On the Complexity of Exact Counting of Dynamically Irreducible ...

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Jun 14, 2017 - NT] 14 Jun 2017. ON THE COMPLEXITY OF EXACT COUNTING OF. DYNAMICALLY IRREDUCIBLE POLYNOMIALS. DOMINGO ...
arXiv:1706.04392v1 [math.NT] 14 Jun 2017

ON THE COMPLEXITY OF EXACT COUNTING OF DYNAMICALLY IRREDUCIBLE POLYNOMIALS ´ ´ ´ ´ MERAI, ´ DOMINGO GOMEZ-P EREZ, LASZL O AND IGOR E. SHPARLINSKI

Abstract. We give an efficient algorithm to enumerate all sets of r ě 1 quadratic polynomials over a finite field, which remain irreducible under iterations and compositions.

1. Introduction For a finite field Fq and a polynomial f P Fq rXs we define the sequence: ` ˘ f p0q pXq “ X, f pnq pXq “ f f pn´1q pXq , n “ 1, 2, . . . .

The polynomial f pnq is called the n-th iterate of the polynomial f . As in [2, 3, 8, 9], we say that a polynomial f P Fq rXs is stable if all iterates f pnq pXq, n “ 1, 2, . . ., are irreducible over Fq . However here, we prefer to use a more informative terminology of Heath-Brown and Micheli [7] and instead we call such polynomials dynamically irreducible. Let q be and odd prime power, and as in [9], for a quadratic polynomial f pXq “ aX 2 ` bX ` c P Fq rXs, a ‰ 0 we define γ “ ´b{p2aq as the unique critical point of f (that is, the zero of the derivative f 1 ). We remark that for q even, it is known that there does not exist quadratic stable polynomials [1]. Let DI q be the set of dynamically irreducible quadratic polynomials over a finite field of q elements Fq and let DIq “ #DI q be their number. It is shown in [10] that for f pXq “ aX 2 ` bX ` c P Fq rXs one can test whether f P DI q in time q 3{4`op1q , see Lemma 2.1 below. G´omez-P´erez and Nicol´as [5], developing some ideas from [10], have proved that for an odd prime power q we have pq ´ 1q2 ď DIq “ Opq 5{2 log qq, 4 where the implied constant is absolute, see also [6] for an upper bound is given on the number of dynamically irreducible polynomials of degree d ě 2 over Fq . (1.1)

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´ ´ ´ D. GOMEZ-P EREZ, L. MERAI, AND I. E. SHPARLINSKI

Here we consider the question of constructing the set DI q and exact evaluating its cardinality DIq . Trivially, using the above test from [10], one can construct the set DI q in time q 15{4`op1q . One can calculate DIq faster, in time q 11{4`op1q if one uses the correspondence between arbitrary and monic dynamically irreducible polynomials, see Lemma 2.2 below. We show that one can do much better. Theorem 1.1. Let q be an odd prime power. Then there exists an algorithm which computes DIq in time q 9{4`op1q and constructs the set DI q in time q 5{2`op1q . We also study the analogue problem in the semigroup generated by several polynomials under the composition. Let f1 pXq, . . . , fr pXq P Fq rXs be polynomials of positive degree. The set tf1 pXq, . . . , fr pXqu is called dynamically irreducible if all the iterates fi1 ˝ . . . ˝ fin , for i1 , . . . , in P t1, . . . , ru and n ě 1 are irreducible. Ferraguti, Micheli and Schnyder [4] have characterized the sets of monic quadratic polynomial to be dynamically irreducible in terms of the unique critical points of the polynomials. Furthermore, HeathBrown and Micheli [7] have given an algorithm to test whether a set of monic polynomials is dynamically irreducible. Here we consider the question to construct the set DI q prq of all sets of r arbitrary pairwise distinct quadratic polynomials over Fq which are dynamically irreducible. We also denote their number by DIq prq “ #DI q prq. In particular, DIq p1q “ DIq . Furthermore, we use DI ˚q prq to denote the subset DI q prq consisting of monic quadratic polynomials and also use DI˚q prq “ #DI ˚q prq for its cardinality. Let Mpqq and M ˚ pqq be the size of the largest set of dynamically irreducible non-monic and monic quadratic polynomials, respectively. We remark that we have renamed Mpqq of [7] to M ˚ pqq. Then HeathBrown and Micheli [7] have shown that M ˚ pqq ď 32qplog qq4 , while for infinitely many fields Fq we have M ˚ pqq ě 0.5plog qq2 . It is easy that the bound (4.4) below implies Mpqq ď q 3{2`op1q . On the other hand, in Example 2.6 we present an explicit family of quadratic polynomials which shows that Mpqq ě pq ´ 1q{2 for infinitely many q (namely for those for which ´1 a square in Fq ). Thus in the case of arbitrary polynomials the gap between upper and lower bounds is less dramatic than the exponential gap in the case of monic polynomials. We note that the proof of Theorem 1.1 is based on the close link between the sets DI q “ DI q p1q and DI ˚q “ DI ˚q p1q, see Lemma 2.2 below. On the other hand, for r ě 2 there does not seem to be any

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close relation between DI q prq and DI ˚q prq. Accordingly, in this case our result is weaker than for r “ 1. We note that throughout the paper op1q denotes the quantity εpqq, which depends only on q (and does not depend on r) with εpqq Ñ 0 as q Ñ 8. Theorem 1.2. Let q be an odd prime power and r ě 2. Then there exists an algorithm which computes DIq prq and constructs the set DI q prq in time q p3{2`op1qqr`5{2 as q Ñ 8 and uniformly over r. As a by-product of the ideas behind our algorithm, we also obtain an analogue of the upper bound (1.1): Theorem 1.3. Let q be an odd prime power and r ě 2. Then DIq prq ď q p3{2`op1qqr`2 as q Ñ 8. 2. Preliminaries We need to recall some notions of the theory of dynamically irreducible quadratic polynomials, mainly introduced by Jones and Boston [8, 9] (we recall that they are called ‘stable’ in [8, 9]). The critical orbit of f is the set Orb pf q “ tf pnq pγq : n “ 2, 3, . . .u, where γ “ ´b{p2aq as the unique critical point of f . We partition Fq into the sets of squares Sq and non-squares Nq , that is Sq “ ta2 : a P Fq u and Nq “ Fq zSq . We recall that for a P Fq one can check whether a P Nq by evaluating its pq ´ 1q{2-th power, as a P Nq if and only if apq´1q{2 “ ´1. By [9, Proposition 3], a quadratic polynomial f P Fq rXs is dynamically irreducible if the adjusted orbit Orb pf q “ t´f pγqu Y Orb pf q satisfies Orb pf q Ď Nq . Clearly, the critical orbit Orb pf q of f is a finite set. Furthermore, by [10, Theorem 1] the size of the critical orbit of a dynamically irreducible quadratic polynomial f admits a nontrivial estimate Lemma 2.1. There is an absolute constant c1 such that for P T M “ c1 q 3{4

for any f P DI q we have we have

#Orb pf q ď M.

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The following result reduces the problem of counting dynamically irreducible polynomials to dynamically irreducible monic polynomials [5, Lemma 2]. Lemma 2.2. For any dynamically irreducible polynomial f pXq and a P F˚q , f paXq fa pXq “ P Fq rXs a is also a dynamically irreducible polynomial. In order to get the upper bound in (1.1), G´omez-P´erez and Nicol´as [5] estimate the number of dynamically irreducible quadratic polynomials by the number of such polynomials that there is no square among the first Oplog qq elements of their critical orbit. Their result can be summarised in the following way. Lemma 2.3. There is an absolute constant c2 such that for R V log q K“ ` c2 2 log 2 we have

# f pXq “ X 2 ` bX ` c P Fq rXs : ( f pnq p´b{p2aqq P Nq , n “ 1, . . . , K “ Opq 3{2 log qq.

In the following we extend some results of Ferraguti, Micheli and Schnyder [4] and Heath-Brown and Micheli [7] about dynamically irreducible sets for non-monic quadratic polynomials. First we need the following result of Jones and Boston [9] (here we state the result in a corrected form, see [7]). Lemma 2.4. Let q be an odd prime and let f pXq “ aX 2 ` bX ` c P Fq rXs and γ “ ´b{p2aq be the unique critical point of f . Suppose that g P Fq rXs is such that g ˝ f pn´1q has degree d, has leading coefficient e and is irreducible over Fq for some n ě 1. Then g ˝ f pnq is irreducible over Fq rXs if and only if p´aqd gpf pnq pγqq{e P Nq . As a corollary, we get the characterization of dynamically irreducible sets of r quadratic polynomials. Corollary 2.5. Let q be an odd prime. Let fi pXq “ ai X 2 ` bi X ` ci P Fq rXs be irreducible quadratic polynomials for 1 ď i ď r. Write γi “ ´bi {p2ai q. Then f1 , . . . , fr form a dynamically irreducible set if and only if for all integers n ě 1 and 1 ď i1 , . . . , in ď r we have (2.1)

a´1 i1 pfi1 ˝ . . . ˝ fin qpγi q P Nq .

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Proof. First assume that f1 , . . . , fr form a dynamically irreducible set, that is, each iterate fi1 ˝ . . . ˝ fin with n ě 1 and 1 ď i1 , . . . , in ď r, is irreducible. Applying Lemma 2.4 with g “ fi1 ˝ . . . ˝ fin´1 and f “ fin we derive (2.2)

e´1 pfi1 ˝ . . . ˝ fin qpγi q P Nq ,

where e is the leading coefficient of fi1 ˝. . .˝fin´1 . By induction, one can n´2 easily get, that e “ ai1 a2i2 . . . a2in´1 . Then (2.1) is equivalent to (2.2). Conversely, if fi1 ˝ . . . ˝ fin is a reducible iterate with the smallest degree, then writing again g “ fi1 ˝ . . . ˝ fin´1 and f “ fin , we see that (2.2) fails, which contradicts (2.1).  We remark that Corollary 2.5 allows us to exhibit a large family of dynamically irreducible set of quadratic polynomials. Example 2.6. Let Fq be of characteristic p ” 1 pmod 4q or be an even power of p and fix b P F˚q . Let fa “ apX ´ bq2 ` b for a P F˚q . Then the set F “ tfa | ab P Nq u of cardinality #F “ pq ´ 1q{2 is dynamically-irreducible. Indeed, let r “ pq ´ 1q{2 and take a1 , . . . , ar such that ai b is a nonsquare in F˚q for all 1 ď i ď r. We first notice that n´1

n

fai1 ˝ ¨ ¨ ¨ ˝ fain pXq “ ai1 a2i2 ¨ ¨ ¨ a2in pX ´ bq2 ` b, 1 ď i1 , . . . , in ď r, and in particular that fai1 ˝¨ ¨ ¨˝fain pbq “ b. We apply now Corollary 2.5 to conclude that the set F is dynamically-irreducible if and only if ´1 ´a´1 i1 b and ai1 b

are nonsquares in Fq for all 1 ď i1 ď r. Since ´1 P Sq is a square, the condition above is equivalent with ai1 b P Nq , which concludes our argument. If f1 , . . . , fr form a dynamically irreducible set of quadratic polynomials, then each polynomial fi has to be dynamically irreducible for 1 ď i ď r. Then in the same way as in the proof of [10, Theorem 1] (applying for f1 ) one may get the following result. Lemma 2.7. There is an absolute constant c3 such that for M “ rc3 q 3{4 s the following holds. If f1 , . . . , fr form a dynamically irreducible set of quadratic polynomials over a finite field Fq of odd characteristic and Γ Ď Fq such that a´1 i1 pfi1 ˝ . . . ˝ fin qpγq P Nq ,

n ě 1, r ě i1 , . . . , in ě 1,

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for all γ P Γ, then #Γ ď M. If not all of the polynomials f1 , . . . , fr are constant multiple of each others, one can get a better bound, which is based on the following result from [7, Lemma 2]. Lemma 2.8. Let g1 , g2 be distinct monic quadratic polynomials over Fq . Suppose that g i1 ˝ . . . ˝ g in “ g j 1 ˝ . . . ˝ g j m with i1 , . . . , in , j1 , . . . , jm P t1, 2u. Then m “ n and ih “ jh for h “ 1, . . . , n. We now obtain a stronger version of Lemma 2.7 in this special case. Lemma 2.9. There is an absolute constant c4 such that for W Sd log log q ` c4 K“ 2 log 2 the following holds. If f1 pXq, f2 pXq P Fq rXs are two quadratic polynomials with f1 pXq{f2 pXq R Fq such that they form a dynamicallyirreducible set and Γ Ď Fq is a set with (2.3) a´1 i1 pfi1 ˝ . . . ˝ fin qpγq P Nq ,

n “ 1, . . . , K, i1 , . . . , in P t1, 2u,

for all γ P Γ, then #Γ ď q 1{2 plog qq1`op1q . Proof. Put ( F “ a´1 i1 fi1 ˝ . . . ˝ fin : 1 ď n ď K, i1 , . . . , in P t1, 2u .

Let χ be the quadratic character of Fq and define χp0q “ 1. Then, by (2.3) 1 ÿ ź (2.4) #Γ ď #F p1 ´ χpF pγqqq. 2 γPF F PF q

Expanding the products and rearranging the terms, we conclude that there are 2#F ´ 1 sums of form p´1qµ ÿ χ pF1 pγq ¨ ¨ ¨ Fµ pγqq , F1 , . . . , Fµ P F , (2.5) 2#F γPF q

with some µ ď #F . As f1 , f2 form a dynamically irreducible set, the polynomials F P F are all irreducible. We also have that they are coprime (that is, they are not constant multiple of each other). Suppose the contrary and let (2.6)

fi1 ˝ . . . ˝ fin “ αfj1 ˝ . . . ˝ fjm

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for some i1 , . . . , in , j1 , . . . , jm P t1, 2u and α P F˚q . Comparing the degrees of both sides, we can assume that n “ m. For a u P F˚q put ξu pXq “ uX. Write gis “ ξu´1 ˝ fis ˝ ξus`1 s

and gjt “ ξv´1 ˝ fjt ˝ ξvt`1 , t

where un`1 “ vn`1 “ 1, us and vt (1 ď s, t ď n, t ‰ 1) are the leading coefficients of fis ˝ . . . ˝ fin and fjt ˝ . . . ˝ fjn , and v1 is the leading coefficient of αfj1 ˝ . . . ˝ fjn . Then clearly, gis , gjt are monic quadratic polynomials with g i1 ˝ . . . ˝ g in “ g j 1 ˝ . . . ˝ g j n , specially, gis “ gjs for all s by Lemma 2.8. Let 1 ď ℓ ď n be the largest index such that fiℓ ‰ fjℓ . Then fiℓ`1 ˝ . . . ˝ fin “ fjℓ`1 ˝ . . . ˝ fjn and thus uℓ`1 “ vℓ`1 . As giℓ “ gjℓ , that is, ˝ fjℓ ˝ ξvℓ`1 ξu´1 ˝ fiℓ ˝ ξuℓ`1 “ ξv´1 ℓ ℓ we get fiℓ “ puℓ {vℓ qfjℓ , which contradicts our assumption f1 pXq{f2 pXq R Fq . Then the product polynomials F1 ¨ ¨ ¨ Fµ in (2.5) are squarefree, thus we can estimate (2.5) by the Weil bound. As F1 ¨ ¨ ¨ Fµ has degree at most µ 2K ď #F 2K we get q #Γ ď #F ` #F 2K q 1{2 . 2 Using that K`1 #F “ 2p 2 q as all compositions in the definition of the set F are distinct and choosing K such that 2#F “ Opq 1{2 q we obtain the result.  Combining the algorithm of [7, Corollary 3] with Lemmas 2.7 and 2.9 one may get in the same way the following result. Proposition 2.10. There is an algorithm to test whether or not a set of r quadratic polynomials over Fq is dynamically irreducible, which takes Oprq 3{4 log qq operations. Moreover if the polynomials are not constant multiple of each other, then the algorithm takes Oprq 1{2plog qq3 q operations. 3. Proof of Theorem 1.1 We present an algorithm which computes the list of all quadratic dynamically irreducible polynomials. For computational reason, the list is not have to be stored, thus to compute DIq one needs to store just the length of this list.

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In the first part, the algorithm computes the set DI ˚q of dynamically irreducible monic quadratic polynomials and its cardinality DI˚q “ #DI ˚q . Let M be as in Lemma 2.1 and let K be as in Lemma 2.3. For each monic polynomial f we check whether its adjusted orbit contains square. If we find a square element, we terminate and start with a new polynomial. If we do not find (any square among the elements ´f pγqu Y tf pnq pγq : n “ 2, 3, . . . , M , we terminate and append fa , defined as in Lemma 2.2, to the list of dynamically irreducible polynomials. In this way the algorithm obtains the list of all dynamically irreducible monic quadratic polynomials. The list of all dynamically irreducible polynomials contains the polynomials fa pXq, where a P F˚q and f is a dynamically irreducible monic quadratic polynomials. We however follow the above strategy in two step, which we describe below. In the first part the algorithm computes K many pq ´ 1q{2-th powers for all monic polynomials. By Lemma 2.3, after this stage there are only at most Opq 3{2 log qq monic polynomials which can potentially be dynamically irreducible. Next, the algorithm computes M many pq ´ 1q{2-th powers for these Opq 3{2 log qq remaining polynomials. Therefore, both steps together can be done in time Kq 2`op1q ` Mq 3{2`op1q “ q 9{4`op1q . This immediately gives the number of dynamically irreducible quadratic polynomials as DIq “ pq ´ 1qDI˚q by Lemma 2.2. Finally, to compute all dynamically irreducible quadratic polynomials, the algorithm computes the polynomials fa pXq, for all a P F˚q and for all dynamically irreducible monic quadratic polynomials f which, by (1.1), requires q 1`op1q DI˚q “ q 5{2`op1q many steps. 4. Proof of Theorem 1.2 The proposed algorithm consists three parts. In the first part we construct all the dynamically irreducible polynomials f1 by Theorem 1.1 in time q 5{2`op1q . Then we construct r ´ 1 many multiples of f1 and test whether these polynomials with f1 form a dynamically irreducible set in time rq 3{4`op1q . Recalling (1.1), we see that this part requires q 5{2`op1q ` DIq ¨ q r´1 ¨ rq 3{4`op1q “ q 5{2`op1q ¨ q r´1 ¨ rq 3{4`op1q “ rq 9{4`r`op1q operations.

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In the second part we construct all sets of the polynomials tf1 , . . . , fr u having the form (4.1)

fi pXq “ ai pX ´ bq2 ` b,

i “ 1, . . . , r.

By Example 2.6, it is a dynamically-irreducible set if and only if ´1 P Sq and b{ai P Nq , i “ 1, . . . , r. This part requires (4.2)

q r`1`op1q

operations. For the third part let K be as in Lemma 2.9. Let f1 , f2 be two quadratic polynomials which are not constant multiple of each other and do not have the form (4.1). We test whether they form a dynamically irreducible set in time rq 1{2`op1q . Clearly, there are at most DIq2 such pairs. For a fixed pair f1 , f2 , we construct the set Γ of elements γ P Fq for which (2.3) holds in time q 1`op1q 2K “ q 1`op1q . By Lemma 2.9, we have #Γ ď q 1{2`op1q . By assumption, f1 , f2 do not have the form (4.1), therefore the set (4.3)

tfi1 ˝ . . . ˝ fin pγj q : i1 , . . . , in P t1, 2u, n ě 0u

consists at least two elements. Let δ1 , δ2 be two different elements of (4.3). Then we construct the polynomials fi pXq “ ai X 2 ` bi X ` ci (i “ 3, . . . , r) such that ˆ ˙ 1 1 1 fi pδ1 q, fi pδ2 q, fi ˝ fi pδ1 q P Γ. ai ai ai Namely, write γi “ ´bi {p2ai q and for α1 , α2 P Γ consider first the solutions of bi ci bi ci δ12 ` δ1 ` “ α1 , δ22 ` δ2 ` “ α2 ai ai ai ai for bi {ai and ci {ai . As the coefficient matrix of the linear equation system has discriminant δ1 ´ δ2 ‰ 0, there exists a unique solution. Choosing ai such that ˆ ˙ 1 1 bi ci fi ˝ fi pδ1 q “ fi pai α1 q “ pai αi q2 ` pai αi q ` “ α3 , ai ai ai ai one can construct fi in at most (4.4)

2#Γ3 “ q 3{2`op1q

ways. Finally, we test whether tf1 , . . . , fr u is dynamically irreducible in time rq 1{2`op1q .

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The third part requires time which is at most ` ˘ q 6 pq 1{2`op1q q ` DIq2 q 1`op1q ` q pr´2qp3{2`op1qq ¨ rq 1{2`op1q

“ rq p3{2`op1qqr`5{2

by (1.1). 5. Proof of Theorem 1.3 Analyzing the counting algorithm of the proof of Theorem 1.2 we see that at the first part we obtain at most q 5{2`op1q`r´1 “ q r`3{2`op1q dynamically irreducible sets. Furthermore, at the second part we get at most q, while at the third part we obtain at most q 5`op1q`pr´2qp3{2`op1qq “ q p3{2`op1qqr`2 dynamically irreducible sets. Since r ě 2 this dominates the count. Acknowledgement The authors are very grateful to Alina Ostafe for several motivating discussions and providing the construction of Example 2.6. Parts of this paper was written during visits of L. M. and I. S. to the Max Planck Institute for Mathematics (Germany) whose support and hospitality are gratefully appreciated. During the preparation of this work D. G-P. is partially supported by project MTM2014-55421-P from the Ministerio de Economia y Competitividad, L. M. is partially supported by the Austrian Science Fund FWF Project F5511-N26 which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications” and I. S. by the Australian Research Council Grant DP140100118. References [1] O. Ahmadi, ‘A note on stable quadratic polynomials over fields of characteristic two’, Preprint , 2009 (available from https://arxiv.org/abs/0910.4556). (p. 1) [2] N. Ali, ‘Stabilit´e des polynˆomes’, Acta Arith., 119 (2005), 53–63. (p. 1) [3] M. Ayad and D. L. McQuillan, ‘Irreducibility of the iterates of a quadratic polynomial over a field’, Acta Arith., 93 (2000), 87–97; Corrigendum: Acta Arith., 99 (2001), 97. (p. 1) [4] A. Ferraguti, G. Micheli and R. Schnyder, ‘On sets of irreducible polynomials closed by composition’, Lecture Notes in Comput. Sci.., 10064, Springer, Berlin (2017), 77–83. (pp. 2 and 4) [5] D. Gomez and A. P. Nicol´ as, ‘An estimate on the number of stable quadratic polynomials’, Finite Fields and Appl., 16 (2010), 329–333. (pp. 1 and 4)

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[6] D. G´ omez-P´erez, A. P. Nicol´ as, A. Ostafe and D. Sadornil, ‘Stable polynomials over finite fields’, Rev. Mat. Iberoam.., 30 (2014), 523–535. (p. 1) [7] D. R. Heath-Brown and G. Micheli, ‘Irreducible polynomials over finite fields produced by composition of quadratics’, Preprint , 2016 (available from http://arxiv.org/abs/1701.05031). (pp. 1, 2, 4, 6, and 7) [8] R. Jones, ‘The density of prime divisors in the arithmetic dynamics of quadratic polynomials’, J. Lond. Math. Soc., 78 (2008), 523–544. (pp. 1 and 3) [9] R. Jones and N. Boston, ‘Settled polynomials over finite fields,’ Proc. Amer. Math. Soc., 140 (2012), 1849–1863. (pp. 1, 3, and 4) [10] A. Ostafe and I. E. Shparlinski, ‘On the length of critical orbits of stable quadratic polynomials’, Proc. Amer. Math. Soc., 138 (2010), 2653–2656. (pp. 1, 2, 3, and 5) D.G.-P.: Department of Mathematics, University of Cantabria, Santander 39005, Spain E-mail address: [email protected] L.M.: Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Straße 69, A4040 Linz, Austria E-mail address: [email protected] I.E.S.: School of Mathematics and Statistics, University of New South Wales. Sydney, NSW 2052, Australia E-mail address: [email protected]