Schwarz maps of algebraic linear ordinary differential equations

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Schwarz maps of algebraic linear ordinary differential equations Camilo Sanabria Malagón 1 Department of Mathematics, Universidad de los Andes, Carrera 1 # 18A–12 Of. H210, Bogotá D.C., 111711, Colombia Received 13 October 2016; revised 24 February 2017

Abstract A linear ordinary differential equation is called algebraic if all its solution are algebraic over its field of definition. In this paper we solve the problem of finding closed form solution to algebraic linear ordinary differential equations in terms of standard equations. Furthermore, we obtain a method to compute all algebraic linear ordinary differential equations with rational coefficients by studying their associated Schwarz map through the Picard–Vessiot Theory. © 2017 Elsevier Inc. All rights reserved.

0. Introduction Since the introduction of the concept of local systems by B. Riemann [30,31], the study of linear ordinary differential equations with rational coefficients has been dominated by his viewpoint. Among the most prominent works based on this perspective are P. Deligne’s work on the Riemann–Hilbert correspondence and Hilbert’s twenty-first problem [8], M. Kashiwara’s work on the Riemann–Hilbert problem for holonomic systems [14], and B. Malgrange’s exposition on the Riemann–Hilbert–Birkhoff correspondence [23]. Although this approach of Riemann is mainly used in the study of equations with regular singularities, i.e. fuchsian systems, introduced by L. Fuchs in [10], it also gave rise to different tools for studying the local behavior around irE-mail address: [email protected]. 1 Partially supported by Vicerrectoría de Investigaciones de la Universidad de los Andes grant PEP P13.160422.030

FAPA – Camilo Sanabria. http://dx.doi.org/10.1016/j.jde.2017.08.002 0022-0396/© 2017 Elsevier Inc. All rights reserved.

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regular singularities. A modern perspective of this theory was introduced by N. M. Katz through his concept of rigid local systems [16]. Another perspective is given by Picard–Vessiot theory, developed by E. Picard [26], E. Vessiot [39] and E. R. Kolchin [21]. With this theory, a differential Galois group is associated to a linear ordinary differential equation. It was proved by J.-P. Ramis [28,29] and J. J. Morales [24,25] that this group is obtained by combining the monodromy, exponential tori and Stoke’s matrices of the local system in a linear algebraic group . The galoisian approach has proved successful at obtaining closed form solutions to the equation. Indeed E. R. Kolchin proved in [21] that the solutions are liouvillian if and only if the differential Galois group is solvable, and for the second order case this result has been systematized by J. Kovacic’s algorithm [22]. M. van Hoeij, J. F. Ragot, M. Singer, F. Ulmer and J.-A. Weil have extended this algorithm to higher order equations [13,35,36,38]. Using those algorithms the solutions are presented as the exponential of the antiderivative of a function which is algebraic over the field of definition of the equation. This algebraic function is described in terms of its minimal polynomial, which is computed using the (semi-)invariants of the differential Galois group. Therefore, the smaller the group the more computations are involved, reaching a maximum when the differential Galois group is finite. This is equivalent to the case when all the solutions to the linear ordinary differential equation are algebraic over its field of definition. These equations are called algebraic linear ordinary differential equations. A classical result of F. Klein [17,18] states that a full system of solutions to an algebraic second order linear ordinary differential equation with rational coefficients can be given in closed form as solutions to hypergeometric equations precomposed with a rational function, multiplied by the solution to a first order linear differential equation. The first modern treatment of Klein’s result was done by B. Dwork and F. Baldassari in [3,4]. To overcome the difficulty of the extensive computations required in the algebraic case in Kovacic’s algorithm, M. Berkenbosch applied in [5] this result of Klein and he generalized it to order three by introducing the concept of standard equations. Furthermore, I in [33] extended it to arbitrary order. Therefore, the problem of finding closed form solutions to algebraic linear ordinary differential equations has been reduced to the problem of first, characterizing the family of standard equations, and second, identifying which standard equation is needed in order to obtain closed form solutions to a given equation. Both of these problems have been solved in this paper. Furthermore, we obtain a method to compute explicitly all irreducible algebraic linear ordinary differential equations. The main tool used in the extensions of Klein’s theorem in [5,33] are Schwarz maps. Schwarz maps offer a third approach to the study of linear ordinary differential equations which was used by H. A. Schwarz [34], F. Klein [19,20] and G. Fano [9]. Their most famous application is the description of tessellations of the Riemann sphere by Schwarz triangles [40, Sections III.6, III.7, III.8]. With the Schwarz approach, instead of studying local systems, a full system of solutions of a linear differential equation is understood as a parametrization of a complex curve in a complex projective space. In this paper, we study Schwarz maps from the perspective of the Picard–Vessiot theory using Compoint’s theorem [7]. In the first section, we construct a method to identify the standard equation needed to obtain closed form solutions to a given algebraic linear ordinary differential equation. We also characterize standard equations by showing that, up to projective equivalence, their associated Schwarz map corresponds to an orbit of a differentiable dynamical system (Theorem 5). On the other hand, in the second section, we show a family of differential dynamical systems whose orbits are Schwarz maps of standard equations (Theorem 6). Moreover, in the third section, we apply this approach to the classical Hesse pencil [11,12], which is a

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one-parameter continuous family of elliptic curves that has gained much attention recently in different areas [2,37,41]. We obtain a one-parameter family of standard equations whose associated Schwarz maps parametrize this family of elliptic curves. We also obtain an equation that explains how these parametrizations deform as we vary the parameter of the Hesse pencil. The computations involved were implemented in MAPLE and the codes, among other examples, can be obtained from my webpage [1]. 1. Schwarz maps and Picard–Vessiot theory for linear ordinary differential equations Consider an irreducible linear ordinary differential equation L(x) = 0, where  L(x) =

d dt

n

 x + an−1

d dt

n−1 x + . . . + a1

d x + a0 x, dt

x = x(t) and an−1 , . . . , a1 , a0 ∈ C(t). A Picard–Vessiot extension for L(x) = 0 is the field  K = C(t)

d dt

i−1

   xj  i, j = 1, . . . , n ,

where xj = xj (t), for j = 1, . . . , n form a full system of solutions [27, Proposition 1.22]. Let J be the kernel of the C(t)-morphism    : C(t) Xij | i, j = 1, . . . , n

1 −→ K det(Xij ) Xij −→

d i−1 xj . dt i−1

An isomorphic representation of the differential Galois group of L(x) = 0 is the algebraic group G ⊂ GLn (C) composed by the elements (gij )ni,j =1 sending the ideal J into itself under the C(t)-automorphisms [27, Observations 1.26]   C(t) Xij | i, j = 1, . . . , n



  1 1 −→ C(t) Xij | i, j = 1, . . . , n det(Xij ) det(Xij ) Xij −→

n 

Xil glj .

l=1

By the Galois correspondence [27, Proposition 1.34] if P ∈ C(t)[Xij | i, j = 1, . . . , n] is G-invariant, then (P ) ∈ C(t). Since L(x) = 0 is irreducible, hence G is reductive [27, Exercise 2.38]. Under these assumptions, we have: Theorem 1 (Compoint’s theorem [6]). If the differential Galois group of L(x) = 0 is reductive, the ideal J is generated by the G-invariants contained in it.

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1.1. Picard–Vessiot theory in the algebraic case From now on we will assume that all the solutions to L(x) = 0 are algebraic over C(t). In particular G is finite and the Picard–Vessiot extension is K = C(t) [x1 , . . . , xn ] . Let I ⊂ C(t)[X1 , . . . , Xn ] be the kernel of the C(t)-morphism  : C(t)[X1 , . . . , Xn ] −→ K Xj −→ xj As above, the Galois correspondence implies that if P ∈ C(t)[X1 , . . . , Xn ] is G-invariant, then (P ) ∈ C(t). We have Theorem 2 (Compoint’s theorem for the algebraic case). Let L(x) = 0 be an algebraic irreducible linear ordinary differential equation. If the differential Galois group of L(x) = 0 is finite, the ideal I is generated by the G-invariants contained in it. Proof. From Theorem 1 we have that J is generated by the G-invariants it contains. Since C(t)[X11 , . . . , X1n ] is closed under the action of G, the contraction of J to C(t)[X11 , . . . , X1n ] is also generated by the G-invariants it contains. The claim follows now by noting that I corresponds to J ∩ C(t)[X11 , . . . , X1n ] where X1j := Xj , for j = 1, . . . , n. 2 Corollary 3. If P1 , . . . , PN ∈ C[X1 , . . . , Xn ] is a set of generators of the G-invariant subring C[X1 , . . . , Xn ]G , then I = P1 − f1 , . . . , PN − fN ,

(1)

where fi = (Pi ), i = 1, . . . , N . 1.2. A first order differential equation In this sections, starting from the solutions x1 , . . . , xn to L(x) = 0, we will construct a nonautonomous first order differential equation with solution X(t) = (x1 , . . . , xn ). Differentiating the generators of I in (1) we obtain the relations n  ∂Pi d d (x1 , . . . , xn ) xj = fi , ∂Xj dt dt

i = 1, . . . , N.

(2)

j =1

Since G is finite, the orbit space Cn /G has dimension n, therefore the derivative of the map of projection into the orbits

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G : Cn −→ Cn /G ⊆ CN X = (X1 , . . . , Xn ) −→ (P1 (X), . . . , PN (X)) is non-singular in a dense subset U ⊆ Cn . In particular, after arranging the indices of the Pi ’s if necessary, we may assume that ⎡

∂P1 /∂X1 ⎢ .. M(X) = ⎣ . ∂Pn /∂X1

... .. . ...

⎤ ∂P1 /∂Xn ⎥ .. ⎦ (X) . ∂Pn /∂Xn

is invertible over U . Using the n first equations of the system (2) we define a G-invariant first order differential equation over U d X = F (t, X) dt

(3)

where ⎡  −1 ⎢ F (t, X) = M(X) ⎣

d dt f1

⎤

.. ⎥ . . ⎦ d dt fn

(4)

Note that F (t, X) in (4) depends upon the choice of solutions x1 , . . . , xn and the invariants P1 , . . . , Pn . 1.3. Schwarz maps, projective equivalence and pullbacks The Schwarz map [40] of L(x) = 0 associated to the solutions x1 , . . . , xn is the analytic extension of the parametric curve in Pn−1 (C) [X] (t) = (x1 : . . . : xn ). The image of the Schwarz map is called the Fano curve [33]. Remark 1. Since the solutions x1 , . . . , xn are algebraic over C(t), the degree of transcendence of C(t)[x1 , . . . , xn ] over C(t) is zero. Therefore the degree of transcendence of C[x1 , . . . , xn ] over C is one. Thus, the algebraic closure of the analytic curve parametrized by (x1 , . . . , xn ) in Cn has dimension one, and so does its projection into Pn−1 (C), which is the Fano curve. Note that the Schwarz map and the Fano curve are uniquely determined up to a rational transformation of Pn−1 (C). Definition 1. Let L(x) = 0 be a linear ordinary differential equation, where  L(x) =

d dt

n

 x + an−1

d dt

n−1 x + . . . + a1

d x + a0 x, dt

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x = x(t) and an−1 , . . . , a1 , a0 ∈ C(t) and let L1 (y) = 0 be a linear ordinary differential equation, where  L1 (y) =

d dt

n

 y + bn−1

d dt

n−1 y + . . . + b1

d y + b0 y, dt

y = y(t) and bn−1 , . . . , b1 , b0 ∈ C(t). We say that L1 (y) = 0 is rationally projectively equivalent to L(x) = 0 if there exists a function f such that: d • f1 dt f ∈ C(t) • every solution to L1 (y) = 0 is of the form y0 (t) = f (t)x0 (t) where x0 is a solution to L(x) = 0.

In particular, when L1 (y) is projectively equivalent to L(x) = 0, the Schwarz map of L1 (y) associated to the solutions f x1 , . . . , f xn coincides with the Schwarz map of L(x) = 0 associated to x1 , . . . , xn . Therefore post-composing these two Schwarz maps with the projection into the projective orbits G,P : Pn−1 (C) −→ P(Cn /G), will result in the same image. Definition 2. Let L(x) = 0 be a linear ordinary differential equation, where  L(x) =

d dt

n

 x + an−1

d dt

n−1 x + . . . + a1

d x + a0 x, dt

x = x(t) and an−1 , . . . , a1 , a0 ∈ C(t) and let L(r) = 0 be a linear ordinary differential equation, where  L(r) =

d ds

n

 r + bn−1

d ds

n−1 r + . . . + b1

d r + b0 r ds

(5)

and bn−1 , . . . , b1 , b0 ∈ C(s). We say that L(x) = 0 is a pullback of L(r) by a rational map if there exists p ∈ C(t), such that every solution to L(x) = 0 is of the form x(t) = r ◦ p(t) where r is a solution to L(r) = 0. 1.4. Quotient Fano curve, minimal Schwarz maps and standard equations From the Galois correspondence it follows that the composition of the Schwarz map with the projection into the projective orbits G,P ◦ [X] (t) = G,P (x1 : . . . : xn ) parametrizes an algebraic curve in P(Cn /G). Therefore we can make an analytic extension of this composition to the whole Riemann sphere to obtain a single-valued algebraic map

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G,P ◦ [X] : P(C) −→ P(Cn /G) t −→ G,P ◦ [X] (t), where we use the same notation for the original composition and the extension. We will call the closure of the image of the map G,P ◦ [X] the quotient Fano curve. If this map is birational we say that the Schwarz map [X] (t) = (x1 : . . . : xn ) is minimal [15] and in such case we say that the equation L(x) = 0 is standard [5]. The importance of standard equations is stated in the following theorem [33, Corollary 12]. Theorem 4. Let L(x) = 0 be an algebraic irreducible linear ordinary differential equation. Then L(x) = 0 is rationally projectively equivalent to the pullback by a rational map of an standard equation. 1.5. Associated first order differential equation, dynamical systems and standard equations Let us consider the first order differential equation (3) built in Section 1.2 from the system in (2). If L(x) = 0 is standard, then this differential equation is a dynamical system. Definition 3. Let L(x) = 0 be an algebraic irreducible linear ordinary differential equation. Let x1 , . . . , xn be n linearly independent solutions to L(x) = 0 and G the representation of the differential Galois group of L(x) = 0 associated to this choice of solutions. Let  ∈ Z>0 be such that the homogeneous elements of C[X1 , . . . , Xn ]G of degree , C[X1 , . . . , Xn ]G  , form a homogeneous coordinate system for P(Cn /G), i.e. n P(C[X1 , . . . , Xn ]G  )  P(C /G).

Let P1 , . . . , Pn ∈ C[X1 , . . . , Xn ]G  be such that ⎡

∂P1 /∂X1 ⎢ .. M(X) = ⎣ . ∂Pn /∂X1

... .. . ...

⎤ ∂P1 /∂Xn ⎥ .. ⎦ (X) .

(6)

∂Pn /∂Xn

is invertible, and let f1 , . . . , fn ∈ C(t) be such that fi = Pi (x1 , . . . , xn ),

i = 1, . . . , n.

The first order differential equation associated to L(x) = 0, given the solutions x1 , . . . , xn , the degree , and the invariants P1 , . . . , Pn is d X = F (t, X) dt

(7)

where ⎡  −1 ⎢ F (t, X) = M(X) ⎣

d dt f1

⎤

.. ⎥ . . ⎦ d dt fn

(8)

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In the following theorem we characterize standard equations, up to projective equivalence, in terms of their associated first order differential equation. Theorem 5. Let L(x) = 0 be an algebraic irreducible linear ordinary differential equation. Then L(x) = 0 is standard if and only if it is rationally projectively equivalent to a linear ordinary differential equation with an associated first order differential equation that is a dynamical system. Proof. Let us assume that L(x) = 0 is standard. We fix n linearly independent solutions x1 , . . . , xn . By the definition of standard equation, the map t → G,P ◦ [X] (t) defines a birational morphism between the Riemann sphere parameterized by t and the quotient Fano curve. Let us denote by C the quotient Fano curve. Then the map G,P ◦ [X] : P(C) −→ C ⊆ P(Cn /G) t −→ G,P ◦ [X] (t) is birational. The homogeneous ideal in C[X1 , . . . , Xn ]G defining C is generated by the homogeneous elements in I ∩ C[X1 , . . . , Xn ]G  h I (C) = I ∩ C[X1 , . . . , Xn ]G = P ∈ C[X1 , . . . , Xn ]G | P is homogeneous and P (x1 , . . . , xn ) = 0. The homogeneous coordinate ring of C is C[X1 , . . . , Xn ]G /I (C). Since C is a curve, there exist n − 2 polynomials Q1 , . . . , Qn−2 ∈ I (C) that are algebraically independent over C. Furthermore, since G,P ◦ [X] is a birational morphism, the associated dual map is an isomorphism from the field of meromorphic functions C(C) to C(t). In particular, there exist two homogeneous polynomials of the same degree, P , Q ∈ C[X1 , . . . , Xn ]G , such that P (x1 , . . . , xn ) = t. Q(x1 , . . . , xn ) Let f = Q(x1 , . . . , xn ) ∈ C(t). Note that f = 0, and that Q1 , . . . , Qn−2 , Q, P can be taken so that they have the same degree . Now we consider L1 (y), the linear ordinary differential equation projectively equivalent to L(x) = 0 with solutions yi (t) = f −1/ deg(Q) xi (t), We have

i = 1, . . . , n.

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Q(y1 , . . . , yn ) = 1 and

9

P (y1 , . . . , yn ) = t, Q(y1 , . . . , yn )

hence Qi (y1 , . . . , yn ) = 0, for i = 1, . . . , n − 2, Q(y1 , . . . , yn ) = 1, P (y1 , . . . , yn ) = t. Therefore, the first order differential equation associated to L1 (y) = 0 given the solutions y1 , . . . , yn , the degree , and the invariants Pi = Qi for i = 1, . . . , n − 2, Pn−1 = Q and Pn = P is a dynamical system. This finishes the first part of the proof. Now we assume that L(x) = 0 is projectively equivalent to L1 (y) = 0 and that the first order differential equation associated to L1 (y) = 0 given the solutions y1 , . . . , yn , the degree  and the invariants P1 , . . . , Pn , is a dynamical system d X = F (X) dt where fi = Pi (y1 , . . . , yn ) ∈ C(t),

i = 1, . . . , n,

and ⎡  −1 ⎢ F (X) = M(X) ⎣

d dt f1

⎤

.. ⎥ . . ⎦ d dt fn

Since the right hand side of the last equation doesn’t depend on t, we have d fi ∈ C, i = 1, . . . , n dt hence fi = αi t + βi , αi , βi ∈ C, i = 1, . . . , n. Let G1 be the representation of the differential Galois group of L1 (y) = 0 given by y1 , . . . , yn , and   1 d = dimC C[X1 , . . . , Xn ]G  . We can complete the elements P1 , . . . , Pn to form a basis (P1 , . . . , Pn , . . . , Pd ) of the C-vector 1 space C[X1 , . . . , Xn ]G  . In particular (P1 : . . . : Pn : . . . : Pd ) is a homogeneous coordinate sysn tem for P(C /G1 ). Let us denote [Y] (t) = (y1 : . . . : yn ). If we denote fi = Pi (y1 , . . . , yn ), for i = n + 1, . . . , d, then the image of the map G1 ,P ◦ [Y] in the homogeneous coordinate system (P1 : . . . : Pn : Pn+1 : . . . : Pd ) is

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G1 ,P ◦ [Y] (t) = (α1 t + β1 : . . . : αn t + βn : fn+1 : . . . : fd ). Hence G1 ,P ◦ [Y] is a birational morphism and L1 (y) = 0 is standard. The proof is completed by noting that the morphism to the quotient Fano curve is the same for two rationally projective linear ordinary differential equation, therefore L(x) = 0 is also standard. 2 1.6. A method to solve all algebraic linear ordinary differential equation in terms of standard equations Let L(x) be an algebraic linear ordinary differential equation. We will generalize the first part of the proof of Theorem 5 so that we can identify explicitly the pullback map and the standard equation from Theorem 4. In this way we will obtain closed form solutions to L(x) in terms of solutions to a now known standard equation. Let us fix n linearly independent solutions x1 , . . . , xn and let G be the representation of the differential Galois group of L(x) = 0 associated to this choice of solutions. Then the map t → G,P ◦ [X] (t) defines a ramified covering of the quotient Fano curve by the Riemann sphere parameterized by t . Let us denote by C the quotient Fano curve. The Lüroth theorem implies that C is birational to a Riemann sphere and therefore it is parametrized by some s = s(t) ∈ C(t). As in the proof of Theorem 5, we can find homogeneous polynomials Q1 , . . . , Qn−2 , P , Q ∈ C[X1 , . . . , Xn ]G , all of same degree and algebraically independent, such that P (x1 , . . . , xn ) =s Q(x1 , . . . , xn ) and Qi (x1 , . . . , xn ) = 0, for i = 1, . . . , n − 2. Let f = Q(x1 , . . . , xn ) ∈ C(t). Consider L1 (y) = 0, the linear ordinary differential equation projectively equivalent to L(x) = 0 with solutions yi (t) = f −1/ deg(Q) xi (t),

i = 1, . . . , n.

We have Qi (y1 , . . . , yn ) = 0, for i = 1, . . . , n − 2, Q(y1 , . . . , yn ) = 1, P (y1 , . . . , yn ) = s.

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We fix a non-singular value t0 of L1 (y) = 0 and we set s0 = s(t0 ). Let (r1 , . . . , rn ) be a solution with initial condition (r1 , . . . , rn )(s0 ) = (y1 , . . . , yn )(t0 ) to the differential dynamical system d X = F (X), ds where ⎡ ⎤ 0 ⎢  −1 ⎢ .. ⎥ ⎥ F (X) = M(X) ⎢.⎥ ⎣0⎦ 1 and M(X) = D(Q1 , . . . , Qn−2 , Q, P )(X) is the total derivative of (X1 , . . . , Xn ) −→ (Q1 , . . . , Qn−2 , Q, P ). Then xi (t) = f 1/ deg(Q) yi (t) = f 1/ deg(Q) ri (s(t)). In the following section we will show how to obtain explicitly the algebraic linear ordinary differential equation L(r) = 0 with full system of solutions r1 , . . . , rn which, by Theorem 5, is standard. 2. Differential dynamical systems invariant under finite linear algebraic group actions In the previous section we showed how, starting with an algebraic irreducible linear ordinary differential equation, we can construct a first order ordinary differential equation having as solutions a full system of solutions to the original equation. Moreover, if the resulting ordinary differential equation is a dynamical system, then the original equation is a standard equation. In this section we will characterize the differential dynamical systems associated to standard equations. Furthermore, we will show that every such dynamical system will give rise to a family of standard equations, corresponding to the different orbits. For the rest of this section we will assume that G ⊂ GLn (C) is a finite group. 2.1. G-homogeneous dynamical systems Definition 4. Let G ⊂ GLn (C) be a finite group and d X = F (X) dt

(9)

a dynamical system such that every coordinate function of F = (F1 , . . . , Fn ) is rational, i.e. Fi ∈ C(X1 , . . . , Xn ), for i = 1, . . . , n. We say that this dynamical system is G-homogeneous of degree  ∈ Z>0 if there exist n homogeneous polynomials P1 , . . . , Pn ∈ C[X1 , . . . , Xn ]G , algebraically independent over C, and of degree  such that if X(t) is a solution of the system, then

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d Pi (X)(t) = αi ∈ C, i = 1, . . . , n. dt Note that in this case ⎧⎡ ⎪ ⎨ ∂P1 /∂X1 ⎢ .. F (X) = ⎣ . ⎪ ⎩ ∂Pn /∂X1

⎫−1 ⎡ ⎤ ⎤ α1 ∂P1 /∂Xn ⎪ ⎬ ⎥ ⎢ .. ⎥ .. ⎦ (X) ⎣ . ⎦. . ⎪ ⎭ αn ∂Pn /∂Xn

... .. . ...

2.2. Linear ordinary differential equations for orbits of G-homogeneous dynamical system In this section we will show how the coordinates x1 , . . . , xn of each solution X(t) to a G-homogeneous dynamical system form a full system of solutions to an algebraic linear ordinary differential equation. Let P1 , . . . , Pn ∈ C[X1 , . . . , Xn ]G be algebraically independent over C and α1 , . . . , αn ∈ C. Then, for any β1 , . . . , βn ∈ C, the C-morphism C[P1 , . . . , Pn ] −→ C(t) Pi −→ αi t + βi , i = 1, . . . , n can be extended to a morphism  : C[X1 , . . . , Xn ]G −→ C(t).

(10)

d Now let dt X = F (X) be a G-homogeneous dynamical system of degree , and let P1, . . . , Pn and α1 , . . . , αn be as in Definition 4. Let us fix a solution X(t) = (x1 , . . . , xn ) to the dynamical system. Then there exist β1 , . . . , βn ∈ C such that

Pi (x1 , . . . , xn ) = αi t + βi ,

i = 1, . . . , n,

hence ⎤ ⎧⎡ x1 ⎪ ∂P1 /∂X1 d ⎢ . ⎥ ⎨⎢ .. ⎣ .. ⎦ = ⎣ . ⎪ dt ⎩ xn ∂Pn /∂X1 ⎡

... .. . ...

⎫−1 ⎡ ⎤ ⎤ α1 ∂P1 /∂Xn ⎪ ⎬ ⎥ ⎢ .. ⎥ .. ⎦ (x1 , . . . , xn ) ⎣ . ⎦. . ⎪ ⎭ αn ∂Pn /∂Xn

Differentiating with respect to t we obtain 

d dt

2

⎛⎧⎡ ⎤ x1 ⎪ ∂P1 /∂X1 ⎜⎨⎢ ⎢ .. ⎥ .. ⎜ ⎣ . ⎦=D⎝ ⎣ . ⎪ ⎩ xn ∂Pn /∂X1 ⎡ ⎤ x1 d ⎢ . ⎥ ⎣ .. ⎦ dt xn ⎡

... .. . ...

⎤⎫−1 ⎡ ⎤⎞ α1 ∂P1 /∂Xn ⎪ ⎬ ⎥ ⎢ .. ⎥⎟ .. ⎦ ⎣ . ⎦⎟ . ⎠ (x1 , . . . , xn ) · ⎪ ⎭ αn ∂Pn /∂Xn

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⎛⎧⎡ ⎪ ∂P1 /∂X1 ⎜⎨⎢ .. ⎜ =D⎝ ⎣ . ⎪ ⎩ ∂Pn /∂X1 ⎧⎡ ⎪ ⎨ ∂P1 /∂X1 ⎢ .. ⎣ . ⎪ ⎩ ∂Pn /∂X1

⎤⎫−1 ⎡ ⎤⎞ α1 ∂P1 /∂Xn ⎪ ⎬ ⎥ ⎢ .. ⎥⎟ .. ⎦ ⎣ . ⎦⎟ . ⎠ (x1 , . . . , xn ) · ⎪ ⎭ αn ∂Pn /∂Xn

... .. . ... ... .. . ...

13

⎫−1 ⎡ ⎤ ⎤ α1 ∂P1 /∂Xn ⎪ ⎬ ⎥ ⎢ .. ⎥ .. (x , . . . , x ) ⎦ 1 ⎣ . ⎦, n . ⎪ ⎭ αn ∂Pn /∂Xn

where D is the total derivative with respect to (X1 , . . . , Xn ). # d $i Repeating this process we obtain rational expressions for dt xj , for i = 0, 1, . . . , n and j = 1, . . . , n in terms of x1 , . . . , xn . Hence for every value of t where the solution X(t) = (x1 , . . . , xn ) is defined we have n + 1 vector in Cn 

d dt

n

⎤ x1 ⎢ .. ⎥ ⎣ . ⎦,

⎡

⎡

... ,

xn

⎤ x1 d ⎢ . ⎥ ⎣ .. ⎦ , dt xn

⎡

⎤ x1 ⎢ .. ⎥ ⎣ . ⎦. xn

To find the linear dependence relation between them, we rewrite their coordinates in terms of (x1 , . . . , xn ) by using the expressions we obtained through iterated differentiation with respect to t . As the coefficients in this linear system are in the field C(x1 , . . . , xn ) so is their solution. Hence there exist Qn , . . . , Q1 , Q0 ∈ C(x1 , . . . , xn ) such that

0 = Qn

⎧ ⎪ ⎨

d ⎪ ⎩ dt

n

⎧ ⎡ ⎤⎫ ⎧⎡ ⎤⎫ ⎤⎫ x1 ⎪ ⎪ ⎪ ⎬ ⎬ ⎬ ⎨ d x1 ⎪ ⎨ x1 ⎪ ⎢ .. ⎥ ⎢ .. ⎥ ⎢ . ⎥ ⎣ . ⎦ + Q0 ⎣ .. ⎦ . ⎣ . ⎦ + . . . + Q1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎭ ⎭ ⎩ dt ⎩ xn xn xn ⎡

(11)

Now, since each P1 , . . . , Pn is G-invariant, and the action of G as well as differentiation are C-linear, the G-action permutes the solutions of the G-homogeneous dynamical system. Therefore, the linear dependence relation in (11) is G-invariant and Qn , . . . , Q1 , Q0 ∈ C(x1 , . . . , xn )G . The evaluations Pi (x1 , . . . , xn ) = αi t + βi , for i = 1, . . . , n can be extended via the C-morphism  in (10) to the field C(x1 , . . . , xn )G . Therefore, we obtain cn , . . . , c1 , c0 ∈ C(t) such that ci = Qi (x1 , . . . , xn ), for i = 1, . . . , n and ⎧ ⎧ ⎡ ⎤⎫ ⎧⎡ ⎤⎫ ⎡ ⎤⎫ ⎪ ⎪ ⎪ ⎨ d n x1 ⎪ ⎨ d x1 ⎪ ⎨ x1 ⎪ ⎬ ⎬ ⎬ ⎢ .. ⎥ ⎢ .. ⎥ ⎢ . ⎥ cn ⎣ . ⎦ + c0 ⎣ .. ⎦ = 0. ⎣ . ⎦ + . . . + c1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dt ⎩ dt ⎩ ⎭ ⎭ ⎭ xn xn xn Hence the coordinates of X(t) = (x1 , . . . , xn ) are solutions to  cn

d dt

n

 x + cn−1

d dt

n−1 x + . . . + c1

d x + c0 x = 0. dt

We now prove that the coordinates of the solutions to G-homogeneous dynamical systems are algebraic by showing that the differential Galois group of the equation is finite.

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d Theorem 6. Let G ⊂ GLn (C) be a finite group and dt X = F (X) a G-homogeneous dynamG ical system. Let P1 , . . . , Pn ∈ C[X1 , . . . , Xn ] and let α1 , . . . , αn ∈ C be as in Definition 4. Let X(t) = (x1 , . . . , xn ) be a solution to this G-homogeneous dynamical system and let β1 , . . . , βn ∈ C be such that

Pi (x1 , . . . , xn ) = αi t + βi ,

i = 1, . . . , n.

Let K0 ⊆ C(t) be the field generated by the image of the extension  in (10). Let  cn

d dt

n

 x + cn−1

d dt

n−1 x + . . . + c1

d x + c0 x = 0 dt

be the linear ordinary differential equation satisfied by x1 , . . . , xn , and let G1 ∈ GLn (C) be the representation of the differential Galois group of this equation over K0 given by the solutions x1 , . . . , xn . Then, if x1 , . . . , xn are linearly independent over C, we have G1 = G. Proof. Under the evaluation K0 -morphism    : K0 Xij | i, j = 1, . . . , n

1 −→ K det(Xij ) Xij −→

d i−1 xj , dt i−1

where  K = K0

d dt

i−1

   xj  i, j = 1, . . . , n ,

every G-invariant polynomial is sent to an element in K0 . Therefore K G = K0 = K G1 . Using the Galois correspondence we obtain G1 = G. 2 3. Example: Hesse pencil In this section, let G ⊂ SL3 (C) be the group of order 27 generated by the matrices [32] ⎡

1 0 ⎣0 ω 0 0

⎤ ⎡ ⎤ 0 0 1 0 0 ⎦ and ⎣ 0 0 1 ⎦ , 1 0 0 ω2

where ω is a primitive 3rd root of unity, i.e. ω2 + ω + 1 = 0. Three algebraically independent invariants in C[X1 , X2 , X3 ] are P3 = X13 + X23 + X33 , Q3 = X1 X2 X3 , P6 = X13 X23 + X23 X33 + X33 X13 .

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15

Hesse pencil is the collection of G-invariant cubic curves  % &  C(μ : λ) = (X1 : X2 : X3 ) ∈ P2 (C) (μP3 + λQ3 )(X1 : X2 : X3 ) = 0 . We have the generic case C(1 : λ) that is an elliptic curve with j-invariant j (1 : λ) = −λ3

(λ3 − 216)3 (λ3 + 27)3

if λ = −3, −3e2πi/3 , −3e−2πi/3 , and the singular cases C(1 : −3), C(1 : −3e2πi/3 ) and C(1 : −3e−2πi/3 ) that are the union of a line and a rational curve, and C(0 : 1) that is the union of three lines. We will apply the methods from this paper to obtain linear ordinary differential equations whose associated Schwarz maps [Xλ ](t) parametrizes C(1 : λ). Let us consider the G-homogeneous dynamical system of degree 6 defined by the equations P3 Q3 (xλ,1 , xλ,2 , xλ,3 ) = −λ

(12)

P6 (xλ,1 , xλ,2 , xλ,3 ) = t

(13)

Q23 (xλ,1 , xλ,2 , xλ,3 ) = 1.

(14)

It yields the family of algebraic linear differential equations  0=

 2 3(6t 2 − λ2 t + 9λ) d x 3 2 2 3 4t − λ t + 18λt − 4λ + 27 dt   8 12t − λ2 d + x 3 2 2 3 9 4t − λ t + 18λt − 4λ + 27 dt d dt

−

3

x+

8 1 x. 27 4t 3 − λ2 t 2 + 18λt − 4λ3 + 27

The generic equation has four singularities. The discriminant of the denominator 4t 3 − λ2 t 2 + 18λt − 4λ3 + 27 is −16(λ3 + 27). In the three singular cases, the discriminant vanishes, the equation has three singularities and the equation factors as the product of one of second order and one of first order. From equations (12), (13), (14) we can obtain two different G-homogeneous dynamical systems of degree 6. The first one differentiating the equations with respect to t d P3 Q3 (X1 , X2 , X3 ) = 0 dt d P6 (X1 , X2 , X3 ) = 1 dt d 2 Q (X1 , X2 , X3 ) = 0 dt 3 and the other one with respect to λ

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d P3 Q3 (X1 , X2 , X3 ) = −1 dλ d P6 (X1 , X2 , X3 ) = 0 dλ d 2 Q (X1 , X2 , X3 ) = 0. dλ 3 With these two G-homogeneous dynamical system we can study the family of Schwarz maps [Xλ ](t) in terms of the deformation parameter λ. Note that the two vector fields defining the two G-homogeneous dynamical systems commute, therefore we can define Xt (λ) = X(λ, t) = Xλ (t) and xt,i (λ) = xi (λ, t) = xλ,i (t), for i = 1, 2, 3. The map [Xt ](λ) is the Schwarz map associated to the equation  0=

3

λt 5 + 12λ2 t 3 + 42λ3 t + 3t 4 + 27λt 2 + 324λ2 − 405t (t 3 + 9λt + 54)(λ2 t 2 + 4λ3 − 4t 3 − 18λt − 27)   4t 5 + 39λt 3 + 216λ2 t + 297t 2 + 2106λ 2 d x + 3 2 2 3 3 9 (t + 9λt + 54)(λ t + 4λ − 4t − 18λt − 27) dλ d dλ

+

x +3



d dλ

2 x

2t 3 − 9λt − 135 4 x. 27 (t 3 + 9λt + 54)(λ2 t 2 + 4λ3 − 4t 3 − 18λt − 27)

The generic equation has five singularities (λ = −(t 3 + 54)/9t is apparent), and the Fano curve parametrized by [Xt ](λ) is the elliptic curve X13 X23 + X23 X33 + X33 X13 − tX12 X22 X33 = 0 with j-invariant j (1 : −t) = t 3

(t 3 + 216)3 , (t 3 − 27)3

for t = 3, 3e2πi/3 , 3e−2πi/3 . The discriminant of the denominator (t 3 + 9λt + 54)(λ2 t 2 + 4λ3 − 4t 3 − 18λt − 27) is 16(5t 3 − 864)2 (t 3 − 27)7 . As before, in the cases t = 3, 3e2πi/3 , 3e−2πi/3 the equation factors. In the case when 5t 3 − 864 = 0, the apparent singularity and a root of λ2 t 2 + 4λ3 − 4t 3 − 18λt − 27 are confluent singularities. References [1] http://matematicas.uniandes.edu.co/~csanabria/AlgLODE.html. [2] Michela Artebani, Igor Dolgachev, The Hesse pencil of plane cubic curves, Enseign. Math. (2) 55 (3–4) (2009) 235–273. [3] Francesco Baldassarri, On second-order linear differential equations with algebraic solutions on algebraic curves, Amer. J. Math. 102 (3) (1980) 517–535. [4] Francesco Baldassarri, Bernard Dwork, On second order linear differential equations with algebraic solutions, Amer. J. Math. 101 (1) (1979) 42–76.

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