Berman. Galois theory of linear ODEs (phd thesis)(157s).pdf: eKnigu

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Berman. Galois theory of linear ODEs (phd thesis)(157s).pdf

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core decision procedure. Chapter 3 is organized as follows: Section 3.1 discusses connections and D-modules and their relation to diﬀerential equations and systems. Section 3.2 presents the results from Section 2 of [BS99]. This section includes Algorithm I, which computes the group of the inhomogeneous equation L(y ) = b, L ∈ C (x)[D] completely reducible, b ∈ C (x). Section 3.3 presents the results from Section 3 of [BS99]. This section includes Algorithm II, which computes the group of L1 (L2 (y )) = 0, L1 , L2 ∈ C (x) completely reducible. Algorithm II ˆ ˆ works by computing an associated inhomogeneous equation L(y ) = b, where L is completely reducible, and applying part of Algorithm I to that equation. Section 3.4 presents Algorithm III, an optimization of Algorithm I: Whereas Algorithm I relies on parameterizing all factorizations of L to compute the group of L(y ) = b, Algorithm III only requires a single expression of L as the least common left multiple of a set of irreducible operators. Note that Algorithm III is presented in terms of inhomogeneous ﬁrst-order systems rather than inhomogeneous equations; the results of Section 3.1 show that the two settings are interchangeable in our case. In Chapter 4, we give a set of criteria to compute the Galois group of a diﬀerential equation of the form y (3) + ay
+...

3. C ∗ = (C \ {0} , ·), the multiplicative group. 4. GLn = GLn (C ), the group of n × n matrices with nonzero determinant. GLn is an open subset of aﬃne n2 -space with coordinates sij , 1 ≤ i, j ≤ n. 5. Any closed subgroup of GLn . Examples: (a) SLn , the group of n × n matrices with determinant 1. (b) PSLn , the quotient of SLn by its center. (c) Tn , the group of upper-triangular nonsingular n × n matrices. We have Tn = {(aij ) ∈ GLn : aij = 0 if j < i} . (d) Un , the group of unipotent upper-triangular n × n matrices. We have Tn = {(aij ) ∈ Tn : aii = 1 for all i} . (e) Dn , the group of diagonal nonsingular n × n matrices. An arbitrary algebraic group G is called a linear algebraic group if it is isomorphic to a subgroup of GLn (C ) for some n. Given an n-dimensional C -vector space V , let E0 = {e1 , e2 , . . . , en } be a ﬁxed basis of V . Then there is a one-to-one correspondence between GL(V ) and GLn whereby φ ∈ GL(V ) corresponds with [φ]E0 ∈ GLn . This correspondence induces an algebraic group structure on GL(V ). One checks that this structure is independent of choice of basis of V , so that GL(V ) is given a unique structure of linear algebraic group. We say that a subgroup G ⊆ GLn is the expression of the subgroup G ⊆ GL(V ) in the basis F , and we write G = [G]F , if G = {[φ]F : φ ∈ G} . It follows from the results of ˜ the previous section that two subgroups G, G ⊆ GLn are conjugate if and only if there ˜ exists a subgroup G ⊆ GL(V ) and a basis B of V such that G = [G]E0 , G = [G]B . Note that the matrix P = [id]E ,F centralizes G = [G]E (i.e., P M P −1 = M for all M ∈ G) if and only if [φ]E = [φ]F for all φ ∈ G. 6. G H = G φ H, the semidirect product of G by H via φ, where G and H are algebraic groups and φ : G × H → G is the mapping corresponding to an algebraic group action of H on G (cf. [Hu, Sec. 8.2]) having the property that φ(•, y ) is an automorphism of G for all y ∈ H. As a set, we have G φ H = G × H. The structure of G φ H is given by (x1 , y1 )(x2 , y2 ) = (x1 φ(x2 , y1 ), y1 y2 ) 6...

extension (resp., the ful l solution space) of Li (y ) = 0 for i = 1, 2. Then, the fol lowing are equivalent: 15...

addition, the pair (L1 , L0 ) given in Item 6 above is unique. Proof. This is Proposition 2.1 of [BS99]; it is an adaptation of Th´or`me 1 of [Ber92]. We ee reproduce it here, with some changes of notation. First, let L1 = 1, L0 = L. We see that Properties 6(ab) are satisﬁed, so that there exists a pair (L1 , L0 ) satisfying those properties with L1 of maximal order. 18...

action of σ on W = VL0 . Restriction to the space W Ct W e make the following additional remarks: GL ....

The claim now follows immediately. In our case, we claim that L1 2 and L2 2 are inequivalent (and therefore that GL = ˜ SL2 (C ) × SL2 (C )). The expanded version of DEtools developed by Mark van Hoeij for MapleV.5 allows one to calculate symmetric powers, LCLM’s and a basis of the ring of Dmodule endomorphisms of D/DL for an operator L ∈ D = C (x)[D]. Using this we proceed as follows. A calculation shows that M = LCLM(L1 2 , L2 2 ) is of order 4. If L1 2 and L2 2 were equivalent then D/DM would be the direct sum of two isomorphic D-modules. The endomorphism ring of D/DM would therefore have dimension 4. Using the eigenring command in DEtools one sees that this ring has dimension 2 and the desired result follows. We now consider the equation ˜ V where H A1 = −AT ⊗ I2 + I2 ⊗ A2 , 1  01 , = x0   0 1 , = 1 −1 − x (0, 1, 0, 0)T . 36
=...

ˆ Computations using the DFactor and ratsols commands in DEtools show that L is irreducible over C (x) and that this equation admits no rational solutions. Thus, the vector space W referred to in the third statement of Lemma 3.3.3 is all of C 4 . We conclude that the Galois group GL is (SL2 (C )) × SL2 (C )) C4....

{id, (2 3), (1 3 2)} . Then G has a faithful parameterization   0 w y dI1     (w, x, y ) →  0 x , y dI2   0 0 y dI3 where d3 = −d1 − d2 and σ (Ij ) = j for 1 ≤ j ≤ 3. Under this parameterization, we h aave (Int y )(w, x) = (y dI1 −dI3 w, y dI2 −dI3 x). The G-invariant subspaces are a1 , a2 nd a1 , a2 , so that n1 = 2, n2 = 1. 74...

Aj = σ −1 (j ), this means that we only need consider the permutations id, (1 3) and (1 2 3). Item 1(c) of the lemma is now clear.
2 If [Ru ]B = U(0,1) , then Lemma 4.3.1 implies that...

mials of degree d in C [x, y ], where x and y are indeterminates. Deﬁne   ab   .(xi y d−i ) = (ax + cy )i (bx + dy )d−i cd   ab  ∈ SL2 and a typical basis vector xi y d−i of Xd , 0 ≤ i ≤ for a typical element  cd d. Then, up to isomorphism, the induced action of SL2 on Xd is the unique irreducible 84...

Suppose G0 = SL2 , so that W = W1 + W2 as in Item 3 of the lemma. Since Φ(SL2 × {id}) commutes with ψ , we see that ψ (W2 ) is a SL2 -invariant subspace of W, which by Item 3 must then be W2 itself. Moreover, ψ |W2 is a SL2 -module endomorphism. Item 2 of the lemma then implies that ψ |W2 = α · id |W2 for some α ∈ C ∗ . It quickly follows that, with respect to some suitable basis B of W,   c−2     [Φ(A)]B =  0     0  0 c 0     ...