Équations Différentielles Linéaires et Transformation de by B. Malgrange

By B. Malgrange

Ces notes reprennent et développent los angeles matière de three exposés
faits à l'IMPA en mai-juin 88. Leur contenu n'est guère unique, et
pour l'essentiel il est bien connu des spécialistes. Mais, malgré son
caractère élémentaire (au moins, au début), il n'est available que
dans des articles très spécialisés, et d'un haut niveau de technicité.
Peut-être y a-t-il là une lacune, que ces notes aideront à combler.

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Extra resources for Équations Différentielles Linéaires et Transformation de Fourier: Une Introduction

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2 Lemma Ir be an element of the opposite big cell Uαmin . Let α ∈ A I(r, n) with α = αmin and let s be the integer such that αs ≤ r and αs+1 > r. If Let A = λ = (αs+1 , αs+2 , . . , αr ), µ = λ − (r) = (αs+1 − r, . . , αr − r), and ν = (1, 2, . . , α ˆ1 , α ˆ2 , . . , α ˆi, . . , α ˆ s , , αs + 1, . . , the complement of {α1 , α2 , . . , αs } in {1, 2, . . , r} arranged in increasing order), then pα (A) = ±pλ,ν (A) = ±pµ,ν (A). Proof. It is clear that the rows of the α-th minor of A which belong to Ir do not contribute any thing towards the value of pα (A) = ± det (α-th minor of A) and pα (A) is in fact determined by those rows of the α-th minor which belong to A.

Proof. Consider the homogeneous coordinate rings S = k[xα , α ∈ I(r, n)] and R = k[pα , α ∈ I(r, n)] of P(∧r ∨) and Gr,n respectively. We have a natural homomorphism Φ : S −→ R xα −→ pα whose kernel is the ideal J. 4) implies that J ′ ⊂ ker Φ = J. Hence we have a surjective homomorphism Φ : S/J ′ −→ R xα −→ pα . Thus, in order to prove that J ′ = J, it is enough to prove that Φ is injective. So let F be any nonzero element of S/J ′ . 9) S/J ′ is generated by standard monomials, and therefore F can be written as a linear combination of distinct standard monomials.

R ) then it is easy to see that γ = (γ1 , γ2 , . . , γr ), where γi = min(αi , βi ), 1 ≤ i ≤ r. 3). (iii) If µ = (µ1 , µ2 , . . , µr ) is a maximal element of Tα − {α}, it differs from α = (α1 , α2 , . . , αr ) just at one place by 1. 5,ii), the assertions in (iii) follow easily. To prove the remaining assertion, first we note that for any α in I(r, n), X(α) = X(Tα ). 8) and (i) above we have X(µ) ∩ X(µ′ ) = X(Tµ ) ∩ X(Tµ′ ) = X(Tµ ∩ Tµ′ ) = X(Tν ) = X(ν), where νi = min(µi , µ′i ), 1 ≤ i ≤ r.

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