On foci of ellipses inscribed in cyclic polygons
Markus Hunziker, Andrei Martinez-Finkelshtein, Taylor Poe, Brian Simanek
OON FOCI OF ELLIPSES INSCRIBED IN CYCLIC POLYGONS
MARKUS HUNZIKER, ANDREI MART´INEZ-FINKELSHTEIN, TAYLOR POE, AND BRIAN SIMANEK
Abstract.
Given a natural number n ≥ a and b in the unit disk D in the complexplane, it is known that there exists a unique elliptical disk having a and b as foci that can also berealized as the intersection of a collection of convex cyclic n -gons whose vertices fill the whole unitcircle T . What is less clear is how to find a convenient formula or expression for such an ellipticaldisk. Our main results reveal how orthogonal polynomials on the unit circle provide a useful toolfor finding such a formula for some values of n . The main idea is to realize the elliptical disk as thenumerical range of a matrix and the problem reduces to finding the eigenvalues of that matrix. Introduction
Suppose n ≥ E is an ellipse in the open unit disk D in the complexplane. A classical result known as Poncelet’s Theorem asserts that if there is an n -gon P inscribedin the unit circle T with every side of P tangent to E , then there are in fact infinitely many such n -gons and the union of the vertices of these n -gons fills T . In this case, the ellipse E is said tobe a Poncelet n -ellipse . A simple argument shows that if a, b ∈ D and n ≥ n -ellipse with foci at a and b . However, if n >
3, then it isnot obvious how to write down a formula for this ellipse or deduce any properties of its size (suchas area, eccentricity, etc.). Some early relevant formulas for this purpose were found by Cayley [5,Chapter 5], but they are not easy formulas to use. Some of the main results in this paper will showhow to write an explicit expression for Poncelet n -ellipses when n = 4 or n = 6. Figure 1.
A Poncelet 3-ellipse and its two foci.To accomplish this task, we must frame this problem in a broader context. The main tool we willuse is numerical ranges of a special class of ( n − × ( n −
1) matrices called completely non-unitarycontractions of defect index 1 (denoted S n − ). The specifics of these objects are provided in Section2 below, but for now it is enough for us to say that the numerical range W ( A ) of a matrix A ∈ S n − Mathematics Subject Classification.
Primary: 30J10, 42C05; Secondary: 14N15, 14H50, 47A12.
Key words and phrases.
Orthogonal polynomials, Poncelet Ellipses, Blaschke products . a r X i v : . [ m a t h . C A ] J a n M. HUNZIKER, A. MART´INEZ-FINKELSHTEIN, T. POE, AND B. SIMANEK is a strictly convex subset of the unit disk with a smooth boundary (see [14]). Furthermore, theboundary of this set ∂W ( A ) has the Poncelet property, meaning that every point on T is the vertexof an n -gon that is inscribed in T having every side tangent to ∂W ( A ) and every point on ∂W ( A )is a point of tangency for such an n -gon (see [4]). Our approach to the problem described in theprevious paragraph has its roots in work of Gau and Wu [10] and aims to realize the desired ellipseas the numerical range of an appropriate matrix in S n − . Theorem 2 below assures us that thisproblem has a solution.One simplification of our problem comes from the fact that instead of finding a matrix in S n − with the desired properties, it suffices to find just the eigenvalues of that matrix. This is becausenumerical ranges are preserved by unitary conjugation and every matrix in S n − is unitarily equiv-alent to a canonical form called a cutoff CMV matrix (see [13, 14]). Such matrices are an importantpart of the theory of orthogonal polynomials on the unit circle and will play an essential role in ouranalysis. Further details are presented in Section 2.Returning to our original problem, notice that Theorem 2 states that both a and b must beeigenvalues of the desired matrix. Thus, one really only needs to determine the remaining n − n = 3. When n = 4, a concise formulafor the one remaining eigenvalue appears in [12], and we give another proof of that formula inSection 3. In [15], Mirman presented a collection of algebraic relationships that must be satisfiedby the eigenvalues we seek. While this finding is significant, it falls short of presenting a completesolution to our problem because the algebraic relationships admit multiple solutions (even in D n − ).In Section 4, we will present a different collection of algebraic relationships that applies in the case n = 6 and admits a unique solution in D , which means that the unique solution to this system isthe collection of 3 eigenvalues that we seek. In Section 5, we will examine some additional propertiesof all of the solutions to Mirman’s system of equations in the case n = 5.The next section is a brief review of the background, notation, and terminology that will berelevant to the remainder of the paper. Many of these topics are discussed in much greater detailin [3, 13], and a thorough introduction to orthogonal polynomials on the unit circle can be foundin [19, 20]. 2. Background & Notation
Our primary objects of study will be numerical ranges of matrices. The numerical range of amatrix A ∈ C n × n is the subset of the complex plane C given by W ( A ) = {(cid:104) x, Ax (cid:105) : x ∈ C n , (cid:107) x (cid:107) = 1 } . For any matrix A , the set W ( A ) is a compact and convex subset of C (a fact known as the Toeplitz-Hausdorff Theorem) that contains the eigenvalues of A . If W ( A ) is bounded by an ellipse, then wewill say that W ( A ) is an elliptical disk .The matrices A that we are most interested in have the following properties:(i) (cid:107) A (cid:107) = 1;(ii) all eigenvalues of A are in D ;(iii) rank( I − AA ∗ ) = rank( I − A ∗ A ) = 1 . The set of all n × n matrices satisfying properties (i-iii) is precisely the set S n that we described inthe Introduction. As we stated there, it is known that the numerical range of a matrix in S n is theconvex hull of an algebraic curve of class n and has the ( n + 1)-Poncelet property, meaning thatevery point on ∂W ( A ) is a point of tangency for a convex ( n + 1)-gon that is circumscribed about W ( A ) and inscribed in T (see [13] for details).There are several canonical forms of matrices from the class S n (see [13, Section 2.3]) and the onethat we will use is that of a cutoff CMV matrix. To define a CMV matrix, first define a sequence N FOCI OF ELLIPSES INSCRIBED IN CYCLIC POLYGONS 3 of 2 × { Θ j } ∞ j =0 by Θ j = (cid:18) ¯ α j (cid:112) − | α j | (cid:112) − | α j | − α j (cid:19) , where α j ∈ D . One then defines the operators L and M by L = Θ ⊕ Θ ⊕ Θ ⊕ · · · , M = 1 ⊕ Θ ⊕ Θ ⊕ · · · where the initial 1 in the definition of M is a 1 × CMV matrix correspondingto the sequence { α n } ∞ n =0 is G := LM = α α ρ ρ ρ . . .ρ − α α − ρ α . . . α ρ − α α α ρ ρ ρ . . . ρ ρ − ρ α − α α − ρ α . . . α ρ − α α . . .. . . . . . . . . . . . . . . . . . , ρ n = (cid:112) − | α n | (2.1)(see [19, Section 4.2]). Since each of L and M is a direct sum of unitary matrices, each of L and M is unitary and hence G is unitary as an operator on (cid:96) ( N ). The principal n × n submatrix of G will also be called the n × n cut-off CMV matrix , which we will denote by G ( n ) . It is easy to seethat G ( n ) ∈ S n .CMV matrices are intimately connected with the theory of orthogonal polynomials on the unitcircle (OPUC). Indeed, if one definesΦ n ( z ) := det( zI n − G ( n ) ) , (2.2)then the polynomial Φ n ( z ) is the degree n monic orthogonal polynomial with respect to the measure µ that is the spectral measure of G and the vector (cid:126)e . Since G is unitary, the formula (2.2) impliesthat all zeros of Φ n are in D . Furthermore, the coefficients { α n } ∞ n =0 that are used to define G arerelated to { Φ n } ∞ n =0 by the Szeg˝o recursion: (cid:18) Φ k +1 ( z )Φ ∗ k +1 ( z ) (cid:19) = (cid:18) z − α k − α k z (cid:19) (cid:18) Φ k ( z )Φ ∗ k ( z ) (cid:19) , (2.3)where if Φ n ( z ) = n (cid:88) j =0 c j z j , then Φ ∗ n ( z ) = n (cid:88) j =0 c j z n − j = z n Φ n (1 /z ) . (2.4)Φ ∗ n is called the reversed polynomial of Φ n . Observe that Φ ∗ n can be of degree strictly less than n .It follows from the Szeg˝o recursion that α n = − Φ n +1 (0) and the sequence { α n } ∞ n =0 is often calledthe sequence of Verblunsky coefficients for the measure µ . For future use, let us define the notationΦ n ( z ) = S α n − (Φ n − ( z )) , Φ ∗ n ( z ) = T α n − (Φ ∗ n − ( z )) . to say that Φ n is related to Φ n − by the Szeg˝o recursion and the parameter α n − .Some of the most important theorems in the study of OPUC come from establishing the followingbijections (see [19, Chapter 1]): • All of the zeros of Φ n,µ ( z ) are in D and any collection { z j } nj =1 ∈ D n is the zero set of Φ n,ν ( z )for some ν supported on T . • The sequence { Φ n,µ (0) } n ∈ N is a sequence in D and every sequence { γ j } j ∈ N ∈ D N satisfiesΦ n,ν (0) = γ n for all n ∈ N and some measure ν supported on T . M. HUNZIKER, A. MART´INEZ-FINKELSHTEIN, T. POE, AND B. SIMANEK
This last fact (known as Verblunsky’s Theorem [19, Section 1.7]) tells us that the sequence { α n } ∞ n =0 completely characterizes the measure µ on T . If it is necessary to make the sequence of Verblunskycoefficients explicit, we will write Φ n ( z ; α , . . . , α n − ). We also note here that the Szeg˝o recursion isinvertible. This means that if we know Φ n ( z ; α , . . . , α n − ), then we can recover Φ j ( z ; α , . . . , α j − )for all j < n and hence we can recover α j for all j = 0 , . . . , n − α n − in the Szeg˝o recursionwith λ ∈ T yields a degree- n paraorthogonal polynomial on the unit circle (POPUC)Φ n ( z ; α , ..., α n − , λ ) = z Φ n − ( z ; α , ..., α n − ) − λ Φ ∗ n − ( z ; α , ..., α n − ) . In contrast to OPUC, the zeros of POPUC are on T . Given Φ n − ( z ), we can define { Φ ( λ ) n } as theset of all degree- n POPUC for Φ n − as λ varies around T . We have already noted that the OPUCΦ n is the characteristic polynomial of a cut-off CMV matrix G ( n ) . Using this parameter λ ∈ T , wecan characterize a family of rank one unitary dilations of G ( n ) . By adding one row and one columnto G ( n ) , we define a unitary ( n + 1) × ( n + 1) matrix whose characteristic polynomial is Φ ( λ ) n +1 ( z ).While the numerical range of G ( n ) has the ( n + 1)-Poncelet property, the numerical ranges of itsunitary dilations are bounded by ( n + 1)-gons inscribed in T and circumscribed around W ( G ( n ) ).The vertices of these ( n + 1)-gons are the eigenvalues of the matrices, or equivalently, the zeros ofthe POPUC.We have already mentioned that the boundary of the numerical range of G ( n ) has the ( n + 1)-Poncelet property. This phenomenon can be reformulated as saying that ∂W ( G ( n ) ) is the envelope ofthe family of circumscribing ( n + 1)-gons. The precise definition of an envelope and its relationshipto numerical ranges is complicated, so we will restrict our attention only to the most relevant factsand refer the reader to [13] for details. The envelope is most easily understood by means of adual curve. If the matrix is G ( n ) , then the dual curve is an algebraic curve of degree exactly n and the dual of that dual is an algebraic curve of class n with multiple components. The largestcomponent is the boundary of the numerical range of G ( n ) and we denote this component by C (tobe consistent with notation in [13]). The other components can be numbered C , . . . , C (cid:100) n/ (cid:101) andthey too have an interpretation in terms of the ( n + 1)-gons that circumscribe ∂W ( G ( n ) ).To understand this interpretation, let us consider the component C , which we call the Pentagramcurve after the pentagram map from [18]. For each ( n + 1)-gon P that circumscribes ∂W ( G ( n ) ), letus order the vertices cyclically on T by P , P , . . . , P n +1 . Consider now the “polygon” obtained byjoining P j to P j +2 for every j = 1 , . . . , n + 1 (where arithmetic is done modulo n + 1). The resultingshape will be a non-convex ( n + 1)-gon if n is even and it will be two ( n + 1) / n is odd. In either case, one can define the envelope of this collection of polygons andthat will be the curve C . A similar construction yields the other curves C j . When n = 6, we willrefer to the curve C as the Brianchon curve (after Brianchon’s Theorem [3, Theorem 5.4]) and wenote that the curve obtained from this procedure can be a single point (see Figures 4 and 5).It was Darboux who first proved that if the curve C is an ellipse, then all of the other curves C j are ellipses, where we consider a single point to be a degenerate ellipse (see [13]). These ellipses allhappen to be from the same package (see [15]). In the context of our motivating problem from theopening paragraph, if C is an ellipse, then the foci of C are eigenvalues of G ( n ) . Darboux’s resultimplies that the other C j curves are ellipses, and it turns out that their foci are also eigenvaluesof G ( n ) . It turns out that this property persists even if C is not an ellipse. More precisely, ifany curve C j is an ellipse, then the foci of that ellipse are eigenvalues of G ( n ) . For this reason,finding matrices G ( n ) for which some components C j are ellipses is an interesting problem relatedto our primary objective, and we will present results of this kind in later sections (see Theorem 1,Theorem 13, and Theorem 16 below). N FOCI OF ELLIPSES INSCRIBED IN CYCLIC POLYGONS 5
We will also work with
Blaschke products B n ( z ) := Φ n ( z )Φ ∗ n ( z ) , (2.5)where Φ n ( z ) = n (cid:89) j =1 ( z − z j ) , | z j | < . With this notation, we will say that the Blaschke product B n ( z ) has degree n . If we need tomake the dependence on the zeros explicit, then we will write B n ( z ; z , . . . , z n ). We will say that aBlaschke product B n ( z ) is regular if B n (0) = 0.Our discussion so far shows that the following sets are in bijection with one another:(i) equivalence classes of matrices in S n − (where equivalence is defined by unitary conjugation)(ii) monic polynomials of degree n − D (iii) regular degree n Blaschke products(iv) D n − (thought of as collections of Verblunsky coefficients)Much of what we will do in Sections 3 and 4 relates properties of the numerical range of a cutoffCMV matrix G ( n − to properties of the corresponding Blaschke product z Φ n − ( z ) / Φ ∗ n − ( z ). Thenext result will be very helpful in that regard. It is essentially an OPUC version of a result from[2]. Theorem 1.
Let n = jk and B n ( z ) be a regular Blaschke product, i.e. B n ( z ) = z Φ n − ( z )Φ ∗ n − ( z ) . Then B n ( z ) can be expressed as a composition of two regular Blaschke products B j ( B k ( z )) (with the degreeof B m equal to m ) if and only if Φ n − ( z ) factors as Φ n − ( z ) = Φ k − ( z ) j − (cid:89) m =1 S ¯ a m (Φ k − ( z )) for some Φ k − having all of its zeros in D and some { a , . . . , a j − } ∈ D j − . If this factorizationholds, then the zeros of Φ k − are the zeros of B k ( z ) /z and { a , . . . , a j − } is the zero set of B j ( z ) /z .Proof. Let B n ( z ) = B j ( B k ( z )). Then B j = z ( z − a ) ... ( z − a j − )(1 − a z ) ... (1 − a j − z )for some { a , . . . , a j − } ∈ D j − . If B k = z Φ k − ( z )Φ ∗ k − ( z ) , then B n ( z ) = B j (cid:32) z Φ k − ( z )Φ ∗ k − ( z ) (cid:33) = z Φ k − ( z )( z Φ k − ( z ) − a Φ ∗ k − ( z )) · · · ( z Φ k − ( z ) − a j − Φ ∗ k − ( z ))Φ ∗ k − ( z )(Φ ∗ k − ( z ) − a z Φ k − ( z )) · · · (Φ ∗ k − ( z ) − a j − z Φ k − ( z ))It follows that B n ( z ) = z Φ k − ( z ) S ¯ a (Φ k − ( z )) · · · S ¯ a j − (Φ k − ( z ))Φ ∗ k − ( z ) T ¯ a (Φ ∗ k − ( z )) · · · T ¯ a j − (Φ ∗ k − ( z ))and hence Φ n − has the desired factorization. Reversing this reasoning shows the converse state-ment. (cid:3) M. HUNZIKER, A. MART´INEZ-FINKELSHTEIN, T. POE, AND B. SIMANEK
Our focus is on finding those A ∈ S n − whose numerical range is bounded not just by an n -Poncelet curve but by an n -Poncelet ellipse. The relationship between numerical ranges of A ∈ S n − and Poncelet ellipses is a subject of significant ongoing research (see [2, 3, 4, 7, 8, 12, 13, 15, 17]).The following theorem (from [17]) is the starting point of our investigation and shows that theellipses that we are looking for do in fact exist. We state it using the terminology that we havedefined so far. Theorem 2. [17]
Suppose f , f ∈ D . There exists a Poncelet n -ellipse with foci at f and f .Furthermore, this ellipse forms the boundary of the numerical range of a matrix A ∈ S n − and f , f are eigenvalues of A . Our path forward is now clear. Given f , f ∈ D , we want to find a matrix A ∈ S n − suchthat ∂W ( A ) is an ellipse with foci at f and f . Theorem 2 tells us that such an A exists, andwe know that we can realize it as a cutoff CMV matrix. Such a matrix has n − D , two of which must be f and f . A priori, there are no other restrictions on the remainingeigenvalues of A other than they must be in D . In Section 3, we will show how to locate the thirdeigenvalue of A when n = 4 and in Section 4 we will consider the case when n = 6 and find a set ofalgebraic equations in three variables whose unique solution marks the locations of the other threeeigenvalues that we seek.The most significant results known for general n come from [15, 16, 17]. The following theoremis a restatement of a result from [15] using the terminology of matrices from S n . Theorem 3.
Suppose E n is a Poncelet n -ellipse in D that is also the boundary of the numericalrange of a matrix A ∈ S n − . Suppose the foci of E n are f and f . Then the eigenvalues of A canbe labeled { w , . . . , w n − } so that w = f , w n − = f , and w j − w j +1 = B ( w j ; f , f ) , j = 2 , , . . . , n − S n − whosenumerical range is bounded by an ellipse with foci at f and f . We will see that this system ofequations has many solutions and only one of them has the desired interpretation. In Section 5,we will consider matrices in S and discover some properties of all of the solutions to the Mirmansystem when n = 5. 3. The quadrilateral case
In this section, we will consider matrices in S to show how our approach to Poncelet ellipses usingOPUC allows us to reformulate, and in some cases strengthen, existing results in the literature. Aclassification of those matrices A ∈ S for which W ( A ) is an elliptical disk is given in [7]. Recall thata Blaschke product is regular if it maps 0 to 0. Fujimura [7] showed that A ∈ S has a numericalrange that is an elliptical disk if and only if there are regular Blaschke products B , C of degree 2such that B A ( z ) := z det( zI − A )det( zI − A ) ∗ = B ( C ( z )) (3.1)(see also [2, 12]). By Theorem 1, we can state the following theorem. Theorem 4.
Suppose A ∈ S . The numerical range of A is an elliptical disk if and only if thereexist a, b ∈ D so that det( zI − A ) = Φ ( z ; a )Φ ( z ; a, b ) . If this condition holds, then the pentagram curve is the single point ¯ a and the foci of ∂W ( A ) arethe zeros of Φ ( z ; a, b ) . N FOCI OF ELLIPSES INSCRIBED IN CYCLIC POLYGONS 7
Remark.
Theorem 4 implies that to any Poncelet 4-ellipse E ⊆ D , one can associate a well-defined point in D that will be the pentagram curve of the matrix A ∈ S that satisfies ∂W ( A ) = E .We will call this point the pentagram point of the ellipse E . Figure 2.
The Poncelet 4-ellipse with foci − .
28 + 0 . i , 0 . . i and pentagrampoint 0 . . i .Theorem 4 gives us a new interpretation of the algorithm for finding a matrix in S with ellipticnumerical range and two prescribed foci (such an algorithm can be found in [12]). Indeed, given { f , f } ∈ D , consider the polynomial Φ ( z ) = ( z − f )( z − f ). Perform the Inverse Szeg˝o recursionto obtain a degree 1 monic polynomial Φ ( z ; ¯ f ) whose zero f is in D . The 3 × { f , f , f } has the desired property.We can even generalize this algorithm to find an A ∈ S with elliptic numerical range havingone prescribed focus and a pentagram curve that is a prescribed point. Theorem 5.
Given { f , f } ∈ D , there exists a unique × cutoff CMV matrix whose numericalrange is bounded by an ellipse with one focus at f and such that the pentagram curve is the singlepoint { f } .Proof. By Theorem 4, this amounts to showing that we can find b ∈ D such that Φ ( z ; ¯ f , b ) vanishesat f . It is easy to see that this can be achieved precisely by setting¯ b = f Φ ( f ; ¯ f )Φ ∗ ( f ; ¯ f ) = f ( f − f )1 − ¯ f f (see also [21]). (cid:3) One can also find an A α ∈ S with ∂W ( A α ) an ellipse and whose pentagram curve is a specifiedpoint. Since this is a weaker set of conditions than was used in Theorem 5, one expects that wewill have many solutions to this problem. Rather than a unique matrix as in Theorem 5, fixingthe pentagram point yields a family of matrices in S parametrized by the Verblunsky coefficientof Φ ( z ; a, b ). Proposition 6.
Given α ∈ D , there exists a one-parameter family A α ∈ S , α ∈ D such that foreach A α , ∂W ( A α ) is an ellipse and the pentagram point of A α is α .Proof. Suppose α is given and consider the polynomial Φ ( z ; ¯ α ). For any α ∈ D , considerΦ ( z ; ¯ α , α ) and define φ ( z ) = Φ ( z ; ¯ α )Φ ( z ; ¯ α , α ) . M. HUNZIKER, A. MART´INEZ-FINKELSHTEIN, T. POE, AND B. SIMANEK
Thinking of φ ( z ) as an OPUC and applying the inverse Szeg˝o recursion allows us to recover theVerblunsky coefficients of φ ( z ) and thus define a cutoff CMV matrix, A α ∈ S , whose characteristicpolynomial is φ ( z ). As φ ( z ) factors into a degree one and degree two OPUC related by the Szeg˝orecursion, Theorem 4 implies that ∂W ( A α ) is an ellipse and the pentagram point of A α is α . (cid:3) Figure 3.
Two Poncelet 4-ellipses with the same pentagram point, 0 . . i .Now suppose we have an A ∈ S with numerical range given by an elliptical disk. Suppose thatthe foci of the ellipse bounding that elliptical disk are { f , f } and the pentagram point of thatellipse is { f } . Then det( zI − A ) = ( z − f )( z − f )( z − f ) . By Theorem 4, we also havedet( zI − A ) = ( z − f )( z ( z − f ) − b (1 − ¯ f z ))for some b ∈ D . Evaluating both of these expressions at 0 and equating them shows b = − f f . Itfollows that( z − f )( z − f ) = z ( z − f ) + f f (1 − ¯ f z ) = z + z ( − f − f f ¯ f ) + f f . (3.2)If we replace z by f in (3.2), we get( f − f )( f − f ) = f f (1 − | f | ) . (3.3)If we look at the reversed polynomials in (3.2) and replace z by f , we get(1 − ¯ f f )(1 − ¯ f f ) = 1 − | f | . (3.4)If we divide (3.3) by (3.4), we recover the Mirman system: f f = B ( f ; f , f ). Solving for f yields the familiar formula for f in terms of f , f : f = f + f − f | f | − f | f | − | f f | . Notice that we have recovered something that the Mirman system does not give us. One can verifythat f = 0 is a solution to the Mirman system, but this does not (in general) give us the matrix in S with elliptical numerical range. Our calculations using OPUC eliminate this extraneous solution. N FOCI OF ELLIPSES INSCRIBED IN CYCLIC POLYGONS 9 The hexagon case
In this section we will consider curves that are realized as the envelope of line segments joiningvertices of hexagons inscribed in the unit circle (see [13, Section 3] for a rigorous discussion ofenvelopes of cyclic polygons). We know that such curves have three components: the largest one(outer) formed by connecting adjacent eigenvalues of the unitary dilations is the
Poncelet curve , themiddle component formed by joining alternate eigenvalues is the pentagram curve , and the smallestcomponent formed by joining opposite eigenvalues is the
Brianchon curve . Our first results are thefollowing two theorems.
Theorem 7.
Let G (5) be a cut-off CMV matrix and Φ ( z ) := det( zI − G (5) ) . The following are equivalent. (i) the pentagram curve of G (5) is an ellipse; (ii) there exist regular Blaschke products { B j } j =2 with deg( B j ) = j such that z Φ ( z )Φ ∗ ( z ) = B ( B ( z ))(iii) There exist α , α , α ∈ D so that Φ ( z ) = Φ ( z ; α , α )Φ ( z ; α , α , α ) . If any of these conditions hold, then the foci of the pentagram curve are the zeros of B ( z ) /z orequivalently, the zeros of Φ ( z ; α , α ) . Figure 4.
A 6-Poncelet curve such that the pentagram curve is an ellipse but theBrianchon curve is not a single point.
Theorem 8.
If we retain the notation from Theorem 7, then the following are equivalent: (i) the Brianchon curve of G (5) is a single point; (ii) there exist regular Blaschke products { B j } j =2 with deg( B j ) = j such that z Φ ( z )Φ ∗ ( z ) = B ( B ( z ))(iii) There exist α , α , γ ∈ D so that Φ ( z ) = Φ ( z ; α )Φ ( z ; α , α )Φ ( z ; α , γ ) If any of these conditions hold, then the Brianchon point is the zero of B ( z ) /z and the zero of Φ ( z ; α ) . The equivalence of (i) and (ii) in both theorems is proven in [2]. The equivalence of (ii) and (iii)in both theorems follows from Theorem 1. By comparing Theorem 8 with Theorem 4 (and theremark after it), one arrives at the following result.
Corollary 9.
Let A ∈ S have eigenvalues { f j } j =1 . Suppose the Brianchon curve of A is the point f . Then { f j } j =1 can be labelled in such a way that both of the following conditions hold: (i) f is the pentagram point of the Poncelet -ellipse with foci at f and f ; (ii) f is the pentagram point of the Poncelet -ellipse with foci at f and f . Using ideas from [13], we can prove the following.
Theorem 10.
If we retain the notation from Theorem 7, then the following are equivalent (i)
The Poncelet curve associated with G (5) is an ellipse. (ii) There exist regular Blaschke products { B j } j =2 with deg( B j ) = j and regular Blaschke prod-ucts { C j } j =2 with deg( C j ) = j such that z Φ ( z )Φ ∗ ( z ) = B ( B ( z )) = C ( C ( z )) . (iii) The pentagram curve of G (5) is an ellipse and the Brianchon curve of G (5) is a single point. Figure 5.
A Poncelet 6-ellipse along with the pentagram ellipse and Brianchon point.
Proof.
The equivalence of (ii) and (iii) is an immediate consequence of Theorems 7 and 8. The factthat (i) implies (iii) is a result of Darboux (see [13, Theorem B]). To see that (iii) implies (i), wewill make use of the dual curve described in Section 2. Notice that [13, Section 3] implies that if C , C , or C is an algebraic curve of degree 1 or 2, then the same is true of the correspondingcomponent of the dual curve and vice versa (counting a single point as having degree 1). The dualcurve in this case has degree 5 (see [9, Section 3]). Thus, if the Brianchon curve has degree 1 andthe pentagram curve has degree 2, then the Poncelet curve has degree 2, which means it is anellipse. (cid:3) Theorem 8 reveals an interesting phenomenon. Suppose we are given an A ∈ S whose Brianchoncurve is a point and whose pentagram curve is not an ellipse (it is easy to find such A ). Take arank one unitary dilation U of A and look at the hexagon with vertices at the eigenvalues of U . N FOCI OF ELLIPSES INSCRIBED IN CYCLIC POLYGONS 11
The diagonals of this hexagon meet in a single point (which is the Brianchon curve of A ), so byBrianchon’s Theorem there exists an ellipse E inscribed in this hexagon. By construction, theellipse E is a Poncelet 6-ellipse, so there is in fact an infinite family of hexagons { H λ } inscribed in ∂ D and circumscribed about E . On the other hand, one can look at any other rank one unitarydilation of A and repeat this process. This gives a second infinite family of hexagons, each of whichis circumscribed in ∂ D and has its diagonals meeting at the point that is the Brianchon curve of A .But these two families of hexagons cannot be the same, for that would imply that the numericalrange of A is bounded by an ellipse and we know it is not. Figure 6.
A 6-Poncelet curve that is not an ellipse having the property that itsBrianchon curve is a single point (Brianchon point). The picture on the right showsthe ellipse that is inscribed in one of the Poncelet hexagons. This ellipse existsby Brianchon’s theorem. Notice that the 6-Poncelet curve contains points in theexterior as well as points in the interior of this ellipse.The next step in our analysis will be very much analogous to that performed in Section 3. If weare given { f , f } ∈ D , we want to find A ∈ S whose numerical range is bounded by an ellipsewith foci at { f , f } . Theorem 2 tells us that such an ellipse exists and is the unique Poncelet5-ellipse with foci at f and f . We will find a cutoff CMV matrix G (5) whose numerical range isbounded by an ellipse with foci at { f , f } , whose pentagram curve is an ellipse (whose foci will becalled { f , f } ), and whose Brianchon curve is a single point (that we will call f ). Theorem 11.
A matrix A ∈ S with eigenvalues { f j } j =1 satisfies all of the following conditions (i) W ( A ) is an elliptical disk (ii) the foci of ∂W ( A ) are { f , f } (iii) the foci of the pentagram curve are { f , f } (iv) the Brianchon curve is the single point f if and only if the complex numbers { f j } j =1 satisfy { f , f } = (cid:26) f − f − f ¯ f , f − f − f ¯ f (cid:27) (4.1) as sets and f + f + f f f ¯ f ¯ f = f + f + f f f + f f f ( ¯ f + ¯ f ) = f f + f f + f f (4.2)Recall that Theorem 8 and Theorem 10 show that if W ( A ) is bounded by an ellipse, then thecharacteristic polynomial factors in a certain way (in terms of OPUC) and the degree 1 polynomialin that factorization vanishes at the Brianchon point. We require the following refinement of thatresult, which tells us about the zeros of the remaining polynomials in that factorization. Lemma 12.
Suppose A ∈ S is such that W ( A ) is an elliptical disk and the eigenvalues of A are { f j } j =1 . Suppose the foci of ∂W ( A ) are { f , f } , the foci of the pentagram curve are { f , f } , andthe Brianchon point is f . Write det( zI − A ) = Φ ( z ; α )Φ ( z ; α , α )Φ ( z ; α , γ ) as in Theorem 8. If Φ ( f ; α , α ) = 0 , then Φ ( f ; α , γ ) = 0 .Proof. Suppose Φ ( f ; α , α ) = 0 and Φ ( f ; α , γ ) (cid:54) = 0. Then Φ ( f ; α , α ) = 0 and henceΦ ( f ; α , γ ) = Φ ( f ; α , γ ) = 0.Since Φ ( f ; α , α ) = 0, we can write( z − f )( z − f )( z − f ) = Φ ( z ; α )Φ ( z ; α , α ) . This means { f , f , f } are the eigenvalues of some A ∈ S that satisfies the hypotheses of Theorem4. Applying the Mirman system in the N = 4 case shows f f = B ( f ; f , f )The Mirman system in the case n = 6 shows shows f f = B ( f ; f , f ) and hence f f = f f .Since f , f are the zeros of Φ ( z ; α , γ ) and γ = − Φ (0; α , γ ), we conclude that γ = α ,which implies Φ ( z ; α , γ ) = Φ ( z ; α , α ). It follows that Φ ( f ; α , γ ) = 0, which gives us acontradiction. (cid:3) Proof of Theorem 11
In Theorem 7 it is stated that the foci of the pentagram ellipse will be thezeros of Φ ( z ; α , α ). Thus, the foci of the Poncelet curve and the Brianchon point must be thezeros of Φ ( z ; α , α , α ). The product of these zeros is then ¯ α and hence we have z ( z − f )( z − f ) − f f f (1 − ¯ f z )(1 − ¯ f z ) = ( z − f )( z − f )( z − f ) (4.3)Equating coefficients of z and z in (4.3) tells us that (4.2) must hold. One can perform a similarcalculation invoking Theorem 8. By equating coefficients of the appropriate polynomials, we findthat (4.1) must hold.For the converse statement, the above calculations show that if (4.1) and (4.2) hold, then theconditions (iii) in Theorems 7 and 8 are satisfied (by equating coefficients of polynomials). Theorem10 then implies that the cutoff CMV matrix G (5) with eigenvalues { f j } j =1 has numerical range thatis bounded by an ellipse, has pentagram curve that is an ellipse with foci { f , f } , and has Brianchonpoint { f } . The foci of ∂W ( G (5) ) are eigenvalues of G (5) (see [13, Section 5]). By [15, Corollary4], we know that one can partition the eigenvalues of G (5) into the Brianchon point, the foci of thepentagram curve, and the foci of the boundary of the numerical range. Thus, by elimination itmust be that the foci of ∂W ( G (5) ) are { f , f } . (cid:3) Recall that the Mirman system in the N = 6 case is f f = B ( f ; f , f ) , f f = B ( f ; f , f ) , f f = B ( f ; f , f ) , (4.4)The conditions (4.1) and (4.2) allow us to recover these relations. Substitute z for f in (4.3) toobtain ( f − f )( f − f ) = f f f (1 − | f | )(1 − f ¯ f ) f − f (4.5)If we replace z by f in the reversed polynomials from (4.3), then we obtain(1 − ¯ f f )(1 − ¯ f f ) = (1 − | f | )(1 − f ¯ f )1 − ¯ f f (4.6) N FOCI OF ELLIPSES INSCRIBED IN CYCLIC POLYGONS 13
If we divide (4.5) by (4.6), and use (4.1), we find B ( f ; f , f ) = f f . Similar reasoning can beused to derive B ( f ; f , f ) = f f . Replacing z by f in (4.3) tells us that f ( f − f )( f − f )(1 − ¯ f f )(1 − ¯ f f ) = f f f If we assume f f f (cid:54) = 0 and we substitute the relations (4.1) for f and f (for an appropriatechoice of which to call f and which to call f ), then we find(1 − ¯ f f )(1 − ¯ f f ) = (1 − ¯ f f )(1 − ¯ f f )In other words (1 − ¯ f f )(1 − ¯ f f ) ∈ R . If we use this fact, then multiplying the expressions in(4.1) gives B ( f ; f , f ) = f f .If one is given ( f , f ) ∈ D , the construction above shows how to find an A ∈ S so that ∂W ( A )is an ellipse with foci at f and f . Our next result shows that one can similarly find A ∈ S with ∂W ( A ) an ellipse and whose pentagram curve has prescribed foci (Theorem 10 assures us that ifthe Poncelet curve of A is an ellipse, then so is its pentagram curve). Theorem 13.
Given ( f , f ) ∈ D , there exists a unique (up to unitary conjugation) A ∈ S sothat ∂W ( A ) is an ellipse and the pentagram curve of A is an ellipse with foci at f and f . Our proof of Theorem 13 requires the following two lemmas.
Lemma 14.
Given two triangles inscribed in ∂ D with interlacing vertices, there is a unique ellipsethat is inscribed in both of them.Proof. Any such ellipse would be a Poncelet 3-ellipse. The lemma is a consequence of Wendroff’sTheorem for POPUC, which includes a uniqueness statement (see [11, Theorem 3.1], [14, Theorem8], or [1]). (cid:3)
Lemma 15.
Let { Φ ( λ )3 } λ ∈ ∂ D be the collection of degree POPUC for the same degree OPUC.Label the zeros of Φ ( λ )3 as { z ( λ ) j } j =1 . For each λ ∈ ∂ D there exists a unique τ ∈ ∂ D so that the linesegments joining z ( λ ) j to z ( τ ) j all meet in a single point independent of j (this assumes an appropriatelabeling of the zeros { z ( λ ) j } j =1 ).Proof. For each τ , let z ( τ )1 be the zero that lies between z ( λ )2 and z ( λ )3 , let z ( τ )2 be the zero that liesbetween z ( λ )3 and z ( λ )1 , and let z ( τ )3 be the zero that lies between z ( λ )1 and z ( λ )2 . Let L ( τ ) j be the linesegment that joins z ( λ ) j to z ( τ ) j for j = 1 , , { z ( λ ) j } j =1 , start with τ = λ and move τ around T counterclockwise. As this happens,consider η j ( τ ) := L ( τ )1 ∩ L ( τ ) j for j = 2 , L ( τ )1 . Define d j ( τ ) = | z ( λ )1 − η j ( τ ) || L ( τ )1 | , j = 2 , . Initially, d ( τ ) is close to 0 and d ( τ ) is close to 1. As τ nears the end of its trip around T , it holdsthat d ( τ ) is close to 0 and d ( τ ) is close to 1. Thus, by the Intermediate Value Theorem, theremust be a value of τ such that d ( τ ) = d ( τ ) as desired. One can see by inspection that this choiceof τ is unique. (cid:3) Proof of Theorem 13
Suppose ( f , f ) ∈ D are given. Consider the Poncelet 3-ellipse with fociat f and f (call it E ). Pick any triangle T ( λ ) that is inscribed in ∂ D and circumscribed about E .By Lemma 15, there exists a unique second such triangle T ( τ ) such that the line segments joiningopposite vertices meet in a single point. By Brianchon’s Theorem, there is an ellipse inscribed in the hexagon whose vertices are the vertices of T ( τ ) and T ( λ ) . Call this ellipse E (cid:48) and suppose itsfoci are f and f .From what we already know, the ellipse E (cid:48) is the unique Poncelet 6-ellipse with foci at f and f , and Theorem 2 tells us that it is associated to a matrix in S . For this matrix, the associatedpentagram curve must be an ellipse and the associated Brianchon curve must be a single point.This pentagram ellipse is a Poncelet 3-ellipse and must be inscribed in the triangles T ( λ ) and T ( τ ) .By Lemma 14, that ellipse must be E . (cid:3) Our next result is an analog of Theorem 13 for the Brianchon curve. Specifically, we will show thatone can find A ∈ S so that ∂W ( A ) is an ellipse and the Brianchon curve is a single predeterminedpoint. The main difference between Theorem 16 and Theorem 13 is the lack of uniqueness. Theorem 16.
Given f ∈ D , there exists a × cutoff CMV matrix A so that ∂W ( A ) is an ellipseand the Brianchon curve of A is the single point f . Furthermore, the set of all possible × cutoffCMV matrices with this property is naturally parametrized by an open triangle.Proof. Suppose f ∈ D is given. Consider the set of all lines passing through f . Each line intersects T in two places. One can choose three distinct lines, thus specifying 6 distinct points of T , labeledcyclically as v j for j = 1 , , ...,
6. By Brianchon’s Theorem, there is an ellipse inscribed in thehexagon whose vertices are { v j } j =1 . Call this ellipse E and its foci f and f . By Theorem 11, E is the boundary of W ( A ) for some A ∈ S . Thus, the Poncelet curve and pentagram curve of A are ellipses and the Brianchon curve of A is a single point, which we see must be f as desired. e it e ix e ix ∗ e iy e iy ∗ f Figure 7.
Given f and the line through 1, f , varying x, y such that 0 < x < y < t parametrizes the set of all A ∈ S with elliptic numerical range and Brianchon curve f .Note that the above procedure yields a well-defined map from collections of distinct triples oflines through f to matrices in S whose numerical range is an elliptical disk and whose Brianchonpoint is f . To see this, suppose one starts with three distinct lines through f . This produces sixpoints on the unit circle by the above procedure. Connecting alternate points forms two trianglesthat must be tangent to the pentagram ellipse of the matrix we seek. By Lemma 14, there is onlyone possible choice for such an ellipse, so its foci f , f are a well-defined output. By Theorem 11and equation (4.1), we can calculate the foci f , f of the Poncelet curve of this matrix. Thus, theeigenvalues { f j } j =1 of the matrix we seek are a computable quantity from any three distinct linesthrough f . Since the eigenvalues determine the matrix in S , this means the map is well-defined. N FOCI OF ELLIPSES INSCRIBED IN CYCLIC POLYGONS 15
To make this map a bijection, we restrict it to triples of lines through f for which one of themalso passes through 1. Given any A ∈ S with W ( A ) an elliptical disk and Brianchon point f ,there exists a hexagon that circumscribes ∂W ( A ) for which 1 is a vertex, so the restricted map isonto. To show that it is injective, suppose H is a hexagon inscribed in T with one vertex at 1 and E is an ellipse that is tangent to every edge of H . From any given point on T (in particular, the point1), there are only two tangents to E through that point. This implies H is the unique hexagon thatincludes 1 and has the required tangency properties. Thus, our restricted map uniquely determinesthe numerical range of the matrix and hence uniquely determines the matrix itself.Suppose that the line through 1 and f also intersects T at e it . We have shown that the space of all5 × { ( x, y ) : 0 < x < y < t } ,which is an open triangle. (cid:3) The pentagon case
To find Poncelet 5-ellipses, it will not be possible to consider compositions of Blaschke productsas in the previous sections, which partially explains why this case has been studied less often in theliterature. Instead, we will revisit the Mirman system and prove a structure theorem about the setof possible solutions. In this setting, the system of equations has multiple solutions and our nextresult describes their relative placement in the plane.
Theorem 17.
Let f , f ∈ D , f f (cid:54) = 0 and set Φ ( z ) = ( z − f )( z − f ) . The system Φ ( z )Φ ∗ ( z ) = wf , Φ ( w )Φ ∗ ( w ) = zf (5.1) has exactly 5 distinct solution pairs ( z, w ) ∈ C : four of them are in D : (0 , f ) , ( f , , ( z , w ) , ( z , w ) , and exactly one solution ( z , w ) satisfies | z | > , | w | > .Moreover, the points z , z and z are collinear, as are w , w and w . More precisely,i) It holds that z − f w − f = z − f w − f = z − f w − f = f Φ ∗ ( f ) f Φ ∗ ( f ) , ii) The points f , w , w and w are collinear, i.e. w i − f w j − f ∈ R , i, j ∈ { , , } , i (cid:54) = j. and the points f , z , z and z are collinear, i.e. z i − f z j − f ∈ R , i, j ∈ { , , } , i (cid:54) = j. Proof.
Let us denote g ( z ) = 1 f ( z − f )( z − f )(1 − z ¯ f )(1 − z ¯ f ) , g ( z ) = 1 f ( z − f )( z − f )(1 − z ¯ f )(1 − z ¯ f ) , so that (5.1) takes the form g ( z ) = w , g ( w ) = z . From here, we get that (5.1) implies that g ◦ g ( z ) = z,g ◦ g ( w ) = w. (5.2) f f w w z z z w Figure 8.
Collinearity of the points f , f , z j ’s and w j ’s, as explained in Theorem 17.Both g ◦ g and g ◦ g have 4 fixed points inside D . Indeed, consider g ◦ g (same analysis for g ◦ g ). Recall that a Blaschke product is analytic in D , maps T → T , D → D and exterior of D onto its exterior. So, | z | = 1 ⇒ | g ( z ) | > ⇒ | g ( g ( z )) | > . By Rouche’s theorem, g ◦ g ( z ) − z and g ◦ g ( z ) have the same number of zeros in D ; it isstraightforward to check that g ◦ g vanishes at 4 points in D .Finally, we claim that the statement of the proposition is equivalent to the just established factthat both g ◦ g and g ◦ g have 4 fixed points inside D .Indeed, the 4 pairs of solutions of (5.1) satisfy (5.2). Reciprocally, let z be a fixed point of g ◦ g and denote τ = g ( z ). Then g ( τ ) = g ( g ( z )) = z , and in consequence, g ( g ( τ )) = g ( z ) = τ ,meaning that τ = g ( z ) is a fixed point of g ◦ g . This shows that the fixed points z and w of g ◦ g and g ◦ g can be paired ( z, w ) in such a way that (5.1) holds.Now we prove the statement about the remaining solution, this time in ( C \ D ) . The identity(1 / ¯ z − f )(1 / ¯ z − f )(1 − ¯ f / ¯ z )(1 − ¯ f / ¯ z ) = 1 ( z − f )( z − f )(1 − z ¯ f )(1 − z ¯ f ) (5.3)allows us to reduce the analysis of (5.1) in ( C \ D ) to the equivalent system( z − f )( z − f )(1 − z ¯ f )(1 − z ¯ f ) = wf , ( w − f )( w − f )(1 − w ¯ f )(1 − w ¯ f ) = zf (5.4)for ( z, w ) ∈ D (we return to the actual solutions outside by the mapping z (cid:55)→ /z , w (cid:55)→ /w ).The advantage of working in D is that again we can use the fixed point argument and Rouche’sTheorem. Indeed, as before, define h ( z ) = f ( z − f )( z − f )(1 − z ¯ f )(1 − z ¯ f ) , h ( z ) = f ( z − f )( z − f )(1 − z ¯ f )(1 − z ¯ f ) , and look for fixed points of h ◦ h and h ◦ h . This time | z | = 1 ⇒ | h ( z ) | < ⇒ | h ( h ( z )) | < , N FOCI OF ELLIPSES INSCRIBED IN CYCLIC POLYGONS 17 and by Rouche’s Theorem, h ◦ h ( z ) − z and f ( z ) = z have the same number of zeros in D , thatis, exactly one.To prove the statements about collinearity, define the linear functions z ( t ) = f − f − f ¯ f + 4 f (1 − | f | )(1 − f ¯ f ) tw ( t ) = f − f − f ¯ f + 4 f (1 − | f | )(1 − f ¯ f ) t (5.5)It is a straightforward calculation to verify that the two polynomials (in the variable t )Φ ( z ( t )) − w ( t ) f Φ ∗ ( z ( t )) , Φ ( w ( t )) − z ( t ) f Φ ∗ ( w ( t ))are scalar multiples of each other so any zero of one is a zero of the other. Notice that both ofthese polynomials have degree 3. One can also check by hand that if t z is such that z ( t z ) = 0, then w ( t z ) (cid:54) = f . Similarly, if t w is such that w ( t w ) = 0, then z ( t w ) (cid:54) = f . Thus, the three zeros { t j } j =1 of Y ( t ) := Φ ( z ( t )) − w ( t ) f Φ ∗ ( z ( t )) (5.6)will be such that ( z ( t j ) , w ( t j )) is a solution to the system (5.1) other than (0 , f ) and ( f , z j , w j ) = ( z ( t j ) , w ( t j )). If we set t = (1 − | f | )(1 − | f | ) and observe that z ( t ) = f and w ( t ) = f , then a short calculation shows z j − f w j − f = z ( t j ) − z ( t ) w ( t j ) − w ( t ) = f Φ ∗ ( f ) f Φ ∗ ( f ) . This proves claim (i) of the theorem.To prove claim (ii), it suffices to show that each t j ∈ R . To this end, a calculation reveals thatif we divide the polynomial Y ( t ) in (5.6) by its leading coefficient, then we obtain a monic degree3 polynomial with real coefficients.Define q ( x, y ; t ) := (cid:18) y + x − f − f x (cid:19) (cid:18) y + x − f − f x (cid:19) − txy (1 − f x )(1 − f x )There exist two distinguished matrices A , A ∈ S such that A has numerical range bounded byan ellipse with foci at { f , f } and the pentagram curve of A is an ellipse with foci at { f , f } . Letus denote the eigenvalues of A j by { f , f , z j , w j } for j = 1 ,
2. Then from [13, Section 5] (and also[15, Equations 29–32]) we know that there exist positive real numbers b and b so that q ( f , z j ; b j ) = q ( w j , f ; b j ) = q ( z j , w j ; b j ) = 0for j = 1 ,
2. In fact, b j is the length of the minor semiaxis of the ellipse associated with A j whosefoci are { f , f } (see [15]).Using q ( f , z j ; b j ) = q ( w j , f ; b j ) = 0 and the formulas (5.5), we find that z j = z ( b j ) and w j = w ( b j ). A short calculation shows (recall t = (1 − | f | )(1 − | f | ) / q ( z ( t ) , w ( t ); t ) = Y ( t )( t − t ) C f ,f Φ ∗ ( z ( t )) , where C f ,f = − f t f . Thus the zeros of Y ( t ) are also zeros of q ( z ( t ) , w ( t ); t ).If t were a zero of Y ( t ), then z = z ( t ) and w = w ( t ) would satisfy (5.1). However, we haveseen that z ( t ) = f and w ( t ) = f , giving 0 = Φ ( f ) = f , which is nonzero by assumption, so Y ( t ) (cid:54) = 0.Thus the zeros of Y ( t ) are zeros of q ( z ( t ) , w ( t ); t ) distinct from t . We know that q ( z ( t ) , w ( t ); t )has zeros at t = b and t = b so these must be zeros of Y ( t ) as well. This gives us two real zeros of Y ( t ). Our earlier observation implies that Y ( t ) has either one or three real zeros, so all zeros of Y ( t ) are real as desired. (cid:3) w z f f w z f f Figure 9.
Pairs ( z , w ) and ( z , w ) as foci of the pentagram and the Ponceletellipses, respectively.The solutions to the Mirman system in this setting have a geometric interpretation. Given { f , f } ∈ D , we have just seen that there are three solution pairs ( z, w ) to the Mirman systemand we denote them by ( z , w ), ( z , w ), and ( z , w ). One of the solution pairs in D (say ( z , w ))will be such that the 4 × { f , f , z , w } will have numericalrange bounded by an ellipse with foci at f and f and the pentagram curve of this matrix will bean ellipse with foci at z and w (see Figure 9, left). For the other solution pair in D (say ( z , w )),it will be true that the 4 × { f , f , z , w } has numericalrange bounded by an ellipse with foci at z and w and the pentagram curve of this matrix will bean ellipse with foci at f and f , see Figure 9, right. The geometric interpretation of the solution( z , w ) ∈ ( C \ ¯ D ) is not clear.Notice that Theorem 17 implies that each line (cid:96) j passing through { z j , w j } for j = 1 , , { f , f } . For j = 1 ,
2, one can obtain this same conclusion from the fact thatthe ellipses C and C corresponding to a matrix A ∈ S are in the same package (see [17]). Thefact that this same conclusion applies to (cid:96) is a new result. Acknowledgments
The second author was partially supported by Simons Foundation Collaboration Grants forMathematicians (grant 710499) and by the Spanish Government–European Regional DevelopmentFund (grant MTM2017-89941-P), Junta de Andaluc´ıa (research group FQM-229 and InstitutoInteruniversitario Carlos I de F´ısica Te´orica y Computacional), and by the University of Almer´ıa(Campus de Excelencia Internacional del Mar CEIMAR).The fourth author graciously acknowledges support from Simons Foundation Collaboration Grant707882.
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Department of Mathematics, Baylor University, Waco TX, USA
Email address : Markus [email protected] (AMF)
Department of Mathematics, Baylor University, Waco TX, USA, and Department of Math-ematics, University of Almer´ıa, Almer´ıa, Spain
Email address : A [email protected] (TP)
Department of Mathematics, Baylor University, Waco TX, USA
Email address : Taylor [email protected] (BS)
Department of Mathematics, Baylor University, Waco TX, USA
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