Distribution results on polynomials with bounded roots

For $d \in \mathbb{N}$ the well-known Schur-Cohn region $\mathcal{E}_d$ consists of all $d$-dimensional vectors $(a_1,\ldots,a_d)\in\mathbb{R}^d$ corresponding to monic polynomials $X^d+a_1X^{d-1}+\cdots+a_{d-1}X+a_d$ whose roots all lie in the open unit disk. This region has been extensively studied over decades. Recently, Akiyama and Peth\H{o} considered the subsets $\mathcal{E}_d^{(s)}$ of the Schur-Cohn region that correspond to polynomials of degree $d$ with exactly $s$ pairs of nonreal roots. They were especially interested in the $d$-dimensional Lebesgue measures $v_d^{(s)}:=\lambda_d(\mathcal{E}_d^{(s)})$ of these sets and their arithmetic properties, and gave some fundamental results. Moreover, they posed two conjectures that we prove in the present paper. Namely, we show that in the totally complex case $d=2s$ the formula \[ \frac{v_{2s}^{(s)}}{v_{2s}^{(0)}} = 2^{2s(s-1)}\binom {2s}s \] holds for all $s\in\mathbb{N}$ and in the general case the quotient $v_d^{(s)}/v_d^{(0)}$ is an integer for all choices $d\in \mathbb{N}$ and $s\le d/2$. We even go beyond that and prove explicit formul\ae{} for $v_d^{(s)} / v_d^{(0)}$ for arbitrary $d\in \mathbb{N}$, $s\le d/2$. The ingredients of our proofs comprise Selberg type integrals, determinants like the Cauchy double alternant, and partial Hilbert matrices.


Introduction
Almost 100 years ago Issai Schur [20,21] introduced and studied what is now called the Schur-Cohn region. For each dimension d ∈ N this region is defined as the set of all coefficient vectors of monic polynomials of degree d each of whose roots lies in the open unit disk. As we will deal with polynomials with real coefficients in the present paper we give the formal definition for this setting. The d-dimensional Schur-Cohn region is defined by E d := {(a 1 , . . . , a d ) ∈ R d : each root ξ of X d + a 1 X d−1 + · · · + a d−1 X + a d satifies |ξ | < 1}.
Sometimes polynomials having all roots in the open unit disk are called contractive polynomials. In the years after Schur's papers were published, Cohn [7] developed an algorithm to check if a polynomial is contractive. In Rahman and Schmeisser [17,Section 11.5] further developments and properties of the Schur-Cohn region are surveyed.
Starting from the 1970s contractive polynomials were studied by Fam and Meditch [10] and their co-authors from the viewpoint of discrete time systems design. In this context Fam [9] calculated the d-dimensional Lebesgue measure λ d (E d ) of E d . In particular, he proved that (here we give a simplified version of [9, equations (2.7) and (2.8)], see also [8, equation (9) (1.1) More recently, the Schur-Cohn region occurred in connection with number systems and so-called shift radix systems (see Akiyama et al. [1] or Kirschenhofer and Thuswaldner [13]). Given a polynomial P ∈ R[X ] of degree d with r real and 2s nonreal roots we call (r, s) the signature of P. Since d = r + 2s for given degree d the signature is determined by the value s of pairs of conjugate nonreal roots. In [2], Akiyama and Pethő studied a partition of E d in terms of the signature of contractive polynomials (see also [3]). In particular, they were interested in the sets E (s) d := {(a 1 , . . . , a d ) ∈ E d : X d + a 1 X d−1 + · · · + a d−1 X + a d has exactly s pairs of nonreal roots} and their volumes v (s) . (1.2) In the quest for formulae for these volumes they unveiled interesting relations of v (s) d with Selberg integrals (introduced in [22]) and their generalizations by Aomoto [6].
In particular, they expressed v (0) d in terms of a classical Selberg integral which led to the simple formula (see [ Besides that, the studies by Akiyama and Pethő in [2] were devoted to arithmetical properties of the quantities v (s) d . It was proved that these numbers are rational for all choices of d and s. Moreover, certain quotients of these numbers were studied. In particular, it was shown that v d /v (0) d is an odd integer for all d ≥ 1 and, based on numerical experiments, the following conjecture was stated. Moreover, for the particular instance d = 2s the following surprisingly simple explicit formula was conjectured. In the present paper we confirm both these conjectures (see Theorem 5.1 and Theorem 2.1 below). We note that a special instance of the first conjecture was confirmed recently by Kirschenhofer and Weitzer [14], where the formula It should be noted that many questions and problems in number theory, combinatorics, CFT, and other mathematical areas are related to integrals of this type (compare e.g. [5,11]). For the treatment of the Selberg type integrals occurring in our proof in the context of random Vandermonde matrices an approach proposed in Alastuey and Jancovici [4] turns out to be very valuable. Besides that, methods from "advanced determinant calculus" (see [15]) are used to obtain closed forms for variants of the Cauchy double alternant that come up along the way. Moreover, we need minors of the Hilbert matrix, which is known in numerics for its numerical instability (see e.g. [23]), to further simplify our formulae in order to finally establish the integrality of the quotients under scrutiny.
The paper is organized as follows. In Sect. 2 we confirm Conjecture 1.2 of Akiyama and Pethő. In particular, we prove (1.3) (see Theorem 2.1) which immediately 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 ideas. Section 3 is devoted to preparatory definitions and results. It contains the proof of a closed form for a certain convolution of a "determinant like" sum that will be needed in the proof of our main result. Section 4 contains a formula for the volumes v 2s . The proof of this conjecture is easier than the proof of Conjecture 1.1 but already contains some of the ideas needed for the general case and serves as a "road 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 deserves to be announced in a separate theorem.
As a main step for the proof of Theorem 2.1 we establish the following result.
Proof We start from [2, Theorem 2.1] for the case of r = 0 real roots and s pairs of complex conjugate roots with B + denoting the upper half of the unit disk. In the following step of computation we adopt an idea used by Physicists in the computation of certain Coulomb potentials (see e.g. [4]). Observe that the integrand in the last integral equals the Vandermonde determinant V (z 1 , z 1 , . . . , z s , z s ) of order 2s. In the following, we denote by S m the symmetric group on m letters. Inserting the definition of the determinant in (2.5) we obtain with σ k : From this, transforming to polar coordinates Observe that the numerators in the last fraction are equal to 0 if σ 2 j−1 − σ 2 j is even and equal to −2 if σ 2 j−1 − σ 2 j is odd. Therefore, only permutations σ ∈ S 2s , for which σ 2 j−1 − σ 2 j is odd for all 1 ≤ j ≤ s give a nonzero contribution to the sum. A permutation of this type has the form where ϕ is a permutation in S o 2s , the set of all permutations of {1, 3, . . . , 2s − 1}, and ψ is a permutation in S e 2s , the set of all prmutations of {2, 4, . . . , 2s}, or has a form where any of the pairs (ϕ(2 j − 1), ψ(2 j)) in the second line are interchanged. Each interchange of the latter type will change the sign of the permutation by the factor −1. Therefore, setting ϕ j := ϕ(2 j − 1), ψ j := ψ ( (2.9) Since ϕ j − ψ j is odd, the term in the round brackets equals 4 Observe that if ϕ and ψ are regarded as elements of S 2s . Therefore, sgn(ϕ, ψ) = sgn(ϕ)sgn(ψ) and we get By the definition of the determinant we have The determinant on the right hand side needs sgn(ϕ) interchanges of columns to be transformed to (2.10) which completes the proof of the lemma.
We finish the proof of Theorem 2.1 by the following product formula for the determinant D s .
Proof We start with the observation that D s is of the form Using the Cauchy double alternant formula (cf. e.g. [15]) 2s . (2.14) From the last equality the lemma follows immediately by induction.
Thus the proof of Theorem 2.1 is complete.

Preparations for the general case
This section contains some definitions and auxiliary results that will be needed in the proof of our main result. Throughout the paper finite (totally) ordered sets will play a prominent role. If X and Y are finite ordered sets we write X ⊆ Y to state that X is a (possibly empty) ordered subset of Y whose elements inherit the order from Y . Moreover Y \ X denotes the finite ordered subset of Y containing all elements of Y that are not contained in X . Also other set theoretic notions will be carried over to finite ordered sets in the same vein. To indicate the order we sometimes write {X 1 < · · · < X r } instead of {X 1 , . . . , X r } for a finite ordered set. We also use the following notation. Given an For r ∈ N let X = {X 1 < · · · < X r } be an ordered set of indeterminates and consider sums of the form We now give a closed form for convolutions of these sums which will immediately imply an identity that is needed in the proof of our main results.
(3.2) Here the sum runs over all ordered subsets K of [r ].
Proof First we note that, according to [12, 3 rd identity on p. 6812], for a finite ordered set X the sum H (X ) defined in (3.1) admits the closed form Indeed, (3.3) is proved by induction using a recurrence relation which is established by splitting the sum in (3.1) w.r.t. the value of σ (m) (see [12, proof of equation (4.9)]). This closed form will be used throughout the proof. Let f r (X 1 , . . . , X r ) and g r (X 1 , . . . , X r ) denote the left and right hand side of (3.2), respectively. To prove the lemma it suffices to show that f r as well as g r satisfy the recurrence for r ≥ 2 with h 1 (X 1 ) = 2. This can be proved using the partial fraction technique. In order to establish the recurrence (3.4) for f r we first observe that, since the summands on the left hand side of (3.2) are invariant under the involution K → K , We evaluate p r +1 (X 1 , . . . , X r , −1) in two ways. Firstly, we set X r +1 = −1 directly in the definition of p r +1 (X 1 , . . . , X r +1 ) and use (3.5) to replace the condition r − 1 ∈ K by a factor 1 2 in the resulting expression. Secondly, we insert the partial fraction expansion in the product containing the indeterminate X r +1 in p r +1 , then set X r +1 = −1, interchange the sums, and reorder the indeterminates appropriately. This yields the identity (δ r,J is 1 if r ∈ J and 0 otherwise) (note that the sum in the second line vanishes by (3.5)). Evaluating the sum q r +1 (X 1 , . . . , X r +1 ) in two ways using the partial fraction expansion w.r.t. X r +1 of (in this case r − 1 / ∈ K holds), by similar reasoning as above we gain the formula (3.7) Adding (3.6) and (3.7) the summands corresponding to indices i with r − i odd vanish. In the remaining summands interchanging the indeterminates X r and X r +1 absorbs the sign (−1) δ r,J and, hence, (3.4) holds for f r .
Let r ∈ N and let Y = {Y 1 < · · · < Y r } be an ordered set of indeterminates. Later we will need sums of the form .
(3.8) Substituting X i = q Y i and letting q tend to 1 it now follows from (3.3) that By the same operations, the following corollary, which will be used later on, is an immediate consequence of (3.9) and Lemma 3.1.

Corollary 3.2 Let
Here the sum runs over all ordered subsets K of [r ].
We now define some notions related to the parity of the elements of a finite ordered set.

Definition 3.3
A finite ordered set X consisting of |X |/2 odd and |X |/2 even numbers is said to be in increasing oe-order, if it is of the shape odd, even, odd, even, …, where the odd and the even ordered subset are both strictly increasing. We denote the sign of the permutation that brings the elements of X from their natural order to the increasing oe-order by oesgn(X ).
Let X be a finite ordered set of strictly ascending positive integers. Define the finite ordered sets X 1 and X 2 such that 2X 1 and 2X 2 −1 constitute the even and odd elements of the finite ordered set X in ascending order, respectively. Then (X 1 , X 2 ) is called the parity splitting of X . The finite ordered set X is called parity balanced if |X 1 | = |X 2 |, i.e., if X contains the same number of odd and even elements.
We will need the following property of the oe-order. For the induction step assume that the lemma is true for all c(c + 1) ≤ a < a 0 and consider a constellation N 1 , N 2 corresponding to the value a = a 0 . Since a 0 > c(c+1) there is x ∈ N with x − 2 ∈ M. Now we switch these two elements, i.e., we put

Formulae for the volumes in the general case
In this section we treat the case of polynomials having s pairs of (nonreal) complex conjugate roots and r real roots within the open unit disk. We will first derive a general formula that expresses the volume v (s) 2s+r of the coefficient space in this instance by means of a sum of determinants. This forms a generalization of Lemma 2.2.

Using the notations introduced in (2.3) and (2.4) the result of [2, Theorem 2.1] states that the volumes in question admit the integral representation
where D r,s denotes the region given by 0 ≤ β j ≤ 1, −2 β j ≤ α j ≤ 2 β j for all 1 ≤ j ≤ s and −1 ≤ ξ j ≤ 1 for all 1 ≤ j ≤ r. Proceeding as in (2.6) with the "complex part" of the integral and bringing the variables ξ 1 , . . . , ξ r in increasing order yields v (s) V 2s,r dx 1 · · · dx s dy 1 · · · dy s dξ 1 · · · dξ r , is the Vandermonde determinant of order 2s +r . Laplace expansion of this determinant by the first 2s columns (which are the columns corresponding to the "complex" part) yields  (4.7) Observing that M 2 ) is the parity splitting of M, and N = {1, . . . , 2s + r } \ M (observe that N consists of r/2 even and r/2 odd numbers). Now we turn our attention to the "real" integral. For the finite ordered set N = {n 1 < · · · < n r } we have where V (ξ 1 , . . . , ξ r ; N ) is given by  It now remains to compute the integrals V (ξ 1 , . . . , ξ r ; N ) dξ 1 · · · dξ r . (4.13) By the tight relation of V (ξ 1 , . . . , ξ r ; N ) to Schur functions mentioned after (4.10) we see that these integrals resemble the Selberg-type integrals studied in [11,Section 2]. Indeed, the Jack polynomials (also called Jack's symmetric functions) used there are generalizations of Schur functions (see [16, p. 379]). The difference is that in our instance we only have skew symmetric rather than symmetric integrands in (4.13). Let us fix k for the moment and expand the determinant by its first k columns. Then the integrals in the kth summand of (4.13) can be rewritten as (4.14) Substituting η i = −ξ i for 1 ≤ i ≤ k in the first integral in (4.14) yields V (u 1 , . . . , u m ; W ) du 1 · · · du m . (4.16) Using this notation, inserting (4.15) into (4.14) and then (4.14) into (4.13), and observing that, because n 1 , . . . , n r are in increasing oe-order the numbers n i j + i j are all even, we finally get In the next step we replace the integrals J m (W ) by the sum S m (W ) from (3.8). This is justified by the following lemma. Proof Recall that (2) ) · · · (w σ (1) + · · · + w σ (m) ) = S m (W ).

(4.18)
For the following considerations we have to make some amendments in case |N | is odd. In particular, we define 2t = r if r is even, r + 1 if r is odd (4.19) and set (4.20) 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 (4.17). ( N 1 , N 2 ) be the parity splitting of N . Then the "real" integrals I r (N ) with N as in (4.11) satisfy 21) where B N 1 ,N 2 is the minor with row indices from N 1 and column indices from N 2 of the matrix

Lemma 4.3 Let
Proof To prove the result for the instance r = 2t we use Corollary 3.2 with the choice Y j = n j , where n j is as in (4.11). Combining this corollary with Lemma 4.2 we obtain Using the double alternant formula (2.13) this yields 1≤ j,k≤t and the result follows because the last determinant equals B N 1 ,N 2 .
In the instance r = 2t − 1 the result follows by the same reasoning. The only difference is that instead of using the double alternant formula (2.13) we have to use (4.22) This identity can in turn be proved along the lines of the usual proof of (2.13). Now we are ready to prove our first main result on the "mixed" volumes.
Proof Using Lemma 3.4 we have to deal with the quantities oesgn(M)oesgn(N ) in (4.12). Observe that oesgn(N ) = oesgn( N ), since 2n is the maximal element. Thus we may apply Lemma 3.4 with N instead of N and thus this lemma holds for odd and even cardinalities of N .
In the first and the third sum in the statement of the theorem expand the determinants simultaneously according to the rows with index in J. The resulting expression is the same as the one we get when we apply Lemma 3.4 and Lemma 4.3 to (4.12). In the second sum of the theorem we have to expand the determinants simultaneously according to the columns with index in J to finish the proof.
Setting s = 0 we get the following expressions for the "totally real" volumes.
Evaluating the determinants by the double alternant (2.13) or its "odd" analogue (4.22), respectively, this corollary yields the formulae for the totally real volumes known from [2, Theorem 4.1] (note the close connection to the Selberg integral S d (1, 1, 1/2) in this reference).

The volume ratios in the general case
Now we are in a position to turn our attention to formulae for the quotients v (s) d /v (0) d and to confirm Conjecture 1.1. In particular, we establish the following result involving determinants of partial Hilbert matrices from which we can infer the integrality of v (s) The binomial coefficient occurs due to the following observation. If J s is the collection of all subsets of cardinality s of the set {1, . . . , n} then a given set K ⊆ {1, . . . , n} is a set of exactly n−|K | s−|K | elements of the collection J s . For the case of s = 0 nonreal roots this specializes to v (0) 2n The determinants d K may be evaluated by the double alternant formula (2.13) as where all products run over the range 1 ≤ j, k ≤ n with the denoted restrictions.
Using the notation j ≡ k (K ) iff ( j, k ∈ K or j, k / ∈ K ), this reads , and, in particular, so that the quotients fulfill For the first product in (5.5) we derive with m!! denoting the double factorial. Therefore, inserting (5.6) and (5.7) in (5.5) yields (5.9) Inserting (5.9) in (5.4) completes the proof of the first formula for the case of even degree in Theorem 5.1. The second formula follows by writing the trinomial coefficient as a product of two binomial coefficients (mind the factor 1/2 in the product in (5.9)) and applying the double alternant formula (2.13) to express the last product by the determinant of the partial Hilbert matrix from the statement of the theorem. Let us now turn to the case of odd degree d = 2n − 1. This time we use the matrices c j,k (J ) from Theorem 4.4. Again using the decomposition (5.2) with X j = 2 j and and, in particular, 1≤ j≤n−1 1≤k≤n Therefore, the quotients fulfill For the the first product in (5.12) we get from (5.6) by replacing n by n − 1 (5.14) We end this section by showing that some results of Kirschenhofer and Weitzer [14] as well as Akiyama and Pethő [2] are simple consequences of Theorem 5.1. The first result refers to the instance of polynomials with exactly one pair of nonreal conjugate zeroes.