Distribution results on polynomials with bounded roots

Transcription

1 Monatsh Math : Distribution results on polynomials with boune roots Peter Kirschenhofer Jörg Thuswalner Receive: 22 September 206 / Accepte: 3 April 207 / Publishe online: 2 April 207 The Authors 207. This article is an open access publication Abstract For N the well-known Schur Cohn region E consists of all - imensional vectors a,...,a R corresponing to monic polynomials X + a X + +a X + a whose roots all lie in the open unit isk. This region has been extensively stuie over ecaes. Recently, Akiyama an Pethő consiere the subsets E s of the Schur Cohn region that correspon to polynomials of egree with exactly s pairs of nonreal roots. They were especially intereste in the -imensional Lebesgue measures := λ E s of these sets an their arithmetic properties, an gave some funamental results. Moreover, they pose two conectures that we prove in the present paper. Namely, we show that in the totally complex case = the formula v 0 = 2 s s Communicate by A. Constantin. Deicate to Professor Johann Cigler on the occasion of his 80th birthay. Both authors are supporte by the Franco-Austrian research Proect I36 Fractals an Numeration grante by the French National Research Agency ANR an the Austrian Science Fun FWF, an by the Doctoral Program W230 Discrete Mathematics supporte by the FWF. The secon author is supporte by the FWF Proect P B Peter Kirschenhofer Peter.Kirschenhofer@unileoben.ac.at Jörg Thuswalner Joerg.Thuswalner@unileoben.ac.at Chair of Mathematics an Statistics, Montanuniversität Leoben, Franz Josef-Straße 8, 8700 Leoben, Austria 23

2 690 P. Kirschenhofer, J. Thuswalner hols for all s N an in the general case the quotient /v0 is an integer for all choices N an s /2. We even go beyon that an prove explicit formulæ for /v0 for arbitrary N, s /2. The ingreients of our proofs comprise Selberg type integrals, eterminants like the Cauchy ouble alternant, an partial Hilbert matrices. Keywors Polynomial with boune roots Selberg integral Cauchy ouble alternant Shift raix system Mathematics Subect Classification Primary 05A0 C08 33B20; Seconary 05A05 C20 Introuction Almost 00 years ago Issai Schur [20,2] introuce an stuie what is now calle the Schur Cohn region. For each imension N this region is efine as the set of all coefficient vectors of monic polynomials of egree each of whose roots lies in the open unit isk. As we will eal with polynomials with real coefficients in the present paper we give the formal efinition for this setting. The -imensional Schur Cohn region is efine by E := {a,...,a R satifies ξ < }. : each root ξ of X + a X + +a X + a Sometimes polynomials having all roots in the open unit isk are calle contractive polynomials. In the years after Schur s papers were publishe, Cohn [7] evelope an algorithm to check if a polynomial is contractive. In Rahman an Schmeisser [7, Section.5] further evelopments an properties of the Schur Cohn region are surveye. Starting from the 970s contractive polynomials were stuie by Fam an Meitch [0] an their co-authors from the viewpoint of iscrete time systems esign. In this context Fam [9] calculate the -imensional Lebesgue measure λ E of E.In particular, he prove that here we give a simplifie version of [9, equations 2.7 an 2.8], see also [8, equation 9] /2 v := λ E = 2 = More recently, the Schur Cohn region occurre in connection with number systems an so-calle shift raix systems see Akiyama et al. [] or Kirschenhofer an Thuswalner [3]. Given a polynomial P R[X] of egree with r real an nonreal roots we call r, s the signature of P. Since = r + for given egree the signature is etermine by the value s of pairs of conugate nonreal roots. In [2], Akiyama an 23

3 Distribution results on polynomials with boune roots 69 Pethő stuie a partition of E in terms of the signature of contractive polynomials seealso[3]. In particular, they were intereste in the sets an their volumes E s := {a,...,a E : X + a X + +a X + a has exactly s pairs of nonreal roots} := λ E s..2 In the quest for formulæ for these volumes they unveile interesting relations of with Selberg integrals introuce in [22] an their generalizations by Aomoto [6]. In particular, they expresse v 0 in terms of a classical Selberg integral which le to the simple formula see [2, Theorem 4. an Lemma 5.] v 0 = 2 +/2 =! 2 2!. Besies that, the stuies by Akiyama an Pethő in[2] were evote to arithmetical properties of the quantities. It was prove that these numbers are rational for all choices of an s. Moreover, certain quotients of these numbers were stuie. In particular, it was shown that v /v 0 is an o integer for all an, base on numerical experiments, the following conecture was state. Conecture. see [2, Conecture 5.]. The quotient /v0 is an integer for all s /2. Moreover, for the particular instance = the following surprisingly simple explicit formula was conecture. Conecture.2 see [2, Conecture 5.2]. For all s N we have v 0 = 2 s..3 s In the present paper we confirm both these conectures see Theorem 5. an Theorem 2. below. We note that a special instance of the first conecture was confirme recently by Kirschenhofer an Weitzer [4], where the formula v v 0 = P was establishe. Here P is the -th Legenre polynomial. Integrality of this quotient was then erive by stanar properties of Legenre polynomials. In our proof of the conectures of Akiyama an Pethő various ingreients are neee. We start with a theorem establishe by these authors viz. [2, Theorem 2.] to write 23

4 692 P. Kirschenhofer, J. Thuswalner as a Selberg-type integral. It shoul be note that many questions an problems in number theory, combinatorics, CFT, an other mathematical areas are relate to integrals of this type compare e.g. [5,]. For the treatment of the Selberg type integrals occurring in our proof in the context of ranom Vanermone matrices an approach propose in Alastuey an Jancovici [4] turns out to be very valuable. Besies that, methos from avance eterminant calculus see [5] are use to obtain close forms for variants of the Cauchy ouble alternant that come up along the way. Moreover, we nee minors of the Hilbert matrix, which is known in numerics for its numerical instability see e.g. [23], to further simplify our formulæ in orer to finally establish the integrality of the quotients uner scrutiny. The paper is organize as follows. In Sect. 2 we confirm Conecture.2 of Akiyama an Pethő. In particular, we prove.3 see Theorem 2. which immeiately implies that this quotient is integral. Although the proof of this result is much shorter than the proof of the general case it contains several of the main ieas. Section 3 is evote to preparatory efinitions an results. It contains the proof of a close form for a certain convolution of a eterminant like sum that will be neee in the proof of our main result. Section 4 contains a formula for the volumes see Theorem 4.4. In Sect. 5 we prove ifferent formulæ for the quotients /v0 an establish their integrality see Theorem 5.. This proves Conecture. of Akiyama an Pethő. We then iscuss how the known special cases prove by Akiyama an Pethő [2] as well as Kirschenhofer an Weitzer [4] follow. 2 The totally complex case To prove the integrality of /v0 turns out to be rather involve. For this reason we ecie to start this paper with the proof of Conecture.2 concerning the quotient /v0. The proof of this conecture is easier than the proof of Conecture. but alreay contains some of the ieas neee for the general case an serves as a roa map for the treatment of the contribution of the nonreal roots in the general proof. Moreover, we think that the nice formula for this quotient eserves to be announce in a separate theorem. Theorem 2. The quotient /v0 is an integer for each s. In particular, v 0 = 2 s. s By [2, Theorem 4. an Corollary 5.] we know that the enominators satisfy 23 v 0 = + /! =

5 Distribution results on polynomials with boune roots 693 Therefore, the assertion of the theorem is equivalent to = 4s s! 2 / = As a main step for the proof of Theorem 2. we establish the following result. Lemma 2.2 For s N we have = 2 3s D s, where D s = et 2 2 2k 2,k s. Proof We start from [2, Theorem 2.] for the case of r = 0 real roots an s pairs of complex conugate roots z = x + iy an z = x iy s, 2.3 where we use the notation X z X z = X 2 α X + β, so that α = 2x an β = x 2 + y2 s. 2.4 With this notation [2, Theorem 2.] states that = z z k z z k z z k z z k α α s β β s, s! D 0,s s where D 0,s enotes the region 0 β, 2 β α 2 β for all s. Since et α,β x, y = 4y = 2iz z, we get = i s s! x,y B + s s z k z z k z z k z z k z s z z x x s y y s 2.5 = with B + enoting the upper half of the unit isk. In the following step of computation we aopt an iea use by Physicists in the computation of certain Coulomb potentials 23

6 694 P. Kirschenhofer, J. Thuswalner see e.g. [4]. Observe that the integran in the last integral equals the Vanermone eterminant V z, z,...,z s, z s of orer. In the following, we enote by S m the symmetric group on m letters. Inserting the efinition of the eterminant in 2.5 we obtain with σ k := σk that = i s s! x,y B + s σ S sgnσ z σ σ z 2 z σ s z s σ x y x s y s. 2.6 From this, transforming to polar coorinates x + iy = r expiθ, we get = i s s! σ S sgnσ s = r =0 π θ =0 r σ 2 +σ 2 exp iσ 2 σ 2 θ θ r. 2.7 For f : R C, weset f t t = Re f t t + i Im f t t. Then expiαt t = expiαt iα for α = 0, an we have = s! σ S sgnσ s = σ2 σ σ 2 + σ 2 σ 2 σ 2 Observe that the numerators in the last fraction are equal to 0 if σ 2 σ 2 is even an equal to 2 ifσ 2 σ 2 is o. Therefore, only permutations σ S,for which σ 2 σ 2 is o for all s give a nonzero contribution to the sum. A permutation of this type has the form σ = ϕ, ψ = , ϕ ψ2 ϕ3 ψ4...ϕ2 ψ2...ϕ ψ where ϕ is a permutation in S o, the set of all permutations of {, 3,..., }, an ψ is a permutation in S e, the set of all prmutations of {2, 4,...,}, or has a form where any of the pairs ϕ2, ψ2 in the secon line are interchange. Each interchange of the latter type will change the sign of the permutation by the factor. Therefore, setting ϕ := ϕ2, ψ := ψ2, we get = s! ϕ,ψ ϕ S o,ψ Se sgnϕ, ψ s = ϕ + ψ ϕ ψ ψ ϕ. ϕ ψ ψ ϕ

7 Distribution results on polynomials with boune roots 695 Since ϕ ψ is o, the term in the roun brackets equals Observe that σ = ϕ, ψ = = s! 2 ϕ,ψ sgnϕ, ψ s ψ 2. ϕ2 = 4 ϕ ψ, which yiels = ϕ ψ, ϕ... ϕ... ϕ s ψ... ψ... ψ s if ϕ an ψ are regare as elements of S. Therefore, sgnϕ, ψ = sgnϕsgnψ an we get = 23s sgnϕ s sgnψ s! ψ 2. ϕ2 ϕ S o ψ S e By the efinition of the eterminant we have ψ S e sgnψ s = ψ 2 ϕ2 = et = 2 2 ϕ 2 k,k s The eterminant on the right han sie nees sgnϕ interchanges of columns to be transforme to D s := et 2 2 2k ,k s Therefore, we have establishe = 23s s! ϕ S o sgnϕsgnϕd s = 2 3s D s, 2.. which completes the proof of the lemma. We finish the proof of Theorem 2. by the following prouct formula for the eterminant D s. Lemma 2.3 D s = 24ss s! 2 / = Proof We start with the observation that D s is of the form D s = et X + Y k,k s, with X = 2 2 an Y k = 2k 2. 23

8 696 P. Kirschenhofer, J. Thuswalner Using the Cauchy ouble alternant formula cf. e.g. [5] et X + Y k,k s = s X X k Y Y k,k s X + Y k 2.3 we fin D s = ss /2 2 s s,k s k 2 k + k + 2 2k k. Therefore, D s+ D s = 2 8s s + 2 4s+3 + 4s+ From the last equality the lemma follows immeiately by inuction. Thus the proof of Theorem 2. is complete Preparations for the general case This section contains some efinitions an auxiliary results that will be neee in the proof of our main result. Throughout the paper finite totally orere sets will play a prominent role. If X an Y are finite orere sets we write X Y to state that X is a possibly empty orere subset of Y whose elements inherit the orer from Y. Moreover Y \ X enotes the finite orere subset of Y containing all elements of Y that are not containe in X. Also other set theoretic notions will be carrie over to finite orere sets in the same vein. To inicate the orer we sometimes write {X < < X r } instea of {X,...,X r } for a finite orere set. We also use the following notation. Given an orere subset I ={i < < i m } of { < < r} we enote I ={ < < r}\i an we write X I ={X i < X i2 < < X im }. For r N let X ={X < < X r } be an orere set of ineterminates an consier sums of the form HX = H r X = σ S r sgnσ r i= X i X σ X σi. 3. We now give a close form for convolutions of these sums which will immeiately imply an ientity that is neee in the proof of our main results. 23

9 Distribution results on polynomials with boune roots 697 Lemma 3. Let [r] ={ < < r}. Then, setting X ={X < < X r },thesums Hfrom3. amit the convolution formula K [r] HX K HX K = 2 r/2 r = +r o + X r X = even Here the sum runs over all orere subsets K of [r]. r k even X X k r k o X X k. 3.2 Proof First we note that, accoring to [2,3 r ientity on p. 682], for a finite orere set X the sum HX efine in 3. amits the close form HX = r X X k X X k. 3.3 Inee, 3.3 is prove by inuction using a recurrence relation which is establishe by splitting the sum in 3. w.r.t. the value of σm see [2, proof of equation 4.9]. This close form will be use throughout the proof. Let f r X,...,X r an g r X,...,X r enote the left an right han sie of 3.2, respectively. To prove the lemma it suffices to show that f r as well as g r satisfy the recurrence h r X,...,X r = + X r X r h r X,...,X r X r X r r i= i+r even + X r X i X r X i h r X,...,X i, X r, X i+,...,x r 3.4 for r 2 with h X = 2. This can be prove using the partial fraction technique. In orer to establish the recurrence 3.4 for f r we first observe that, since the summans on the left han sie of 3.2 are invariant uner the involution K K, HX K HX K = 2 HX K HX K = 2 HX K HX K. K [r] Now set K [r] r K p r+ X,...,X r+ = q r+ X,...,X r+ = K [r] r K K [r] r / K K [r] r / K HX K HX K i K HX K HX K i K X i X r+ X i X r+, X i X r+ X i X r We evaluate p r+ X,...,X r, in two ways. Firstly, we set X r+ = irectly in the efinition of p r+ X,...,X r+ an use 3.5 to replace the conition r K 23

10 698 P. Kirschenhofer, J. Thuswalner by a factor 2 expansion in the resulting expression. Seconly, we insert the partial fraction X r X r+ i K i =r X i X r+ X i X r+ = X 2 i X i K r X i X i X r+ i =r X X r + X r X r+ X X r K \{r } K \{i,r } X X i X X i in the prouct containing the ineterminate X r+ in p r+, then set X r+ =, interchange the sums, an reorer the ineterminates appropriately. This yiels the ientity δ r,j is if r J an 0 otherwise 2 f r X,...,X r = p r+ X,...,X r, r 2 X i = + X r HX J HX X i= i X J δ r,j r J {< <i <r <i+< <r 2<r} r J + HX J HX J δ r,j J {< <r 2<r} + + X r X r X r X r J {< <r 2<r } r J HX J HX J 3.6 note that the sum in the secon line vanishes by 3.5. Evaluating the sum q r+ X,...,X r+ in two ways using the partial fraction expansion w.r.t. X r+ of X i X r+, X r X r+ X i X r+ i K in this case r / K hols, by similar reasoning as above we gain the formula 2 f r X,...,X r = q r+ X,...,X r, r 2 = + X r i= + + X r X r X r X r 23 X i HX J HX X i X J r i+δ r,j r J {< <i <r <i+< <r 2<r} r J J {< <r 2<r } r J HX J HX J. 3.7

11 Distribution results on polynomials with boune roots 699 Aing 3.6 an 3.7 the summans corresponing to inices i with r i o vanish. In the remaining summans interchanging the ineterminates X r an X r+ absorbs the sign δ r,j an, hence, 3.4 hols for f r. To show that g r also satisfies 3.4 we start with g r+ X,...,X r+. Inee, to get the left han sie of 3.4 ust set X r+ =. For the right han sie expan g r+ X,...,X r+ /X r+ X r in partial fractions w.r.t. X r+, multiply by X r+ X r, an set X r+ = again. Let r N an let Y ={Y < < Y r } be an orere set of ineterminates. Later we will nee sums of the form SY = S r Y := sgnσ Y σ Y σ + Y σ2 Y σ + +Y σr. σ S r Substituting X i = q Y i an letting q ten to it now follows from 3.3 that S r Y = Y Y n r 3.8 Y k Y Y k + Y. 3.9 By the same operations, the following corollary, which will be use later on, is an immeiate consequence of 3.9 an Lemma 3.. Corollary 3.2 Let [r] ={ < < r}. Then, setting Y ={Y < < Y r },thesums S m from 3.8 amit the convolution formula K [r] S K Y K S r K Y K = 2 r r k even Y k Y r = o Y r k o Y k + Y. Here the sum runs over all orere subsets K of [r]. We now efine some notions relate to the parity of the elements of a finite orere set. Definition 3.3 A finite orere set X consisting of X /2 o an X /2 even numbers is sai to be in increasing oe-orer, if it is of the shape o, even, o, even,, where the o an the even orere subset are both strictly increasing. We enote the sign of the permutation that brings the elements of X from their natural orer to the increasing oe-orer by oesgnx. Let X be a finite orere set of strictly ascening positive integers. Define the finite orere sets X an X 2 such that 2X an 2X 2 constitute the even an o elements of the finite orere set X in ascening orer, respectively. Then X, X 2 is calle the parity splitting of X. The finite orere set X is calle parity balance if X = X 2, i.e.,ifx contains the same number of o an even elements. These concepts are best illustrate by an example. Let X ={ < 3 < 4 < 5 < 6} be given. Then the elements of X are orere increasingly. To get X in increasing 23

12 700 P. Kirschenhofer, J. Thuswalner oe-orer { < 4 < 3 < 6 < 5} we have to o two transpositions. Thus this reorering is achieve by an even permutation an, hence, oesgnx =. The parity splitting of X is {2 < 3}, { < 2 < 3}, an since {2 < 3} an { < 2 < 3} have ifferent carinality we see that X is not parity balance. We will nee the following property of the oe-orer. Lemma 3.4 For ν N consier the orere set X ={ < 2 < < 2ν}. Let M be a parity balance orere subset of X an let N = M. If N, N 2 is the parity splitting of N then oesgnmoesgnn = m N m+ n N n Proof Fix the carinality c = N = N 2 an let a = m N m + n N 2 n.note that a cc +. We prove the lemma by inuction on a. For the inuction start observe that if a = cc + then N = N 2 ={,...,c} an, hence, oesgnm = oesgnn = the oe-orer an the natural orer coincie an a =. Thus the lemma hols in this instance. For the inuction step assume that the lemma is true for all cc + a < a 0 an consier a constellation N, N 2 corresponing to the value a = a 0. Since a 0 > cc+ there is x N with x 2 M. Now we switch these two elements, i.e., we put x in the set M an x 2inthesetN.Ifx N then oesgnm remains the same uner this change, an oesgnn changes its sign. If x M then oesgnn remains the same uner this change, an oesgnm changes its sign. Thus in any instance, oesgnmoesgnn changes the sign. Summing up, in the new constellation we have a = a 0 an a total sign change of in oesgnmoesgnn compare to the original constellation. Thus the lemma follows by inuction. 4 Formulæ for the volumes in the general case In this section we treat the case of polynomials having s pairs of nonreal complex conugate roots an r real roots within the open unit isk. We will first erive a general formula that expresses the volume +r of the coefficient space in this instance by means of a sum of eterminants. This forms a generalization of Lemma 2.2. Using the notations introuce in 2.3 an 2.4 the result of [2, Theorem 2.] states that the volumes in question amit the integral representation +r = z z k z z k z z k z z k s!r! D r,s s z ξ k z ξ k ξ k ξ r α α s β β s ξ ξ r, s, k r r 4. where D r,s enotes the region given by 0 β, 2 β α 2 β for all s an ξ for all r. Proceeing as in 2.6 with the 23

13 Distribution results on polynomials with boune roots 70 complex part of the integral an bringing the variables ξ,...,ξ r in increasing orer yiels +r = i s s! where x,y B + s ξ ξ r V,r x x s y y s ξ ξ r, 4.2 V,r := V z, z,...,z s, z s,ξ,...,ξ r 4.3 is the Vanermone eterminant of orer +r. Laplace expansion of this eterminant by the first columns which are the columns corresponing to the complex part yiels V,r = M {,...,+r} M = V M,{,...,},r V N,{+,...,+r},r M ++ 2, 4.4 where the sum runs over finite orere sets M, the orere set N is efine as the complement N ={,..., + r} \M, an V I,J,r is the minor of the Vanermone matrix 4.3 with the rows having inex in I an the columns having inex in J. Inserting 4.4in4.2 we obtain +r = i s s! M M ++ 2 x,y B + s V M,{,...,},r ξ ξ r x y V N,{+,...,+r},r ξ ξ r. 4.5 Proceeing with the first integral as in 2.7, 2.8, an 2.9 we see that this integral equals zero if M is not parity balance in the sense of Definition 3.3. Furthermore, letting M, M 2 be the parity splitting of M see again Definition 3.3, with the same arguments as in the instance r = 0 in Sect. 2 for the efinition of oesgnm see also Definition 3.3 for parity balance M the first integral in 4.5 satisfies 2 s i s s! where D M,M 2 s x,y B + s V M,{,...,},r x y = 2 3s oesgnmd M,M 2 s, 4.6 is the minor of the eterminant D s from 2.0, i.e., D M,M 2 s := et 2 2 2k 2 M,k M

14 702 P. Kirschenhofer, J. Thuswalner Observing that M ++ 2 = M 2 ++s = since M 2 =s, wehave +r = 23s M {,...,+r}, M = M parity balance oesgnmd M,M 2 s V N,{+,...,+r},r ξ ξ r, ξ ξ r 4.8 where M, M 2 is the parity splitting of M, an N ={,..., + r}\m observe that N consists of r/2 even an r/2 o numbers. Now we turn our attention to the real integral. For the finite orere set N = {n < < n r } we have I r N := = ξ ξ r ξ ξ r V N,{+,...,+r},r ξ ξ r V ξ,...,ξ r ; N ξ ξ r, 4.9 where V ξ,...,ξ r ; N is given by V ξ,...,ξ r ; N := et ξ n k,k r. 4.0 These eterminants are strongly relate to the well-known Schur functions, viz., V ξ,...,ξ r ; N is ust a prouct of a Schur function an a Vanermone eterminant see [6, p. 40, equation 3.]. Let N ={n,...,n r } 4. be the orere set containing the arrangement of the elements of N in increasing oe-orer. Then 4.8 an 4.9 imply the following result. Lemma 4. The volume +r satisfies +r = 23s M {,...,+r}, M = M parity balance oesgnmoesgnnd M,M 2 s I r N. 4.2 Here N ={,..., + r}\m, an N is a finite orere set containing the elements of N in increasing oe-orer see Definition 3.3 for the terminology. It now remains to compute the integrals I r N = 23 r k=0 ξ ξ k 0 0 ξ k+ ξ r V ξ,...,ξ r ; N ξ ξ r. 4.3

15 Distribution results on polynomials with boune roots 703 By the tight relation of V ξ,...,ξ r ; N to Schur functions mentione after 4.0we see that these integrals resemble the Selberg-type integrals stuie in [, Section 2]. Inee, the Jack polynomials also calle Jack s symmetric functions use there are generalizations of Schur functions see [6, p. 379]. The ifference is that in our instance we only have skew symmetric rather than symmetric integrans in 4.3. Let us fix k for the moment an expan the eterminant by its first k columns. Then the integrals in the kth summan of 4.3 can be rewritten as ξ ξ k 0 0 ξ k+ ξ r V ξ,...,ξ r ; N ξ ξ r = k i + k+ 2 V ξ,...,ξ k ; K ξ ξ k K ={n i,...,n i } N ξ ξ k 0 k V ξ k+,...,ξ r ; N \ K ξ k+ ξ r ξ k+ ξ r Substituting η i = ξ i for i k in the first integral in 4.4 yiels V ξ,...,ξ k ; K ξ ξ k ξ ξ k 0 k = n i k V η,...,η k ; K k η η k = η η k 0 0 η k η k n i + k+ = k+ k n i + k 2 V η,...,η k ; K k η η k 2 Jk K, 4.5 where for a finite orere set W ={w,...,w m } we set J m W := V u,...,u m ; W u u m u u m Using this notation, inserting 4.5into4.4 an then 4.4into4.3, an observing that, because n,...,n r are in increasing oe-orer the numbers n i + i are all even, we finally get I r N = J K K J r K N \ K. 4.7 K N In the next step we replace the integrals J m W by the sum S m W from 3.8. This is ustifie by the following lemma. Lemma 4.2 For each m N an each orere set W ={w < <w m } of complex numbers with positive real part we have J m W = S m W, 23

16 704 P. Kirschenhofer, J. Thuswalner where S m W an J m W are efine in 3.8 an 4.6, respectively. Proof Recall that J m W = 0 u u m etu w k,k r u u m. With the substitution u k := v k v k+ v m, k m, the integral turns to J m W = 0 v,v 2,...,v m etv w k v w m,k m v2 v2 3 vm m v v m = sgnσ σ S 0 v,v 2,...,v m m = sgnσ w σ w σ + w σ2 w σ + +w σm σ S m = S m W. v w σ v w σ+w σ2 2 v w σ+ +w σm m v v m 4.8 For the following consierations we have to make some amenments in case N is o. In particular, we efine 2t = { r r + if r is even, ifr is o 4.9 an set Ñ = { N N {2s + t} if r is even, if r is o We now apply Corollary 3.2 in combination with the last lemma to the evaluation of the sums K J K K J r K N \ K occurring in 4.7. Lemma 4.3 Let Ñ, N 2 be the parity splitting of Ñ. Then the real integrals I r N with N as in 4. satisfy I r N = 2 r B Ñ,N 2, 4.2 where B Ñ,N 2 is the minor with row inices from Ñ an column inices from N 2 of the matrix 2k 2 +2k,k s+t B = 2k 2 +2k, s + t = s + t 2k,,k s+t if r = 2t even, if r = 2t o. 23

17 Distribution results on polynomials with boune roots 705 Proof To prove the result for the instance r = 2t we use Corollary 3.2 with the choice Y = n, where n is as in 4.. Combining this corollary with Lemma 4.2 we obtain I r N = 2 r r k even n k n r = o n r k o n k +. n Using the ouble alternant formula 2.3 this yiels I r N = 2 r et = 2 r et n 2 n 2k n 2k n 2 n 2k r n,k t k= k k o,k t an the result follows because the last eterminant equals B Ñ,N 2. In the instance r = 2t the result follows by the same reasoning. The only ifference is that instea of using the ouble alternant formula 2.3 wehavetouse X X k Y Y k et X +Y k, n n n =., = n X + Y k,k n n, k n 4.22 This ientity can in turn be prove along the lines of the usual proof of 2.3. Now we are reay to prove our first main result on the mixe volumes. Theorem 4.4 Let enote the volume of the coefficient space of real polynomials of egree = + r with s pairs of nonreal complex conugate roots an r real roots in the open unit isk. Then, for s n, 2n 2 2n+s = J {,...,n} J =s et c,k J,k n = J {,...,n} J =s et c,k J,k n with c,k J = 2 2 2k 2 for J, for / J, 2k 2 + 2k an, for s n, c,k J = 2 2 2k 2 for k J, for k / J, 2k 2 + 2k 2n 2 2n+s = J {,...,n } J =s et c,k J,k n 23

18 706 P. Kirschenhofer, J. Thuswalner with 2 2 2k 2 for J, c,k J = for / J, = n, 2k 2 + 2k for = n. 2k Proof Using Lemma 3.4 we have to eal with the quantities oesgnmoesgnn in 4.2. Observe that oesgnn = oesgnñ, since 2n is the maximal element. Thus we may apply Lemma 3.4 with Ñ instea of N an thus this lemma hols for o an even carinalities of N. In the first an the thir sum in the statement of the theorem expan the eterminants simultaneously accoring to the rows with inex in J. The resulting expression is the same as the one we get when we apply Lemma 3.4 an Lemma 4.3 to 4.2. In the secon sum of the theorem we have to expan the eterminants simultaneously accoring to the columns with inex in J to finish the proof. Setting s = 0 we get the following expressions for the totally real volumes. Corollary 4.5 The volumes v r 0 of the coefficient space of real polynomials of egree r with all roots real an in the open unit isk fulfill = et 2n 2k 2 + 2k,k n = et 2k 2 +2k, n 2n = n v 0 2n 2 v 0 2n 2 2k,,k n if r = 2n an if r = 2n. Evaluating the eterminants by the ouble alternant 2.3 or its o analogue 4.22, respectively, this corollary yiels the formulæ for the totally real volumes known from [2, Theorem 4.] note the close connection to the Selberg integral S,, /2 in this reference. 5 The volume ratios in the general case Now we are in a position to turn our attention to formulæ for the quotients /v0 an to confirm Conecture.. In particular, we establish the following result involving eterminants of partial Hilbert matrices from which we can infer the integrality of immeiately. /v0 23

19 Distribution results on polynomials with boune roots 707 Theorem 5. The quotients of the volumes.2 fulfill, for s /2, v 0 /2 K = s+ K + 2k s K 2 2k, 2k, 2k K {,..., /2 } k K /2 K = s+ K + 2k 4k 2k et s K 4k 2k K {,..., /2 } k K /2 K = s+ K k et s K 2 + 2k 2 K {,..., /2 } k 2 k k. 5. Note that n k = 0 for k < 0 by convention. Moreover, for K = the eterminant has to be assigne the value. In particular, vs is an integer. v 0 Proof We start with the instance of even egree = 2n. Observe that, setting X = 2 an Y k = 2k, the entries of the matrices c,k J from Theorem 4.4 are,k n of the form X c,k J = 2 Y k 2 Y k X + Y k for J, for / J. Using the partial fraction ecomposition X 2 Y k 2 = + 2Y k X + Y k X + Y k 5.2 an the multilinearity of the eterminant function we fin, since J =s, J := et c,k J,k n = s 2 s K, 2k k n K J K := et,k K,k n with,kk = 2 + 2k 2 + 2k where for K, for / K. 5.3 Inserting the last result in Theorem 4.4 yiels 2n 2 2n+s = J {,...,n} J =s s J = 2 s 2k k n K {,...,n} n K K. s K 23

20 708 P. Kirschenhofer, J. Thuswalner The binomial coefficient occurs ue to the following observation. If J s is the collection of all subsets of carinality s of the set {,...,n} then a given set K {,...,n} is a set of exactly n K s K elements of the collection Js. For the case of s = 0 nonreal roots this specializes to v 0 2n 2 2n = 2k, so that finally k n 2n n K v 0 = s K with K from s K 2n K {,...,n} The eterminants K may be evaluate by the ouble alternant formula 2.3 as K = 2 k,k / K K, all k 2k K,k / K 2 + 2k 2k + / K, all k / K,k K 2 k 2k 2 + 2k where all proucts run over the range, k n with the enote restrictions. Using the notation k K iff, k K or, k / K, this reas K = 2 2n 2 { :k K, all } kk K, all k k kk 2 + 2k k + k / K, all k 2 + 2k,, an, in particular, so that the quotients fulfill k 2 = 2 2n k,,k K { :k K, all } = For the first prouct in 5.5 we erive kk 23 kk k + k k + k = {k< : / K,k K } n + k! k!2k k K K, all k 2 + 2k 2 + 2k. 5.5 n k k!n k! =k k k +, 5.6

21 Distribution results on polynomials with boune roots 709 while the secon prouct in 5.5 satisfies K, all k 2 + 2k 2 + 2k = k K, all = k K 2 + 2k 2 2k 2n + 2k!! 2k!! 2 2n 2k!! k 5.7 with m!! enoting the ouble factorial. Therefore, inserting 5.6 an 5.7 in5.5 yiels K Since = { :k K, all } + {k< : / K,k K } k K { :k K, all } + {k< : / K,k K } + {k< : } = n k K 2 2n + 2k 2n 2k, 2k, 2k 2n + 2k!2 n k 2 2k k 2 n+k 2n 2k!2k! 2 n k + =k k 2 k { < k :k K, all } + {k < : / K, k K } + {k < :, k K } = { = k : k K, all } = K n, equation 5.8 implies that K = K k K 2 2n + 2k 2n 2k, 2k, 2k k 2 k Inserting 5.9 in5.4 completes the proof of the first formula for the case of even egree in Theorem 5.. The secon formula follows by writing the trinomial coefficient as a prouct of two binomial coefficients min the factor /2 in the prouct in 5.9 an applying the ouble alternant formula 2.3 to express the last prouct by the eterminant of the partial Hilbert matrix from the statement of the theorem. Let us now turn to the case of o egree = 2n. This time we use the matrices c,k J from Theorem 4.4. Again using the ecomposition 5.2 with X = 2 an Y k = 2k we get 2n 2 2n+s = s 2 s 2k v 0 2n 2 2n = k n k n 2k e, K {,...,n } n K e K, s K 23

22 70 P. Kirschenhofer, J. Thuswalner where for J, 2 + 2k e K = ete,k K with e,k K = for / J, = n, 2 + 2k for = n. 5.0 Therefore, 2n v 0 = s 2n K {,...,n } n K s K ek e with e K from Evaluating the eterminants with the variant 4.22 of the Cauchy ouble alternant formula yiels e K = 2 k n,k / K K, k n = 2 n 2 + n 2 { :k K, all } 2k n K,k / K 2 + 2k n kk K, k n k 2k + / K,k K n, k n / K n kk 2 + 2k 2 k 2 + 2k k + / K, k n n n k 2 + 2k, 2k an, in particular, e = 2 n 2 + n 2 Therefore, the quotients fulfill e K { :k K, all } = e n n kk k n k n k + k n 2 + 2k K, k n k k 2 + 2k. 5.2 For the the first prouct in 5.2 we get from 5.6 by replacing n by n n kk 23 k + = { k< n : / K,k K } n + k! k k!2k k K n k k!n k! =k k k +,

23 Distribution results on polynomials with boune roots 7 whereas the secon prouct in 5.2 equals the result of 5.7. Therefore, we get e K e = K k K 2 2n + 2k 2n 2k, 2k, 2k k 2 k Inserting 5.3 into5. completes the proof of the first formula for the case = 2n in Theorem 5.. The secon formula follows again by applying the ouble alternant formula 2.3 to express the last prouct by the eterminant of a partial Hilbert matrix. The thir formula in the theorem follows immeiately from + 2k 4k 2k et 4k 2k + k k K = + 2! et 22! 2k!2k! + k K k K + 2! = et 2 + 2k 2k!2!2k! k = et k 2 Remark 5.2 Using the secon sum for 2n /22n+s in Theorem 4.4 yiels the following alternative formula for the volume ratios in the case = 2n for all s n. 2n v 0 2n = n K s K s K k K 2 2n + 2k 2n 2k +, 2k, 2k K {,...,n} = n K 2n + 2k 4k 3 s K 2k K {,...,n} s K 4k 2 2k k K = n K 2n k 3 s+ K et K {,...,n} s K 2 + 2k k 2 k + 2. et + k We en this section by showing that some results of Kirschenhofer an Weitzer [4] as well as Akiyama an Pethő[2] are simple consequences of Theorem 5.. The first result refers to the instance of polynomials with exactly one pair of nonreal conugate zeroes. Corollary 5.3 cf. [4]. For N we have v v 0 = P 3 2, 4 where P enotes the -th Legenre polynomial. 23

24 72 P. Kirschenhofer, J. Thuswalner Proof From 5. with s = wehave v v 0 = /2 + /2 k= + 2k 4k 4k k On the other han the Legenre polynomials may be efine as [9, p.66] P x = + 2 x, =0 so that P 3 = = Furthermore, since see e.g. [8, p. 58] P x = P x, we have P = = P =, 2 = yieling + 2 = o 2 = + = even Therefore, setting = 2k for even, 5.7 can be rewritten as so that finally P = 4 P /2 k= = / k 4k /2 k= + 2k 4k 4k, 5.8 2k 4k, 5.9 2k which coincies with 5.5 from above. As in.letv enote the volume of the coefficient space E of the polynomials of egree with all zeroes in the open unit isk. Summing up over all possible signatures in Theorem 5. we get the results on the quotients v /v 0 containe in [2, equation 9]. 23

25 Distribution results on polynomials with boune roots 73 Corollary 5.4 cf. [2, equation 9]. For N we have Proof Let v v 0 = 2n 2 n =n+ 2n 2 n =n 2 2 n 2 for = 2n, = 2 for = 2n. n = p K := + 2k 4k k 2 4k 2k k k K Then by equation 5. in Theorem 5. we have, setting D = /2, v v 0 = D s=0 = v 0 = K {,...,D} D s=0 K {,...,D} p K D K k=0 D K s K p K s K D K k = p,...,d = p,..., /2. Using 5.20 thisyiels For = 2n we get v v 0 = 2 /2 whereas for = 2n we fin /2 k= v 2n v 0 = 2n 2 2 n 2n =n+ v 2n v 0 2n = 2n 2 n k = K {,...,D} + 2k!k! 2 k! 2 2k!2k! 2 2k! 2. =n 2 n = n = 2, 2 p K δ D, K which conclues the proof. The integrality of the ratios v /v 0 is establishe in [2] by a careful analysis of the ivisors of the numerators an enominators of the fractions occurring in Corollary 5.4. Of course, the integrality is now immeiate from Theorem 5. an, furthermore, we gain the following alternative expressions for the ratios in consieration. 23

26 74 P. Kirschenhofer, J. Thuswalner Corollary 5.5 The ratios v /v 0 of the volumes from Corollary 5.4 are integers an are given by the alternative formula v k v 0 = et 2 + 2k 2 Proof From the proof of Corollary5.4 we have But from 5.4 we see that v v 0 = p, 2,..., / k p, 2,..., /2 = et 2 + 2k ,k /2.,k /2 Acknowlegements Open access funing provie by Montanuniversität Leoben. We are grateful to the anonymous referees whose comments le to a consierable improvement of the exposition of this paper. In particular, one of the referees suggeste to replace our original proof of Corollary 3.2 by introucing HX efine in 3. an ientity 3.2 together with the reference to 3.3 an the iea of using partial fraction techniques. Open Access This article is istribute uner the terms of the Creative Commons Attribution 4.0 International License which permits unrestricte use, istribution, an reprouction in any meium, provie you give appropriate creit to the original authors an the source, provie a link to the Creative Commons license, an inicate if changes were mae. References. Akiyama, S., Borbély, T., Brunotte, H., Pethő, A., Thuswalner, J.M.: Generalize raix representations an ynamical systems. I. Acta Math. Hungar. 08, Akiyama, S., Pethő, A.: On the istribution of polynomials with boune roots, I. Polynomials with real coefficients. J. Math. Soc. Jpn. 66, Akiyama, S., Pethő, A.: On the istribution of polynomials with boune roots II. Polynomials with integer coefficients. Unif. Distrib. Theory 9, Alastuey, A., Jancovici, B.: On the classical two-imensional one-component Coulomb plasma. J. Phys. 42, Anrews, G.E., Askey, R., Roy, R.: Special Functions, vol. 7 of Encyclopeia of Mathematics an Its Applications. Cambrige University Press, Cambrige Aomoto, K.: Jacobi polynomials associate with Selberg integrals. SIAM J. Math. Anal. 8, Cohn, A.: Über ie Anzahl er Wurzeln einer algebraischen Gleichung in einem Kreise. Math. Z. 4, Dubickas, A.: Counting integer reucible polynomials with boune measure. Appl. Anal. Discret. Math. 0, Fam, A.T.: The volume of the coefficient space stability omain of monic polynomials. Proc. IEEE Int. Symp. Circuits Syst. 2, Fam, A.T., Meitch, J.S.: A canonical parameter space for linear systems esign. IEEE Trans. Autom. Control 23,

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