les théorèmes de Minkowski

Rn
k·k ΛRn
λ1(Λ) = min{kvk, v ∈Λ, v 6= 0}.
k∈ {1,··· , n}λk(Λ) r
kRΛr
ΛRn
vol(Λ)
n∈N∗Cn>0
ΛRn
λ1(Λ) ≤Cndet(Λ)1/n.
Cn= (2/√π)Γ(n/2 + 1)1/n
γn
γn≤Cn
n∈N∗ΛRn
n
pλ1···λn(Λ) ≤√γndet(Λ)1/n.
ΛRn
(a)SRnvol(S)>vol(Λ)
s1, s2∈S s1−s2∈Λ
(b)SRn−S=S
vol(S)>2nvol(Λ) s∈Λ∩S
(c)S
n∈N∗VnRn
(a)
Vn=Vn−1·Z1
0
(1 −t)(n−1)/2
√tdt.
(b)
B(x, y) = Z1
0
ux−1(1 −u)y−1du, Γ(x) = Z+∞
0
tx−1e−tdt.
x, y > 0
B(x, y) = Γ(x)Γ(y)
Γ(x+y).
Vn=Vn−1Γ((n+ 1)/2)Γ(1/2)/Γ(n/2 + 1)

(c) Γ(1/2) = √π
Vn=(√π)n
Γ(n/2 + 1).
(d)n= 1,2,3
(e)
(f)γ2
ΛRn
(a)R(x1,··· , xn) Λ
λi(Λ) = kxiki∈ {1,··· , n}
(b) (x∗
1,··· , x∗
n)T
xix∗
ii∈ {1,··· , n}T(Λ)
det(Λ)/(λ1···λn(Λ))
(c)w∈Λv=T(w)k λk(Λ) ≤ kwk
(x1,··· , xk, w)
(d)kvk2≥1λ1(T(λ)) ≥1
(a)p k ∈Zk2=−1
mod pR2(1, k),(0, p)a, b ∈Z
a2+b2=p
(b)TZ2
1/2
T
(c)L1,··· , LnRnc1,···cn>0c1···cn>det(M)
Zn
|Li(x)|< ci, i ∈ {1,··· , n}.
(d)qRnM
x∈Znq(x)< γndet(M)1/n