The Bell locus of rational functions and problems of Goldberg

Transcription

1 Communications of JSSAC (2012) Vol. 1, pp The Bell locus of rational functions and problems of Goldberg Masayo Fujimura National Defense Academy Mohaby Karima Nara Women s University Masahiko Taniguchi Nara Women s University (Received 04/Feb/2010 Accepted 26/Apr/2010) Abstract The spaces of Möbius equivalence classes of rational functions can be represented by the Bell loci. Noting this fact, we solve problems of Goldberg, namely, determine several kinds of the non-generic loci for the map from the Bell locus CB d to the space of the sets of critical points explicitly when the degree is small. The symbolic and algebraic computation system is crucial for the results. 1 Introduction A general form of a rational function of degree d is P(z) Q(z) with polynomials P(z) and Q(z) of degree at most d, where P(z) and Q(z) have no common nonconstant factors and one of them has d as the degree, and the canonical family C d of rational functions of degree d is defined as the totality of those with monic Q(z) of degree d: { R(z) = P(z) } : deg Q = d, Resul(P, Q) 0, Q is monic. Q(z) Writing P(z) = p d z d + + p 0, Q(z) = z d + q d 1 z d q 0, we call the vector (p d,, p 0, q d 1,, q 0 ) the system of coefficient parameters for C d. Also see Theorem 2.4 in [5]. On the other hand, from the viewpoint of the geometric function theory, special attentions have been paid to rational functions of the overlap locus whose overlap type contains an integer not masayo@nda.ac.jp iam_ karima@cc.nara-wu.ac.jp The third author is partially supported by Grant-in-Aid for Scientific Research (C) tanig@cc.nara-wu.ac.jp c 2012 Japan Society for Symbolic and Algebraic Computation

2 68 Communications of JSSAC Vol. 1 less than 3. (Cf. Definition 3.7 and Theorem 3.8 in [5].) Actually, every non-degenerate d-ply connected planar domain can be mapped biholomorphically onto a domain d 1 W w := z C : u k z + z v k < 1 with some complex vectors w = (u 1,..., u d 1, v 1,..., v d 1 ). See, [7]. Such domains W w are a new type of canonical planar domains with connectivity d, which are called Bell representations. Here recall that, every overlap locus C{n 1,, n p } admits so-called decomposition parameters: Letting ˆP(z) = P(z) zq(z) and assuming that is not a fixed point, Q(z)/ ˆP(z) has a unique partial fractions decomposition α 1,n1 (z ζ 1 ) n 1 k=1 + + α 1,1 + α 2,n 2 z ζ 1 (z ζ 2 ) n + + α p,1, 2 z ζ p where, ζ k are fixed points of R(z), α k,nk 0 for every k, and p k=1 α k,1 = 1. The set {ζ k } of fixed points and the set {α k,l } of coefficients give a system of coordinates for C{n 1,, n p }, and is called the system of decomposition parameters for C{n 1,, n p }. The Bell locus CB d is the union of the overlap locus C{n 1,, n p } with n 1 3 such that the corresponding ζ 1 =. Bell representations are those in the Bell locus CB d with generic polar divisors. Proposition 1 The Bell locus CB d has a system of coordinates consisting of coefficients in the representation with z + ˆP(z) Q(z), ˆP(z) = a d 2 z d a 0, Q(z) = z d 1 + b d 2 z d b 0. Next, we state Goldberg s theorem. Definition 2 We say that two rational functions R 1 and R 2 are Möbius equivalent if there is a Möbius transformation M : Ĉ Ĉ such that R 2 = M R 1. We denote by X d the set of all Möbius equivalence classes of rational functions of degree d. Then Goldberg showed Proposition 3 ([6] Thoerem 1.3) Every (2d 2)-tuple B of points in Ĉ is the set of critical points of at most C(d) classes in X d, where C(d) is the Catalan number: C(d) = 1 d ( 2d 2 d 1 The maximal is attained on a Zariski open subset of the space Ĉ 2d 2 of all B. Remark 4 Every set of critical points of a rational function is admissible, namely, every point has multiplicity at most d 1. ).

3 Communications of JSSAC Vol In the sequel, we will assume that is non-critical, for the other cases can be treated similarly and more easily. Then, the Bell locus represents the corresponding subset of X d faithfully. Proposition 5 Let X d is the sublocus of X d consisting of those classes of rational functions such that is noncritical. Then every point in X d is represented by a point in CB d. Moreover, different functions in CB d correspond to different points in X d. Consider a polynomial map Φ d of CB d to C 2d 2 defined from the equation by sending to Q 2 (z) + ˆP (z)q(z) ˆP(z)Q (z) = z 2d 2 + c 2d 3 z 2d c 1 z + c 0 (a, b) = (a d 2,, a 0, b d 2,, b 0 ) (c) = (c 2d 3,, c 0 ). Then Goldberg s theorem implies that Φ d is C(d)-valent on a Zariski open subset of C 2d 2. And the problems in [6] 143p mean Problem 6 1) Describe in detail the ramification set of the map Φ d. 2) For every point (c) in C 2d 2, determine the number of points in the preimage Φ 1 d ((c)). Note that to solve these problems are actually very important. For instance, Eremenko and Gabrielov proves the Shapiro conjecture for two-dimensional case by using the fact that, for any given 2d 2 distinct points on the real line, there exist exactly C(d) distinct Möbius equivalent classes of rational functions of degree d with these critical points. See [3] and [4]. (Also cf. [8].) In the next section, we will determine for Φ d, the ramification locus, the exceptional locus, and the degeneration locus where the number of points in the preimage by Φ d is less than C(d), for d = 2, 3, and 4 explicitly. Here the exceptional locus E(d) is the set of all non-admissible points (c) C 2d 2. Recall that Goldberg also asked whether there is a rational function having an arbitrarily given admissible set of points as the set of critical points. We answer this question for d = 2, 3, and 4. 2 The explicit descriptions of the loci for the map Φ d In this section, we use risa/asir, a symbolic and algebraic computation system, to obtain the defining equations of the loci considered. Definition 7 Let ˆP(z) and Q(z) be as before. Set R(d) = { (a, b) C 2d 2 : Resul( ˆP, Q) = 0 }, which is the locus where Φ d is not defined. (In other words, CB d can be identified with C 2d 2 R(d).) First, we begin with

4 70 Communications of JSSAC Vol. 1 Proposition 8 If d = 2, then the map Φ 2 : CB 2 C 2 E(2) is bijective, and the exceptional locus E(2) is the algebraic curve defined by c 2 1 4c 0 = 0. Proof The map Φ 2 is given by (a 0, b 0 ) (b 2 0 a 0, 2b 0 ). The locus R(2) is given by a 0 = 0 and corresponds to the locus E(2) defined by c 2 1 4c 0 = 0. Conversely, each point (c) on the locus c 2 1 4c 0 = 0 coincides with a non-admissible set, because the equation z 2 + c 1 z + c2 1 4 = 0 always has a double root. Therefore c 2 1 4c 0 = 0 gives a defining equation of E(2). If d = 3, we have the following. Theorem 9 If d = 3, the ramification locus of Φ 3 is a 1 = b 2 1 4b 0, Φ 3 (CB 3 ) = C 4 E(3), and Φ 3 is 2-valent on the set of points in C 4 E(3) satisfying that c 2 2 3c 1c c 0 0, E 0 0. Moreover, the exceptional locus E(3) is the algebraic variety defined by E 0 = E 1 = 0. Here E 1 = 216c c 2c 3 c 1 + (216c c 2)c c 3 2, E 0 = 27c ( 4c c 2c 3 )c (( 6c c 2)c 0 + c 2 2 c2 3 4c3 2 )c2 1 +( 192c 3 c (18c 2c c2 2 c 3)c 0 )c c 3 0 +( 27c c 2c c2 2 )c2 0 + ( 4c3 2 c c4 2 )c 0. Proof The Jacobian of the map Φ 3 is b b 0 2b 1 = 4(a 1 b b 0). b 0 b 1 a 0 a 1 + 2b 0 Also, for (c) = (c 3, c 2, c 1, c 0 ) in C 4 E(3), every (a 1, a 0, b 1, b 0 ) in Φ 1 3 ((c)) is a solution of 12b 2 0 4c 2b 0 + c 1 c 3 4c 0 = 0 b 1 = 1 2 c 3 a 1 = 1 4 (8b 0 + c 2 3 4c 2) a 0 = 1 2 (c 3b 0 c 1 ) which has exactly 2 solutions except for the set defined by the discriminant Here, the locus R(3) is given by c 2 2 3c 1c c 0 = 0. a 2 0 b 1a 1 a 0 + a 2 1 b 0 = 0. (2) Eliminating four variables a 0, a 1, b 0, b 1 from the equation r = a 2 0 b 1a 1 a 0 + a 2 1 b 0 by using (1), we have the following equation 432r 2 + E 1 r + E 0 = 0, (3) (1)

5 Communications of JSSAC Vol where E 0, E 1 are as in the theorem. Then, R(3) corresponds to the condition that (3) has 0 as a solution, and there are no rational functions of degree 3 corresponding to (c) if and only if the equation (3) has 0 as a unique solution. By using the relation between coefficients and solutions, we can check that the equation z 4 + c 3 z 3 + c 2 z 2 + c 1 z + c 0 = 0 has a solution of multiplicity at least 3, for every point (c) in the set defined by E 0 = E 1 = 0. Therefore E 0 = E 1 = 0 gives a set of defining equations of E(3). Theorem 10 If d = 4, the ramification locus of Φ 4 is given by (b 2 a 2 b b 1b 2 9b 0 )a 1 b 1 a ( 3a 0 + b 1 b b 0b 2 5b 2 1 )a 2 +(b 2 2 3b 1)a 0 4b 0 b b2 1 b b 0b 1 b 2 4b b2 0 = 0. Moreover, for a given (c) in C 6 E(4), b 1 is a solution of algebraic equation of degree 5, and other coefficients are determined from b 1 and (c) uniquely. In particular, Φ 4 (CB 4 ) = C 6 E(4), and Φ 4 is 5-valent on the set of points in C 6 E(4) satisfying D 0 and E 0 0, where D is the discriminant given explicitly in the proof and E 0 is stated in Remark 12, where the description of the exceptional set E(4) is given. Proof The Jacobian of the map Φ 4 is 8((b 2 a 2 b b 1b 2 9b 0 )a 1 b 1 a ( 3a 0 + b 1 b b 0b 2 5b 2 1 )a 2 +(b 2 2 3b 1)a 0 4b 0 b b2 1 b b 0b 1 b 2 4b b2 0 ). For (c) in C 6 E(4), every (a 2, a 1, a 0, b 2, b 1, b 0 ) in Φ 1 4 ((c)) is a solution of B 1 = 1296b c 4b (216c 5c 3 432c c 2 4 )b3 1 +( 144c 4 c 5 c 3 + (24c c 4)c 2 216c 5 c c 0 48c 3 4 )b2 1 +(9c 2 5 c2 3 + ( 36c 5c c c 2 4 c 5)c 3 + ( 8c 4 c c2 4 )c 2 +( 4c c 4c 5 )c 1 + (144c c 4)c 0 )b 1 3c 4 c 2 5 c2 3 +((2c c 4c 5 )c 2 + ( 6c c 4)c 1 )c 3 8c 2 5 c c 5c 1 c 2 72c (4c4 5 48c 4c c2 4 )c 0 = 0 B 0,1 = 12c 5 b (72b 0 + 4c 4 c 5 )b 1 + (4c c 4)b 0 12c 1 + 4c 5 c 2 c 2 5 c 3 = 0 B 0,2 = 12b c 4b (4c 5b 0 + 4c 2 c 5 c 3 )b b 2 0 6c 3b 0 12c 0 = 0 (4) b 2 = 1 2 c 5 a 2 = 1 4 (8b 1 + c 2 5 4c 4) a 1 = 1 2 (c 5b 1 + 2b 0 c 3 ) a 0 = 1 12 (12b2 1 4c 4b 1 + 2c 5 b 0 + c 5 c 3 4c 2 ). There is a unique common root b 0 of B 0,1 = B 0,2 = 0 if and only if b 1 is a solution of B 1 = 0, since Resul b0 (B 0,1, B 0,2 ) = 48B 1. Therefore, there are five solutions of B 1 = 0 except for the locus defined by the following discriminant D: (6c 3 5 c 4 c c 3c c 2c 5 324c 1 ) 2 12(108c 0 c 2 1 c 4 648c 2 0 c 1c 5 +( 108c 0 c 1 c 2 27c 3 1 )c 3+32c 0 c c 2 1 c2 2 )c (2916c3 0 c2 5 + ((972c2 0 c c 0 c 2 1 )c 3 234c 0 c 1 c c3 1 c 2)c c 0 c 1 c ( 27c 0c c 2 1 c 2)c ( 51c 1c c2 0 c 1)c 3 + 8c c2 0 c c 0c 2 1 c 2 162c 4 1 )c5 4 + (49572c2 0 c 1c 3 +

6 72 Communications of JSSAC Vol c 2 0 c c 0c 2 1 c c 4 1 )c2 5 + ( 972c2 0 c3 3 + (8316c 0c 1 c c 3 1 )c2 3 + ( 2412c 0c c2 1 c c 3 0 )c 3 + 8c 1 c c2 0 c 1c c 0 c 3 1 )c c 2 1 c c 1c 2 2 c3 3 + ( 9c c2 0 c c 0 c 2 1 )c2 3 ( 13500c 0c 1 c c3 1 c 2)c c 0 c c2 1 c c3 0 c c 2 0 c2 1 )c4 4 4((18225c 3 0 c c 2 0 c 1c 2 849c 0 c 3 1 )c3 5 + ((6156c2 0 c c 0 c 2 1 )c2 3 + ( 378c 0c 1 c c3 1 c 2)c 3 330c 0 c c2 1 c c3 0 c c 2 0 c2 1 )c2 5 +(486c 0c 1 c 4 3 +( 162c 0c c2 1 c 2)c 3 3 +( 324c 1c c 2 0 c 1)c 2 3 +(51c c2 0 c c 0c 2 1 c c 4 1 )c c 0 c 1 c c3 1 c c3 0 c 1)c 5 + ( 972c 0 c 1 c 2 81c 3 1 )c3 3 +(324c 0c c2 1 c c3 0 )c2 3 +(603c 1c c2 0 c 1c c 0 c 3 1 )c 3 96c c2 0 c c 0c 2 1 c c4 1 c c 4 0 )c ((20250c3 0 c c 2 0 c2 1 )c4 5 + ( 43335c 2 0 c 1c (6345c2 0 c c 0c 2 1 c 2 492c 4 1 )c c 0 c 1 c c3 1 c c3 0 c 1)c (2916c 2 0 c4 3 + ( 6804c 0c 1 c c 3 1 )c3 3 + (1836c 0c c2 1 c c3 0 )c2 3 + ( 369c 1c c 2 0 c 1c c 0 c 3 1 )c 3+54c c2 0 c c 0c 2 1 c c4 1 c c 4 0 )c2 5 +(243c2 1 c c 1 c 2 2 c4 3 +(27c c2 0 c c 0 c 2 1 )c3 3 +(22680c 0c 1 c c3 1 c 2)c 2 3 +( 6750c 0c c2 1 c c 3 0 c c 2 0 c2 1 )c 3 468c 1 c c2 0 c 1c c 0c 3 1 c c 5 1 )c 5 486c 2 1 c 2c (324c 1 c c2 0 c 1)c ( 54c c2 0 c c 0c 2 1 c c 4 1 )c2 3 + (55242c 0c 1 c c 3 1 c c3 0 c 1)c c 0 c c2 1 c c3 0 c c2 0 c2 1 c c 0 c 4 1 )c (50625c 3 0 c 1c (30375c3 0 c2 3 + (29700c2 0 c 1c c 0 c 3 1 )c c 2 0 c c 0c 2 1 c c4 1 c c 4 0 )c4 5 + ((9720c2 0 c c 0 c 2 1 )c3 3 + (756c 0c 1 c c3 1 c 2)c ( 972c 0c c2 1 c c 3 0 c c 2 0 c2 1 )c c 1 c c2 0 c 1c c 0c 3 1 c 2 600c 5 1 )c3 5 + (729c 0c 1 c ( 243c 0 c c2 1 c 2)c 4 3 +( 513c 1c c2 0 c 1)c 3 3 +(81c c2 0 c c 0c 2 1 c c 4 1 )c2 3 + ( 7182c 0 c 1 c c3 1 c c3 0 c 1)c c 0 c c2 1 c c3 0 c c2 0 c2 1 c c 0 c 4 1 )c2 5 + (( 2916c 0c 1 c c 3 1 )c4 3 + (972c 0c c2 1 c c3 0 )c3 3 + (2079c 1c c 2 0 c 1c c 0 c 3 1 )c2 3 + ( 324c c2 0 c c 0c 2 1 c c4 1 c c 4 0 )c c 0 c 1 c c3 1 c c3 0 c 1c c 2 0 c3 1 )c c 0 c 2 1 c4 3 + ( 5832c 0c 1 c c 3 1 c 2)c (486c 0c c2 1 c c3 0 c c 2 0 c2 1 )c2 3 + (972c 1c c2 0 c 1c c 0 c 3 1 c c 5 1 )c c 2 0 c c 0c 2 1 c c4 1 c c4 0 c c 3 0 c2 1 )c c 4 0 c6 5 2((101250c3 0 c c 2 0 c2 1 )c c 2 0 c 1c c 0c 3 1 c 2 32c 5 1 )c5 5 +3(4050c2 0 c 1c ( 12150c 2 0 c c 0c 2 1 c 2 144c 4 1 )c2 3 +(4860c 0c 1 c c3 1 c c3 0 c 1)c 3 216c 0 c c2 1 c c 3 0 c c2 0 c2 1 c c 0 c 4 1 )c4 5 6(1458c2 0 c5 3 + ( 162c 0c 1 c 2 162c 3 1 )c4 3 + ( 54c 0c c 2 1 c c3 0 )c3 3 +( 351c 1c c2 0 c 1c c 0 c 3 1 )c2 3 +(54c c2 0 c c 0c 2 1 c c 4 1 c c 4 0 )c c 0 c 1 c c3 1 c c3 0 c 1c c 2 0 c3 1 )c3 5 27(27c2 1 c6 3 18c 1 c 2 2 c5 3 + (3c c2 0 c c 0 c 2 1 )c4 3 + ( 612c 0c 1 c c3 1 c 2)c (180c 0c c2 1 c c 3 0 c c 2 0 c2 1 )c2 3 +(276c 1c c2 0 c 1c c 0c 3 1 c c 5 1 )c 3 48c c2 0 c c 0 c 2 1 c c4 1 c c4 0 c c 3 0 c2 1 )c (18c2 1 c 2c ( 12c 1c c2 0 c 1)c (2c c2 0 c c 0c 2 1 c 2 +75c 4 1 )c3 3 +( 390c 0c 1 c c3 1 c c3 0 c 1)c 2 3 +(72c 0c c2 1 c c 3 0 c c2 0 c2 1 c c 0 c 4 1 )c 3 48c 1 c c2 0 c 1c c 0c 3 1 c c5 1 c c 4 0 c 1)c 5 81(108c 3 1 c5 3 72c2 1 c2 2 c4 3 + (12c 1c c2 0 c 1c c 0 c 3 1 )c3 3 + (108c2 0 c c 0 c 2 1 c c4 1 c c 4 0 )c2 3 + (432c 0c 1 c c3 1 c c3 0 c 1c c 2 0 c3 1 )c 3 144c 2 1 c c3 0 c c2 0 c2 1 c c 0c 4 1 c c c5 0 ) = 0. Remark 11 If d = 5, then we can see that b 2 is a solution of an algebraic equation of degree 14 for every (c) in C 8 E(5). Remark 12 The locus R(4) is given by r = 0, where r = b 2 0 a3 2 + ( b 0b 1 a 1 + ( 2b 0 b 2 + b 2 1 )a 0)a (b 0b 2 a ( b 1b 2 + 3b 0 )a 0 a 1 +(b 2 2 2b 1)a 2 0 )a 2 b 0 a b 1a 0 a 2 1 b 2a 2 0 a 1 + a 3 0. (5)

7 Communications of JSSAC Vol Eliminating six variables in (a, b) from the equation (5) by using (4), we have the following equation r E 4 r E 3 r 3 864E 2 r 2 +16E 1 E 0 r E 2 0 = 0, (6) where each E j ( j = 0, 1, 2, 3, 4) is a polynomial in C[c 0,, c 5 ]. And, there are no rational functions of degree 4 corresponding to (c) if and only if the equation (6) has 0 as a unique solution. On the other hand, by using the relation between coefficients and solutions, we can check that the equation z 6 + c 5 z 5 + c 4 z 4 + c 3 z 3 + c 2 z 2 + c 1 z + c 0 = 0 has a solution of multiplicity at least 4, for every point (c) in the set defined by E 4 = E 3 = E 2 = E 0 = 0. Therefore E 4 = E 3 = E 2 = E 0 = 0 gives a set of defining equations of E(4). Finally, E 0 is given by 3125c 4 0 c6 5 + ( 2500c3 0 c 1c 4 + ( 3750c 3 0 c c 2 0 c2 1 )c c 2 0 c 1c c 0c 3 1 c c 5 1 )c5 5 + ((2000c 3 0 c 2 50c 2 0 c2 1 )c2 4 +(2250c3 0 c2 3 +( 2050c2 0 c 1c c 0 c 3 1 )c 3 900c 2 0 c c 0c 2 1 c c4 1 c c 4 0 )c 4 900c 2 0 c 1c 3 3 +(825c2 0 c c 0c 2 1 c 2 128c 4 1 )c2 3 +( 630c 0c 1 c c3 1 c c3 0 c 1)c c 0 c c2 1 c c3 0 c c2 0 c2 1 c c 0 c 4 1 )c4 5 + (( 1600c3 0 c c 2 0 c 1c 2 36c 0 c 3 1 )c3 4 + (1020c 2 0 c 1c (560c2 0 c c 0c 2 1 c c 4 1 )c c 0 c 1 c 3 2 6c3 1 c c3 0 c 1)c (( 630c2 0 c c 0 c 2 1 )c3 3 +(356c 0c 1 c c3 1 c 2)c 2 3 +( 72c 0c c2 1 c c3 0 c c 2 0 c2 1 )c c 2 0 c 1c c 0 c 3 1 c c 5 1 )c 4+108c 2 0 c5 3 +( 72c 0c 1 c 2 +16c 3 1 )c4 3 +(16c 0c 3 2 4c2 1 c c3 0 )c3 3 +(1980c2 0 c 1c 2 208c 0 c 3 1 )c2 3 + ( 120c2 0 c c 0c 2 1 c c4 1 c c 4 0 )c c 0 c 1 c c3 1 c c3 0 c 1c c 2 0 c3 1 )c3 5 + (256c3 0 c5 4 + ( 192c2 0 c 1c 3 128c 2 0 c c 0c 2 1 c 2 27c 4 1 )c4 4 + ((144c2 0 c 2 6c 0 c 2 1 )c2 3 + ( 80c 0 c 1 c c3 1 c 2)c c 0 c 4 2 4c2 1 c c3 0 c c 2 0 c2 1 )c3 4 + ( 27c2 0 c4 3 + (18c 0c 1 c 2 4c 3 1 )c3 3 + ( 4c 0c c2 1 c c3 0 )c2 3 + (10152c2 0 c 1c 2 682c 0 c 3 1 )c c 2 0 c c 0c 2 1 c c 4 1 c c 4 0 )c2 4 + (3942c2 0 c 1c ( 4536c2 0 c c 0c 2 1 c c 4 1 )c2 3 + (3272c 0c 1 c c 3 1 c c3 0 c 1)c 3 576c 0 c c2 1 c c3 0 c c2 0 c2 1 c c 0 c 4 1 )c c 2 0 c 2c ( 108c 0 c 1 c c3 1 c 2)c 3 3 +(24c 0c 4 2 6c2 1 c c3 0 c c 2 0 c2 1 )c2 3 +(16632c2 0 c 1c c 0c 3 1 c c 5 1 )c 3 192c 2 0 c c 0c 2 1 c3 2 50c4 1 c c4 0 c c 3 0 c2 1 )c2 5 + ((6912c3 0 c 3 640c 2 0 c 1c c 0 c 3 1 )c4 4 +( 4464c2 0 c 1c 2 3 +( 2496c2 0 c c 0c 2 1 c 2 630c 4 1 )c 3 96c 0 c 1 c c3 1 c c3 0 c 1)c ((2808c 2 0 c 2 108c 0 c 2 1 )c3 3 +( 1584c 0c 1 c c3 1 c 2)c 2 3 +(320c 0c c2 1 c c3 0 c c 2 0 c2 1 )c c 2 0 c 1c c 0c 3 1 c c 5 1 )c2 4 + ( 486c2 0 c5 3 + (324c 0c 1 c 2 72c 3 1 )c4 3 + ( 72c 0c c2 1 c c 3 0 )c3 3 +( 22896c2 0 c 1c c 0 c 3 1 )c2 3 +( 5760c2 0 c c 0c 2 1 c c4 1 c c 4 0 )c 3 640c 0 c 1 c c3 1 c c3 0 c 1c c 2 0 c3 1 )c c 2 0 c 1c 4 3 +(5832c2 0 c c 0c 2 1 c 2 900c 4 1 )c3 3 + ( 4464c 0 c 1 c c3 1 c c3 0 c 1)c (768c 0c c2 1 c c3 0 c c2 0 c2 1 c c 0 c 4 1 )c c 2 0 c 1c c 0c 3 1 c c5 1 c c 4 0 c 1)c c 3 0 c6 4 + (768c2 0 c 1c c 2 0 c c 0c 2 1 c c 4 1 )c5 4 + (( 576c2 0 c c 0 c 2 1 )c2 3 + (320c 0c 1 c c3 1 c 2)c 3 64c 0 c c 2 1 c c3 0 c 2 192c 2 0 c2 1 )c4 4 + (108c2 0 c4 3 + ( 72c 0c 1 c c 3 1 )c3 3 + (16c 0c 3 2 4c2 1 c c3 0 )c2 3 + ( 5760c 2 0 c 1c 2 120c 0 c 3 1 )c c 2 0 c c 0c 2 1 c c4 1 c c 4 0 )c3 4 + (5832c2 0 c 1c (8208c 2 0 c c 0c 2 1 c c 4 1 )c2 3 +( 2496c 0c 1 c c3 1 c c3 0 c 1)c c 0 c c2 1 c c 3 0 c c2 0 c2 1 c c 0 c 4 1 )c2 4 + (( 4860c2 0 c c 0 c 2 1 )c4 3 + (2808c 0c 1 c c3 1 c 2)c ( 576c 0 c c2 1 c c3 0 c c 2 0 c2 1 )c2 3 + ( 3456c2 0 c 1c c 0c 3 1 c c 5 1 )c c 2 0 c c 0c 2 1 c c4 1 c c4 0 c c 3 0 c2 1 )c 4+729c 2 0 c6 3 +( 486c 0c 1 c c 3 1 )c5 3 + (108c 0 c c2 1 c c3 0 )c4 3 +(21384c2 0 c 1c c 0 c 3 1 )c3 3 +( 8640c2 0 c c 0c 2 1 c c4 1 c c 4 0 )c2 3 + (6912c 0c 1 c c3 1 c c3 0 c 1c c 2 0 c3 1 )c c 0 c c2 1 c c 3 0 c c2 0 c2 1 c c 0c 4 1 c c c5 0. References [1] D. Cox, J. Little, and D. O Shea, Ideals, Varieties, and Algorithms, UTM, Springer-Verlag, 1998.

8 74 Communications of JSSAC Vol. 1 [2] D. Cox, J. Little, and D. O Shea, Using Algebraic Geometry, GTM 185, Springer-Verlag, [3] A. Eremenko and A. Gabrielov, Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry, Ann. of Math., 155 (2002), [4] A. Eremenko and A. Gabrielov, Elementary proof of the B. and M. Shapiro conjecture for rational functions, preprint. [5] M. Fujimura and M. Taniguchi, Stratification and coordinate systems for the moduli space of rational functions, Conform. Geom. Dyn., 14 (2010), [6] L. Goldberg Catalan numbers and branched covering by the Riemann sphere, Adv. Math., 85 (1991), [7] M. Jeong and M. Taniguchi, Bell representation of finitely connected planar domains, Proc. AMS., 131 (2003), [8] I. Scherbak, Rational functions with prescribed critical points, GAFA, 12 (2002), Masayo Fujimura, Department of Mathematics, National Defense Academy, Yokosuka , JAPAN (masayo@nda.ac.jp) Mohaby Karima, Graduate School of Humanities and Sciences, Nara Women s University, Nara , JAPAN (iam_karima@cc.nara-wu.ac.jp) Masahiko Taniguchi, Department of Mathematics, Nara Women s University, Nara , JAPAN (tanig@cc.nara-wu.ac.jp)