# Computer algebra recipes: An introductory guide by Enns R.H., McGuire G.C.

By Enns R.H., McGuire G.C.

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Additional info for Computer algebra recipes: An introductory guide

Example text

Let KZ be the Picard-Vessiot extension of LZ . By the definition of a standard operator we can Z(G) write KZ = C(Z). Let VZ ⊂ KZ be the solution space of LZ . In the compositum ´ E ´ MATHEMATIQUE ´ SOCIET DE FRANCE 2006 M. BERKENBOSCH 32 K of K and KZ over C(Z), we have the identity VZ = f V , for the appropriate f, as defined in the construction of LZ . 7.

T = NA2 + NB1 . Now the local exponents at some point We can make a decomposition N 2 p ∈ k¯ are the solutions of the equation λ(λ−1) = λ satisfy λ(λ − 1) = A(p) N2 (p)2 . A·(x−p)2 |x=p . N22 So the local exponents For the D2 -case we search for points with local exponent difference in 12 +Z. Because 3−2n L is in normal form, the local exponents of L at such a point are 2n+1 , for 4 , 4 some n ∈ Z. Therefore we get the system of equations: D2 -case : (3 + 4n − 4n2 )N2 (p)2 + 16A(p) = 0 N2 (p) = 0.

Then φF = φf ◦ φh , and we only need to show that f is Gal(k/k) −1 invariant. But we have that φσ(f ) = φσ(F ) ◦ φ−1 = φf σ(h) = φF ◦ φS(σ) ◦ (φh ◦ φS(σ) ) and therefore f ∈ k(x). 14. — The above corollary states that every differential operator ∂x2 − r, with r ∈ k(x) is the pullback of a differential operator over k(x) with three singularities, and with the same local exponents as the corresponding standard operator (use Norm(φh (StG ))). So we can see this corollary as a “rational version” of Klein’s theorem.