reads

ı ≤ 1 for n = 1, 2, 3, ı ≤ 2 for n = 4, 5, ı ≤ 4, 8, 16 for n = 6, 7, 8, respecvtively. However, disregarding the lattices D4, D6, E7 and E8, the bounds above can be improved to ı ≤ 1, 3, 4, 8 for n = 4, 6, 7, 8, respectively; and since the lattices above have bases of minimal vectors, we have H = M for them. Finally, we reduce ourselves to consider cyclic quotients of order 3 or 4 for n = 6 and n = 7, and for n = 8, quotients which are cyclic of order 3, 4, 5, 6

• r of type 4 · 2.

This is somewhat general: a “large” or “small” index is rather easily dealt with.

Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 10 / 12

Explicit bounds for the index (2)

Just a glance at dimension 9 ! Cohn-Elkies’ bound of γ9 implies ı ≤ 30. The conjecture γ9 = γ(Λ9) implies ı ≤ 22. [K-M-S] proves ı ≤ 16, and more precisely [Λ : Λ′] ≤ 10 or = 12 (all possible structures),

• r Λ/Λ′ is of type 42, 4 · 22 or 24.

The list of codes relative to index 6, 7, 8, 9 is a long list !

Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 11 / 12

Sketch of proof for n ≤ 6

Relies on an identity of Watson: let e = a1e1+···+anen

d

with ai ≥ 1 and d ≥ 2. Then n

i=1 ai

• N(e − ei) − N(ei)
• =

ai

• − 2d
• N(e) .

Applied with minimal ei, say, N(ei) = 1, this implies (1) ai ≥ 2d, and (2) if ai = 2d, then the e − ei are minimal. Thus n = 6 ⇒ ı ≤ 4.

Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 12 / 12

Sketch of proof for n ≤ 6

Relies on an identity of Watson: let e = a1e1+···+anen

d

with ai ≥ 1 and d ≥ 2. Then n

i=1 ai

• N(e − ei) − N(ei)
• =

ai

• − 2d
• N(e) .

Applied with minimal ei, say, N(ei) = 1, this implies (1) ai ≥ 2d, and (2) if ai = 2d, then the e − ei are minimal. Thus n = 6 ⇒ ı ≤ 4. n = 6, d = 3 : (e − e1, e1, e3, e4, e5, e6) is a basis of minimal vectors; n = 6, d = 22 : the binary code defines D6. Hence H = M in both cases. n = 6, d = 4 : write e =

e1+···+em1+2(em1+1+···+e6) 4

=

f+em1+1+···+e6 2

with f =

e1+···+em1 2

. We have m1 ≥ 4, m1 + 2m2 ≥ 8 hence (m1, m2) = (4, 2), whence e = f+e5+e6

2

, and an index 2 shows up in dimension 3, a contradiction. Finally we are left with index 2, for which H M ≤ n 4 ... THE END.

Jacques Martinet (Universit´ e de Bordeaux, IMB/A2X) Hermite versus Minkowski Luminy, November–December, 2009 12 / 12