Invariant hyperbolic curves: determinantal representations and applications to the numerical range
IINVARIANT HYPERBOLIC CURVES:DETERMINANTAL REPRESENTATIONS ANDAPPLICATIONS TO THE NUMERICAL RANGE
FAYE PASLEY SIMON AND CYNTHIA VINZANT
Abstract.
Here we study the space of real hyperbolic plane curves that are invariant underactions of the cyclic and dihedral groups and show they have determinantal representationsthat certify this invariance. We show an analogue of Nuij’s theorem for the set of invarianthyperbolic polynomials of a given degree. The main theorem is that every invariant hy-perbolic plane curve has a determinantal representation using a block cyclic weighted shiftmatrix. This generalizes previous work by Lentzos and the first author, as well as by Chienand Nakazato. One consequence is that if the numerical range of a matrix is invariant underrotation, then it is the numerical range of a block cyclic weighted shift matrix. Introduction
Here we study properties of a real plane curve that can be certified by a Hermitian de-terminantal representation, in particular, hyperbolicity and invariance under the action of afinite group. A real homogeneous polynomial is hyperbolic with respect to a point in R n ifit is positive at the point and has real-rooted restrictions on every line through that point.Hyperbolic polynomials were introduced in the mid-20th century by Petrovsky and G˚arding,in the context of partial differential equations. Since then they have appeared in a wide rangeof areas and applications, including convex optimization [23, 37], combinatorics [6, 7, 24, 32],convex and complex analysis [3, 5], and operator theory [28, 33].A fundamental example is given by the determinant. On the real vector space of Hermitianmatrices, the determinant is hyperbolic with respect to the identity matrix. More generally,given a linear matrix pencil A ( x ) = (cid:80) ni =1 x i A i where the matrices A , . . . , A n are Hermitianand the matrix A ( e ) is positive definite, the polynomial f ( x ) = det( A ( x )) is hyperbolic withrespect to e ∈ R n . This determinantal representation certifies the hyperbolicity of f and wesay that A ( x ) is a definite determinantal representation of f . See [43] for more.For n = 3, V C ( f ) is a plane curve in P ( C ). Determinantal representations are a classicalobject of study [4, 17]. In 1902, Dixon showed that every plane curve has a symmetric Figure 1.
The variety of a quartic invariant hyperbolic form in P ( R ) and R . a r X i v : . [ m a t h . AG ] F e b FAYE PASLEY SIMON AND CYNTHIA VINZANT determinantal representation over the complex numbers [16]. Almost a hundred years later,Helton and Vinnikov proved the Lax conjecture, showing that every hyperbolic plane curvehas a definite determinantal representation with real symmetric matrices [28]. Proving theexistence of such representations involves the existence of two-torsion points on the Jaco-bian of the curve with certain real structure. Showing the existence of definite Hermitianrepresentations is less delicate. This was done, for example, by Fiedler [19] and later studiedconstructively by Plaumann and Vinzant [36].Here we study this question in the context of curves invariant under the action of a finitegroup, in particular the cyclic or dihedral groups. We say that a polynomial f ∈ R [ t, x, y ] isinvariant under a group Γ ⊂ GL( R ) if f ( γ · ( t, x, y )) = f for all γ ∈ Γ . We will be interestedin the cyclic and dihedral groups C n = (cid:104) rot (cid:105) and D n = (cid:104) rot , ref (cid:105) on R , given by(1) rot · txy = π/n ) sin(2 π/n )0 − sin(2 π/n ) cos(2 π/n ) txy and ref · txy = tx − y . Our first main theorem is an analogue of Nuij’s theorem on the structure of the set ofhyperbolic polynomials of a given degree.
Theorem (Theorem 2.2) . The set of polynomials in R [ t, x, y ] d that are hyperbolic with respectto (1 , , and invariant under the action of the cyclic or dihedral group (of any order) iscontractible and equal to the closure of its interior in the Euclidean topology on R [ t, x, y ] d . This is a key step in the proof that all such polynomials have an invariant definite deter-minantal representation. Such representations were first studied by Chien and Nakazato inthe context of numerical ranges [12]. Given a matrix A ∈ C d × d , define the polynomial(2) F A ( t, x, y ) = det( tI + x ( A + A ∗ ) / y ( A − A ∗ ) / i ) ∈ R [ t, x, y ] d . Since the matrices I , ( A + A ∗ ) / A − A ∗ ) / i are Hermitian and the identity matrixis positive definite, F A has a definite determinantal representation and is hyperbolic withrespect to (1 , , A ∈ C n × n is a complex cyclic weightedshift matrix , then F A is invariant under the action of the cyclic group C n and if additionally A has real entries then F A is invariant under the action of the dihedral group D n [12]. Theyalso show that for n = 3 and 4 any hyperbolic, invariant polynomial f ∈ R [ t, x, y ] n has sucha representation. This was generalized by Lentzos and Pasley [31] who show this for all n .Here we generalize this to block cyclic weighted shift matrices , as defined in Definition 3.1.For such a matrix A , the polynomial F A is hyperbolic with respect to (1 , ,
0) and invariantunder C n or D n , if additionally the matrix is real. Moreover, any invariant hyperbolicpolynomial has such a representation when its degree is an integer multiple of n . Theorem (Theorem 6.1) . Let d ∈ n Z + and suppose f ∈ R [ t, x, y ] d is hyperbolic with respectto (1 , , , with f (1 , ,
0) = 1 , and invariant under the action of Γ . (a) If Γ = C n , then f = F A for some block cyclic weighted shift matrix A ∈ C d × d . (b) If Γ = D n , then f = F A for some block cyclic weighted shift matrix A ∈ R d × d . The original motivation of Chien and Nakazato was to understand invariance of numericalranges. Formally, the numerical range of a matrix A ∈ C d × d is W ( A ) = (cid:8) v ∗ A v : v ∈ C d , || v || = 1 (cid:9) ⊂ C . NVARIANT HYPERBOLIC PLANE CURVES 3 - - - - - - - - -
50 0 50 100 - - - Figure 2.
The curve V R ( F A ) and its dual curve bounding W ( A ) for A ∈ C × .The Toeplitz-Hausdorff theorem states that this is a convex body in C ∼ = R [27, 41]. Thisset appears in applications related to engineering, numerical analysis, and differential equa-tions [1, 8, 18, 19, 22]. Theorem (Kippenhahn [29]) . Let X ∗ be the dual variety to X = V C ( F A ) . The numericalrange of A ∈ C d × d is the convex hull of the real, affine part of X ∗ . That is, W ( A ) = conv ( { x + iy : [1 : x : y ] ∈ X ∗ ( R ) } ) . Using the above theorem on invariant determinantal representations, we show the followingabout numerical ranges that are invariant under the action of the cyclic or dihedral group.
Theorem (Theorem 7.1) . Let A ∈ C d × d and let W ( A ) denote its numerical range. If W ( A ) is invariant under multiplication by n -th roots of unity, then there exists a block cyclicweighted shift matrix B of size ≤ n · (cid:100) d/n (cid:101) so that W ( A ) = W ( B ) . Moreover if W ( A ) isinvariant under conjugation, then the entries of B can be taken in R . The paper is organized as follows. In Section 2, we introduce the theory of hyperbolicpolynomials and prove an invariant analogue of Nuij’s theorem on the topology of this set.The precise definition of block cyclic weighted shift matrices and their connection to invari-ant hyperbolic polynomials is discussed in Section 3. In Sections 4 and 5, we prove parts (a)and (b) of Theorem 6.1 under some genericity conditions on the curve V C ( f ) and in Section 6we address the degenerate cases to complete the proof. Applications to numerical ranges aregiven in Section 7. Finally we conclude with a discussion of open problems in Section 8. Acknowledgements.
We thank Ricky Liu, Daniel Plaumann and Rainer Sinn for helpfulcomments and discussions. Part of this work was done while both authors were partici-pants at the Fall 2018 Nonlinear Algebra program at the Institute for Computational andExperimental Research in Mathematics. Both authors were partially supported by the USNSF-DMS grant
Invariant hyperbolic polynomials
For a field F = R or C , we use F [ t, x, y ] d to denote the F -vectorspace of polynomials invariables t, x, y that are homogeneous of degree d . Given f ∈ F [ t, x, y ] d , we use V F ( f ) todenote the variety of f in the projective plane P ( F ) over F . A finite group Γ of GL( R ) FAYE PASLEY SIMON AND CYNTHIA VINZANT defines an action on the vector space F [ t, x, y ] given by γ · f = f ( γ · ( t, x, y )). Let F [ t, x, y ] Γ denote the subring of invariant polynomials f satisfying f ( γ · ( t, x, y )) = f for all γ ∈ Γ andlet F [ t, x, y ] Γ d denote homogeneous elements of degree d in this ring. By a classical theoremof Hilbert, the invariant ring F [ t, x, y ] Γ is finitely generated. For the group actions of C n and D n given in (1), we can explicitly find these generators. See e.g. [20]. For F = R or C , F [ t, x, y ] C n = R (cid:2) t, x + y , R [( x + iy ) n ] , I [( x + iy ) n ] (cid:3) , and F [ t, x, y ] D n = R (cid:2) t, x + y , R [( x + iy ) n ] (cid:3) where(3) R [( x + iy ) n ] = ( x + iy ) n + ( x − iy ) n I [( x + iy ) n ] = ( x + iy ) n − ( x − iy ) n i . We will be particularly interested in the set of invariant hyperbolic polynomials.
Definition 2.1.
A polynomial f ∈ R [ t, x, y ] d is hyperbolic with respect to a point e ∈ R if f ( e ) > f ( λ e − v ) ∈ R [ λ ] is real-rooted for every choice of v ∈ R . We call f strictlyhyperbolic with respect to e if f is hyperbolic with respect to e and the roots of f ( λ e − v )are distinct for every v ∈ R \ ( R e ). By [36, Lemma 2.4], an equivalent definition is that f ishyperbolic with respect to e and its real projective variety V R ( f ) is smooth.A polynomial g ∈ R [ t, x, y ] d − is interlaces f with respect to e ∈ R if both are hyperbolicwith respect to e and the roots of g ( λ e − v ) interlace the roots of f ( λ e − v ) for every v ∈ R .We say that g strictly interlaces f with respect to e if g interlaces f with respect to e theroots of g ( λ e − v ) and f ( λ e − v ) are all distinct for every v ∈ R \ ( R e ).For Γ = C n , D n denote the set of hyperbolic, invariant forms of degree d by H Γ d = (cid:8) f ∈ R [ t, x, y ] Γ d : f (1 , ,
0) = 1 , f is hyperbolic with respect to (1 , , (cid:9) . The subset of hyperbolic polynomials without any real singularities we denote by( H ◦ ) Γ d = (cid:8) f ∈ H Γ d : V R ( f ) ⊂ P ( R ) is smooth (cid:9) . As noted above, these are exactly the strictly hyperbolic forms in H Γ d . Polynomials in ( H ◦ ) Γ d may have complex singularities and indeed may be forced to do for some choices of d , asdiscussed in Section 8.1.Nuij [34] showed that the set of hyperbolic polynomials of a given degree is contractiblein R [ t, x, y ] d ∼ = R ( d +22 ) and equal to the closure of its interior, which consists of strictlyhyperbolic polynomials. Here we show an analogous statement for hyperbolic polynomialsinvariant under the cyclic and dihedral groups. Theorem 2.2.
For
Γ = C n or D n and any d ∈ Z + , both ( H ◦ ) Γ d and H Γ d are contractible.Moreover, ( H ◦ ) Γ d is a full-dimensional, open subset of the set of polynomials in R [ t, x, y ] Γ d with coefficient of t d equal to and its closure equals H Γ d . The proof requires developing an invariant version of techniques used in [34]. To un-derstand how the sets ( H ◦ ) Γ d and H Γ d relate, we introduce the following linear operator oninvariant polynomials. For s ∈ R , define the linear map T s : R [ t, x, y ] d → R [ t, x, y ] d by T s ( f ) = f − s ( x + y ) ∂ f∂t . NVARIANT HYPERBOLIC PLANE CURVES 5
Lemma 2.3.
For any s ∈ R > , the map T s preserves invariance under Γ and hyperbolicity.That is, T s ( H Γ d ) ⊂ H Γ d . Moreover for any f ∈ H Γ d , the polynomial T ds ( f ) , obtained by applying T s d times to f , is strictly hyperbolic with respect to (1 , , . That is T ds ( H Γ d ) ⊂ ( H ◦ ) Γ d .Proof. First, note that if f ∈ R [ t, x, y ] Γ d , then so are x + y and ∂ f∂t , meaning that T s preserves invariance under Γ.For the other claims, consider the operator on univariate polynomials T : R [ t ] → R [ t ]where T ( p ) = p − s p (cid:48)(cid:48) . We claim that for any real-rooted polynomial p ∈ R [ t ], T ( p ) isalso real rooted and the roots of T d ( p ) where d = deg( p ) are simple. To see this, considerthe maps T ± : R [ t ] → R [ t ] where T ± ( p ) = p ± sp (cid:48) for some s ∈ R . The roots of T ± ( p )have multiplicity one less than those of p , any repeated roots of T ± ( p ) are also repeatedroots of p and any added roots of T ± ( p ) are simple by the lemma of [34]. Let T = T + ◦ T − so T ( p ) = p − s p (cid:48)(cid:48) . The roots of T ( p ) have multiplicity two less than those of p , and anyrepeated roots are also repeated roots of p . Any other roots of T ( p ) are simple. If d = deg( p ),this implies every root of T d ( p ) is simple.Since for any ( a, b ) ∈ R , the restriction T s ( f )( t, a, b ) equals the image of p ( t ) = f ( t, a, b )under the univariate operator T , the polynomial T s ( f ) is hyperbolic with respect to (1 , , T ds ( f ) is strictly hyperbolic. (cid:3) Proof of Theorem 2.2.
We follow the proof of the main theorem in [34]. Since strict hy-perbolicity with respect to (1 , ,
0) is an open condition on R [ t, x, y ] d , it suffices to showthat ( H ◦ ) Γ d is non-empty. An explicit example is t δ · (cid:81) Di =1 ( t − r i ( x + y )) where D = (cid:98) d (cid:99) , δ ∈ { , } depending on the parity of d , and r < . . . < r D ∈ R + .The set H Γ d is closed in the hyperplane in R [ t, x, y ] Γ d of polynomials with coefficient of t d equal to one. To see that it is the closure of ( H ◦ ) Γ d , let f ∈ H Γ d . By Lemma 2.3, for s > T ds ( f ) is strictly hyperbolic with respect to (1 , , T ds ( f ) belongs to ( H ◦ ) Γ d .The limit at s = 0 is exactly f . For s ∈ R , consider the linear map G s : R [ t, x, y ] d → R [ t, x, y ] d given by G s f ( t, x, y ) = f ( t, s x, s y ) . This map preserves hyperbolicity and invariance forany s ∈ R as well as strict hyperbolicity when s (cid:54) = 0.For f ∈ H Γ d , consider the path in R [ t, x, y ] Γ d parametrized by s (cid:55)→ T d − s G s f for s in[0 , s = 1, this gives T d G f = f and at s = 0, this gives T d G f = T d t d , which isindependent of the choice of f . Note that for s ∈ [0 , T d − s G s f ∈ ( H ◦ ) Γ d . The map[0 , × R [ t, x, y ] d → R [ t, x, y ] d given by ( s, f ) (cid:55)→ T d − s G s f in the Euclidean topology definesa deformation retraction of both H Γ d and ( H ◦ ) Γ d onto the point T d t d . (cid:3) Cyclic weighted shift matrices and invariance
One way of producing hyperbolic polynomials that are invariant under the actions of C n or D n is via cyclic weighted shift matrices. Definition 3.1.
We call A ∈ F d × d a block cyclic weighted shift matrix of order n if A ij = 0 if j − i (cid:54) = 1 mod n. Let C F ( n, d ) denote the set of such matrices. Remark 3.2.
The term “block cyclic weighted shift matrix” is justified after a permutationof the rows and columns of A . Consider the permutation of [ d ] that groups numbers bytheir image modulo n and otherwise keeps them in order. For example, for n = 3 , d = 5, weconsider the permutation (1 , , , , (cid:55)→ (1 , , , , r × s FAYE PASLEY SIMON AND CYNTHIA VINZANT where r, s ∈ {(cid:98) dn (cid:99) , (cid:100) dn (cid:101)} , indexed by pairs ( i, j ) of equivalence classes modulo n , where theblock corresponding to ( i, j ) is the zero matrix whenever j − i (cid:54) = 1 modulo n . Example 3.3.
An arbitrary matrix A ∈ C F (3 ,
5) has the form A = a a a a a a a a and P AP T = a a
00 0 a a
00 0 0 0 a a a a where P is the permutation matrix representing (1 , , , , (cid:55)→ (1 , , , , V R ( F A (1 , x, y )) and numerical range W ( A ) for such a matrix are shown in Figure 2.The set of matrices C C ( n, n ), also called cyclic weighted shift matrices, have been studiedextensively especially with respect to their numerical range [13, 20, 40]. In general, thenumerical range of any matrix in C C ( n, d ) is invariant under multiplication by n th roots ofunity. To see this, define the group homomorphism ρ : C n → GL( C d ) by(4) ρ (rot) = Ω ∗ where Ω := diag (cid:0) , ω, ω , . . . , ω d (cid:1) and ω = e πi/n . This induces an action of the cyclic group on d × d matrices by rot · A = Ω ∗ A Ω. Note thatthe ( i, j )th entry of Ω ∗ A Ω is ω j − i A ij . Therefore, for any matrix A ∈ C C ( n, d ), the cyclicgroup acts by scaling by the n th root of unity. That is,Ω ∗ A Ω = ωA.
Since the matrix Ω is unitary and numerical ranges are invariant under conjugation byunitary matrices, we see that the numerical range of A is invariant under multiplication by n th roots of unity, i.e. W ( A ) = W ( ωA ). Chien and Nakazato [12] show that for matrices A ∈ C F ( n, n ), the polynomials F A are invariant under the cyclic group (for F = C ) anddihedral group (for F = R ). Here we generalize this observation to matrices of arbitrary size.To do this it is useful to rewrite the polynomial F A as F A ( t, x, y ) = det (cid:0) tI + ( x + iy ) A ∗ + ( x − iy ) A (cid:1) . This is particular convenient as the group C n acts diagonally on the linear forms t , x + iy ,and x − iy . Specifically, each action fixes t and we haverot · ( x + iy ) = e − πi/n ( x + iy ) ref · ( x + iy ) = x − iy rot · ( x − iy ) = e πi/n ( x − iy ) ref · ( x − iy ) = x + iy. Proposition 3.4.
For any A ∈ C d × d , the polynomial F A is hyperbolic with respect to (1 , , .If A ∈ C C ( n, d ) , then F A belongs to H C n d and if A ∈ C R ( n, d ) , then F A belongs to H D n d . Here H Γ d denotes the set of invariant hyperbolic polynomials as in Section 2. Proof.
By definition, the polynomial F A is the determinant of a linear matrix pencil thatequals the identity matrix at (1 , , F A then follows from the factthat all of the eigenvalues are real. For invariance, it suffices to check that rot · F A = F A for A ∈ C C ( n, d ) and ref · F A = F A for A ∈ C R ( n, d ). Following [12] and using the simplification NVARIANT HYPERBOLIC PLANE CURVES 7 of F A above, we apply rotation to giverot · F A ( t, x, y ) = det (cid:0) tI + ( x + iy )( ωA ) ∗ + ( x − iy )( ωA ) (cid:1) = det (cid:0) tI + ( x + iy )(Ω ∗ A Ω) ∗ + ( x − iy )(Ω ∗ A Ω) (cid:1) = det(Ω ∗ ) · det (cid:0) tI + ( x + iy ) A ∗ + ( x − iy ) A (cid:1) · det(Ω) = F A ( t, x, y ) . Similarly, if A has real entries then A ∗ = A T andref · F A ( t, x, y ) = F A ( t, x, − y ) = det (cid:0) tI + ( x − iy ) A T + ( x + iy ) A (cid:1) = det (cid:16)(cid:0) tI + ( x + iy ) A T + ( x − iy ) A (cid:1) T (cid:17) = F A ( t, x, y ) . (cid:3) Chien and Nakazato asked the converse question and provided a positive answer for thecase when d = n = 3 ,
4. The authors of [15, 25] studied rotational symmetry of the numericalrange of matrices of size d = 3 ,
4. We will provide a converse in the case d = qn inTheorem 7.1. The difficulty for arbitrary d comes from the fact that for many values of d ,all forms in ( H ◦ ) Γ d define curves with complex singularities, as discussed in Section 8.4. A constructive proof for smooth curves
In this section we aim to prove Theorem 6.1, but with some added assumptions about ofa curve in H Γ d and an interlacer. Throughout Sections 4 and 5 we will assume that A1. f ∈ ( H ◦ ) Γ d and V C ( f ) is smooth, A2. g ∈ H Γ d − interlaces f with respect to (1 , , A3. V C ( f ) and V C ( g ) intersect transversely, and A4. (cid:12)(cid:12) V C ( f, g, t ) (cid:12)(cid:12) = (cid:40) n is odd, d if n is even.Specifically, we prove the following theorem. Theorem 4.1.
Let d = qn for some q ∈ Z + . Let f and g satisfy (A1)–(A4). (a) If Γ = C n , then there exists a matrix A ∈ C C ( n, d ) so that f = F A . (b) If Γ = D n , then there exists a matrix A ∈ C R ( n, d ) so that f = F A . In order to construct the matrix A and corresponding determinantal representation of f ,we first construct the adjugate of this matrix, which will be a d × d matrix of forms of degree d − ≤ V C ( f ). Following [16, 31, 36], we take g to be the (1 , V C ( f ) and V C ( g ).The next lemma is a general statement about complex points in the intersection of V C ( f )and V C ( g ). This will allow us to split these intersection points into disjoint sets determinedby orbits under the action of rotation. Lemma 4.2.
Let f and g satisfy Assumptions (A1)–(A4). Any point [ t : x : y ] in V C ( f, g ) satisfies | x + iy | (cid:54) = | x − iy | .Proof. For the sake of contradiction, suppose that | x + iy | = | x − iy | . If y (cid:54) = 0, then x + iy = z ( x − iy ) with | z | = 1 and z (cid:54) = 1. Solving for x/y gives xy = − i (1 + z )1 − z · (1 − z )(1 − z ) = − i (1 − z + z − zz ) | − z | = − i ( − z + z ) | − z | = 2 · Im( z ) | − z | ∈ R . FAYE PASLEY SIMON AND CYNTHIA VINZANT
By homogeneity of f , f ( t/y, x/y,
1) = 0, meaning that t/y is a root of the polynomial f ( λ, x/y, ∈ R [ λ ] where x/y ∈ R is fixed. The hyperbolicity of f then implies that t/y ∈ R .Since both x/y and t/y are real, the point [ t : x : y ] belongs to P ( R ). By [36, Proposition 4.3],any real intersection point of V C ( f ) and V C ( g ) is non-transverse, contradicting (A3).Similarly, if y = 0, then x (cid:54) = 0 since f (1 , ,
0) = 1. Then f ( t/x, ,
0) = 0, implying that t/x ∈ R and [ t : x : y ] = [ t : x : 0] belongs to P ( R ), again contradicting (A3). (cid:3) Corollary 4.3.
Let f and g satisfy Assumptions (A1)–(A4). Then each C n -orbit in V C ( f, g ) is disjoint from its image under conjugation.Proof. Let O be a C n -orbit of points in V C ( f, g ) and suppose p = [ t : x : y ] ∈ O ∩ O . Thenrot (cid:96) · p = p for some (cid:96) ∈ [ n ]. If rot (cid:96) · p = [ t : a : b ], then [ t : ω − (cid:96) ( x + iy ) : ω (cid:96) ( x − iy )] = [ t : a + ib : a − ib ]. In particular, if rot (cid:96) · p equals p , then(5) (cid:2) t : ω − (cid:96) ( x + iy ) : ω (cid:96) ( x − iy ) (cid:3) = [ t : x + iy : x − iy ] = [ t : x − iy : x + iy ] . The cross ratio the the last two coordinates gives x − iy · ω (cid:96) ( x − iy ) = x + iy · ω − (cid:96) ( x + iy ).Taking the modulus of both sides shows that | x − iy | = | x + iy | , contradicting Lemma 4.2. (cid:3) Define the linear map(6) ϕ : C [ t, x, y ] → C [ t, x, y ] where h ( t, x, y ) (cid:55)→ h (rot − · ( t, x, y )) . The eigenvectors of this map have the form t l ( x + iy ) j ( x − iy ) k , each with eigenvalue ω j − k . The restriction ϕ | d of ϕ to C [ t, x, y ] d has a finite number of eigenvectors equal todim C ( C [ t, x, y ] d ) = (cid:0) d +22 (cid:1) . For each (cid:96) = 0 , , . . . , n −
1, denote by(7) Λ( ω (cid:96) ) d = (cid:8) f ∈ C [ t, x, y ] d : ϕ ( f ) = ω (cid:96) f (cid:9) the eigenspace of the restriction ϕ | d associated to eigenvalue ω (cid:96) . Notice Λ( ω ) d = C [ t, x, y ] C n d and we can write C [ t, x, y ] d as a decomposition of eigenspaces C [ t, x, y ] d = n − (cid:77) (cid:96) =0 Λ( ω (cid:96) ) d . We will be interested in the dimension of each eigenspace. In particular, for d = qn , we wantat least q elements in each eigenspace Λ( ω (cid:96) ) d − in order to choose linearly independent setof elements in C [ t, x, y ] d − for the first row of the adjugate matrix we wish to construct. Lemma 4.4.
Let d = qn for some q ∈ Z + . The dimension of the eigenspace Λ( ω (cid:96) ) d − is dim C (cid:0) Λ( ω (cid:96) ) d − (cid:1) = dq + q if n is odd dq + q if n is even and (cid:96) is odd dq if n is even and (cid:96) is even . Proof.
The monomials t d − − j − k ( x + iy ) j ( x − iy ) k where j − k ≡ (cid:96) mod n form a basis forthe vectorspace (cid:0) Λ( ω (cid:96) ) d − (cid:1) . Thus the dimension of (cid:0) Λ( ω (cid:96) ) d − (cid:1) is the number points in thesimplex { ( j, k ) ∈ Z ≥ : j + k ≤ d − } with j − k ≡ (cid:96) mod n . Note that the such first pointson the j and k axes will be ( (cid:96),
0) and (0 , n − (cid:96) ).For any 0 ≤ j ≤ d −
1, the number of integer points of the form ( a + j, a ) in this simplexis given by (cid:100) d − j (cid:101) . Similarly the number of integer points of the form ( a, a + k ) is (cid:100) d − k (cid:101) . We NVARIANT HYPERBOLIC PLANE CURVES 9 are interested in these values when j = (cid:96) + an and k = n − (cid:96) + an . That is,dim (cid:0) Λ( ω (cid:96) ) d − (cid:1) = q − (cid:88) a =0 (cid:24) d − ( (cid:96) + an )2 (cid:25) + (cid:24) d − ( n − (cid:96) + an )2 (cid:25) . When n is odd, (cid:96) and n − (cid:96) have different parities, meaning exactly one of d − ( (cid:96) + an )2 and d − ( n − (cid:96) + an )2 will be an integer. When n is even the parities of d − ( (cid:96) + an ) and d − ( n − (cid:96) + an )depend only on the parity of (cid:96) . They are odd if and only if (cid:96) is odd. Let δ = 1 when n isodd, 2 when n is even and (cid:96) is odd, and zero otherwise. Then dim (cid:0) Λ( ω (cid:96) ) d − (cid:1) equals q − (cid:88) a =0 d − ( (cid:96) + an ) + d − ( n − (cid:96) + an ) + δ q − (cid:88) a =0 d − n + δ − an q d − n + δ − n q ( q − , where the last equality is obtained by summing the arithmetic sequence. Recalling that d = qn , we see that this dimension simplifies to q ( qn + δ ) / q ( d + δ ) /
2, as desired. (cid:3)
For f and g that satisfy (A1)–(A4), we will split the points of V C ( f, g ) into S ∪ S basedon orbits under rotation. The next lemma helps enumerate conditions imposed by the setof orbit representatives, and accurately count dimensions later in Lemma 4.7. Lemma 4.5.
Let d = qn for some q ∈ Z + . If n and (cid:96) are even, then each monomial in Λ( ω (cid:96) ) d − has a factor of t .Proof. Let t d − − j − k ( x + iy ) j ( x − iy ) k be an arbitrary monomial in Λ( ω (cid:96) ) d − . Then j − k ≡ (cid:96) mod n . Since n and (cid:96) are even, j − k is even and so is j + k = j − k + 2 k . Moreover d = qn is even and d − d − − ( j + k ) of t is odd and ≥ (cid:3) By Corollary 4.3, V C ( f, g ) may be split into two disjoint sets according to orbits invariantunder the action of C n . More explicitly, write V C ( f, g ) = S ∪ S as the union of two disjointconjugate sets. Define ˜ S to be a minimal set of orbit representatives from S so that(8) S = (cid:110) rot (cid:96) · p | p ∈ ˜ S, (cid:96) ∈ [ n ] (cid:111) .The next proposition gives the maximum number of possible conditions imposed by ˜ S on anelement of Λ( ω (cid:96) ) d − . Proposition 4.6.
Let d = qn for some q ∈ Z + and suppose f and g satisfy (A1)–(A4).Then the number of distinct orbits in S is (cid:12)(cid:12) ˜ S (cid:12)(cid:12) = (cid:40) q ( d − / if n is odd qd/ if n is even.Proof. Since f (1 , , (cid:54) = 0, each point [ t : x : y ] ∈ V C ( f, g ) with t (cid:54) = 0 generates a C n -orbit ofsize n , since rot (cid:96) fixes such a point if and only if (cid:96) ≡ n . When n is odd, all points in V C ( f, g ) have t (cid:54) = 0, so the d ( d − / S split up into d ( d − / n = q ( d − / C n . When n is even, the d ( d − / S with t (cid:54) = 0 split up into q ( d − / V C ( f, g, t ) generates a C n -orbit of size n/ n/ acts asthe identity on points of the form [0 : x : y ]. Thus the d total points of V C ( f, g, t ) contribute( d/ / ( n/
2) = q orbits to S which means S has a total of qd/ C n -orbits. (cid:3) Denote the space of forms in C [ t, x, y ] d − vanishing on points S and ˜ S from (8) by(9) I ( S ) d − and I ( ˜ S ) d − respectively. Now we can show there are enough elements in each eigenspace to choose alinearly independent forms in C [ t, x, y ] d − for the first row in our desired adjugate matrix. Lemma 4.7.
Let d = qn for some q ∈ Z + . There exist q linearly independent polynomialsin each eigenspace Λ( ω (cid:96) ) d − that vanish on the points S . That is, dim C (cid:0) Λ( ω (cid:96) ) d − ∩ I ( S ) d − (cid:1) ≥ q. Proof.
An element of Λ( ω (cid:96) ) d − vanishes on S if and only if it vanishes on ˜ S . Then for any (cid:96) ,dim (cid:0) Λ( ω (cid:96) ) d − ∩ I ( S ) d − (cid:1) = dim (cid:16) Λ( ω (cid:96) ) d − ∩ I (cid:0) ˜ S (cid:1) d − (cid:17) ≥ dim (cid:0) Λ( ω (cid:96) ) d − (cid:1) − (cid:12)(cid:12)(cid:12) ˜ S (cid:12)(cid:12)(cid:12) . In the cases when n is odd or n is even with (cid:96) odd, this count is straightforward due toLemmas 4.4 and 4.6. By Lemma 4.5, when n and (cid:96) are even, every monomial in Λ( ω (cid:96) ) d − has a factor of t . Thus every element of Λ( ω (cid:96) ) d − will already vanish at points with t = 0without adding additional constraints from those in V C ( f, g, t ). In this case we do not takeinto account the q orbits at infinity and using Lemmas 4.4 and 4.6 we havedim (cid:0) Λ( ω (cid:96) ) d − ∩ I ( S ) d − (cid:1) ≥ dim (cid:0) Λ( ω (cid:96) ) d − (cid:1) − (cid:12)(cid:12)(cid:12) ˜ S (cid:12)(cid:12)(cid:12) + q ≥ q. (cid:3) The final piece to the construction is an invariant version of Max Noether’s Theorem ondivisors on smooth plane curves, appearing in [31].
Lemma 4.8 (Lemma 3.7 [31]) . Suppose f, g ∈ Λ( ω ) and h ∈ Λ( ω (cid:96) ) are homogeneous with V C ( f ) smooth where deg( h ) > deg( f ) , deg( g ) and g and h have no irreducible componentsin common with f . If V C ( f, g ) consists of distinct points and V C ( f, g ) ⊆ V C ( f, h ) , then thereexists homogeneous a, b ∈ Λ( ω (cid:96) ) with deg( a ) = deg( h ) − deg( f ) and deg( b ) = deg( h ) − deg( g ) so that h = af + bg . Moreover, if f , g , and h are real, then a and b can be chosen real. The construction below is similar to Construction 4.1 from [31]. For normalization of thecoefficient matrix of t , however, we must be more careful. Now the variable t appears inoff-diagonal entries of the determinantal representation, so we must first block diagonalizethe coefficient matrix of t , then normalize with respect to each block separately in order topreserve the desired matrix structure. Construction 4.9.
Let d = qn for some q ∈ Z + and Γ = C n .Input: Two plane curves f and g satisfying (A1)–(A4).Output: A ∈ C C ( n, d ) with f = F A .(1) Set g = g .(2) Split up the distinct d ( d −
1) points of V C ( f, g ) into two disjoint, conjugate sets S ∪ S of C n -orbits such that rot( S ) = S .(3) Extend g to a linearly independent set { g , g , . . . , g d } ⊂ C [ t, x, y ] d − vanishingon all points of S with g j ∈ Λ( ω − j ) d − for all j ∈ [ n ] and set g j = g j for each j .(4) For 1 < i ≤ j , choose g ij ∈ Λ( ω i − j ) d − so that g g ij − g i g j ∈ (cid:104) f (cid:105) and g ii ∈ R [ t, x, y ].(5) For i < j , set g ji = g ij and define G = ( g ij ) i,j ∈ ( C [ t, x, y ] d − ) d × d .(6) Define M = (1 /f d − ) · adj( G ). NVARIANT HYPERBOLIC PLANE CURVES 11 (7) For (cid:96) ∈ [ d ], write (cid:96) − an + b for some integers a and b with 0 ≤ b ≤ n − P be the permutation matrix that takes (cid:96) = an + b + 1 to bq + a + 1. Define M (cid:48) = P M P T as a matrix with q × q blocks M (cid:48) = ( M (cid:48) kl ) nk,l =1 ( M (cid:48) kl ) ij ∈ Λ( ω k − l ) for k, l ∈ [ n ] and i, j ∈ [ q ] . (8) For each k compute the Cholesky decomposition of each diagonal block M (cid:48) kk andwrite ( M (cid:48) ) − kk = U k U ∗ k for some U k ∈ C q × q .(9) Define U = diag( U , U , . . . , U k ) and output A = (cid:0) P T U ∗ M (cid:48) U P (cid:1) (0 , , i ). Proof of Theorem 4.1.
Our goal is to show each step of Construction 4.9 can be completedand produces a matrix A ∈ C C ( n, d ) such that f = F A as in (2). Let g = g . By Corollary 4.3,we can write V C ( f, g ) as a disjoint union S ∪ S where rot( S ) = S . For Step 3, Lemma 4.4allows us extend g to a linearly independent set { g , g , . . . , g d } where g j ∈ Λ( ω − j ) d − vanishes on S for every j ∈ [ n ]. Now let g j = g j . By Lemma 4.8, we can choose g ij such that g ij ∈ Λ( ω i − j ) d − for 1 < i < j and g g ij − g i g j = af for some homogeneous a ∈ C [ t, x, y ] to complete Step 4. Since f , g , g i g i ∈ R [ t, x, y ], we can choose g ii ∈ R [ t, x, y ]as well. Let g ji = g ij for i < j and define G = ( g ij ) i,j be the d × d complex matrix of formsof degree d −
1. By Theorem 4 . G ) will be divisible by f d − andStep 6 is valid. The entries in G have degree d −
1, so entries of its adjugate have degree( d − . Then f d − has degree d ( d − M are linear in t, x, and y . By [36,Theorem 4 . M (1 , ,
0) is positive definite and det( M ) is a nonzero scalar multiple of f .Let Ω = diag(1 , ω, . . . , ω d − ). Applying the map ϕ to the ( i, j )-th entry of M = ( m ij ) ij gives ϕ ( m ij ) = (1 /f d − ) · adj( ϕ ( G )) ij = (1 /f d − ) · adj(Ω G Ω ∗ ) ij = (1 /f d − ) · (adj(Ω ∗ )adj( G )adj(Ω)) ij = (Ω M Ω ∗ ) ij = ω j − ω i − m ij = ω i − j m ij .Therefore, m ij ∈ Λ( ω i − j ) for each i, j . The restriction of ϕ to C [ t, x, y ] has eigenvalues1 , ω , and ω n − with associated eigenspaces Λ( ω ) , Λ( ω ) , and Λ( ω n − ) . This implies m ij = 0 if i − j (cid:54)≡ , ± n . For m ij such that i − j ≡ n − n , we have m ij ∈ Λ( ω n − ) , showing that m ij is a scalar multiple of x − iy . Similarly, since M isHermitian, this implies m ji ∈ Λ( ω ) must be a scalar multiples of x + iy . If i − j ≡ n then i ≡ j mod n and m ij ∈ Λ( ω ) ∩ R [ t, x, y ] is a multiple of t .Next we will show by permuting rows and columns of M we may get the identity matrixas the coefficient of t in our representation. Consider M as a matrix of n × n blocks. Eachblock is a cyclic weighted shift matrix and there are q blocks in total. For (cid:96) ∈ [ d ], write (cid:96) − an + b for some integers a and b with 0 ≤ b ≤ n −
1. Let P be the permutationmatrix that takes (cid:96) = an + b + 1 to bq + a + 1, as in Remark 3.2. Define M (cid:48) = P M P T as amatrix with q × q blocks M (cid:48) = ( M (cid:48) kl ) nk,l =1 with( M (cid:48) kl ) ij ∈ Λ( ω k − l ) for k, l ∈ [ n ] and i, j ∈ [ q ] . It follows that M (cid:48) (1 , ,
0) is a block diagonal matrix. Moreover, since M (1 , ,
0) is pos-itive definite, so is M (cid:48) (1 , , k ∈ [ n ] we can decompose M (cid:48) kk (1 , ,
0) so that M (cid:48) kk (1 , , − = U ∗ k U k for some U k ∈ C q × q . Define U = diag( U , U , . . . , U n ). Then M (cid:48)(cid:48) = U M (cid:48) U ∗ is a desired representation of f since M (cid:48)(cid:48) (1 , ,
0) = I d , ( M (cid:48)(cid:48) kl ) ij ∈ Λ( ω k − l ) for k, l ∈ [ n ] and i, j ∈ [ q ], and f = (1 /λ ) · det ( U M (cid:48) U ∗ ) for λ = det( U ) · det( U ∗ ). Lastly, applythe inverse permutation so f = (1 /λ ) · det( P T M (cid:48)(cid:48) P ) and evaluating ( P T M (cid:48)(cid:48) P )(0 , , i ) givesa cyclic weighted shift matrix of order n . (cid:3) Example 4.10. [ d = 6, n = 3, q = 2] For n = 3 and d = 6, we see that V C ( f, g ) consistsof d ( d −
1) = 30 points, which split into 10 orbits, each of size 3. These orbits come inconjugate pairs, of which we take half to form the set S , which will have size 15. The set ˜ S of orbit representatives in S has size q ( d − / (cid:96) , Λ( ω (cid:96) ) d − has dimension q ( d + 1) / S imposes a linear condition on forms in Λ( ω (cid:96) ) d − ,we can find 7 − g j , g j + n ) ∈ Λ( ω − j ) that vanish on˜ S . Ranging over j = 0 , , G . The entries of the linearmatrix M = ( m ij ) ij satisfy m ij ∈ Λ( ω i − j ) . In particular, the ( i, j ) entry of M (1 , ,
0) zerowhenever i (cid:54) = j mod 3. Evaluating the matrix M (cid:48) = P M P T at ( t, x, y ) = (1 , ,
0) thereforeresults in a block diagonal matrix of three 2 × f . A detailed example of this construction can befound in [39, Example 3.1.9]. Example 4.11 ( d = 12, n = 4, q = 3) . For n = 4 and d = 12, we see that V C ( f, g ) consistsof d ( d −
1) = 132 points. Of these, 120 have t (cid:54) = 0 and split up into q ( d −
2) = 30 orbits ofsize 4. Since g has a factor of t , there are an additional 12 points with t = 0, splitting upinto 6 orbits, each of size two. Splitting these 132 into conjugate pairs S ∪ S , we see that S has 66 points, consisting of 15 orbits of size 4 and three orbits of size two. The set ˜ S of orbitrepresentatives has size qd/ ω (cid:96) ) d − is qd/ (cid:96) is evenand qd/ q = 21 when (cid:96) is odd. Note that when (cid:96) is even, elements of Λ( ω (cid:96) ) d − have a factorof t and so automatically vanish on those points in ˜ S with t = 0. Each of the 15 remainingpoints imposes a linear condition on Λ( ω (cid:96) ) d − , leaving a three-dimensional subspace of formsin Λ( ω (cid:96) ) d − that vanish on S . Similarly, if (cid:96) is odd, then dim C Λ( ω (cid:96) ) d − − | ˜ S | = 21 −
18 = 3.Therefore for each j = 0 , , ,
3, we can choose linearly independent g j , g j + n ) , g j +2 n ) inΛ( ω − j ) d − that vanish on S . 5. Dihedral Invariance
In this section, we modify Construction 4.9 to include the invariance under reflection andproduce a matrix in C R ( n, d ). We divide the points of V C ( f, g ) based on orbits under rotation,then split according to reflection. Specifically, we require not only that V C ( f, g ) = S ∪ S where rot( S ) = S , but also ref (cid:0) S (cid:1) = S meaning that if p ∈ S , then ref( p ) ∈ S . Corollary 5.1.
Every C n -orbit in V C ( f, g ) is disjoint from its image under reflection when f and g satisfy (A1)–(A4).Proof. Let O be a C n -orbit in V C ( f, g ). Suppose p = [ t : x : y ] ∈ O ∩ ref( O ). Thenref( p ) = rot (cid:96) · p for some (cid:96) ∈ [ n ], giving that(10) [ t : x − iy : x + iy ] = (cid:2) t : ω − (cid:96) ( x + iy ) : ω (cid:96) ( x − iy ) (cid:3) . Then ( x − iy ) · ω (cid:96) ( x − iy ) = ( x + iy ) · ω − (cid:96) ( x + iy ). Taking the modulus of both sides shows that | x − iy | = | x + iy | , which contradicts Lemma 4.2. Therefore O ∩ ref( O ) must be empty. (cid:3) NVARIANT HYPERBOLIC PLANE CURVES 13
Remark 5.2.
Corollaries 4.3 and 5.1 imply that a C n -orbit O ∈ V C ( f, g ) is disjoint fromboth conj( O ) and ref( O ). However, this tells us nothing about the intersection of orbitsconj( O ) and ref( O ). Their intersection may be nonempty, hence D n -orbits in V C ( f, g ) donot always have the same cardinality.When the matrix A has real entries, both the linear matrix with determinant F A and itsadjugate have entries in R [ t, x + iy, x − iy ]. Therefore to reverse engineer this process andproduce a matrix in C R ( n, d ), we amend the construction to use forms in R [ t, x + iy, x − iy ]. Remark 5.3.
Complex conjugation, denoted conj acts on C [ t, x, y ] by conjugating the co-efficients of a polynomial in the basis of monomials t l x j y k . We claim that the invariant ringof the composition ref ◦ conj is given by C [ t, x, y ] (cid:104) ref ◦ conj (cid:105) d = R [ t, x + iy, x − iy ] d . Indeed, anyelement in C [ t, x, y ] is a C -linear combination of forms t l ( x + iy ) j ( x − iy ) k . Then(ref ◦ conj) · (cid:88) l + j + k = d c ljk t l ( x + iy ) j ( x − iy ) k = ref · (cid:88) l + j + k = d c ljk t l ( x − iy ) j ( x + iy ) k = (cid:88) l + j + k = d c ljk t l ( x + iy ) j ( x − iy ) k , meaning that the polynomial is invariant if any only if its coefficients c ljk with respect tothis basis are real. Lemma 5.4. If S ⊂ P ( C ) is fixed under ref ◦ conj , i.e. ref( S ) = S , then the intersection ofthe subspace Λ( ω (cid:96) ) d − in (7) with I ( S ) d − has a basis in R [ t, x + iy, x − iy ] d − .Proof. We will argue that each linear subspace is invariant under ref ◦ conj separately, henceso is their intersection. The subspace Λ( ω (cid:96) ) d − is invariant under ref ◦ conj since in spannedby monomials t d − − j − k ( x + iy ) j ( x − iy ) k , which are invariant. The subspace I ( S ) d − isinvariant under ref ◦ conj because I ( S ) d − = I (ref( S )) d − = (ref ◦ conj)( I ( S ) d − ) . Since both Λ( ω (cid:96) ) d − and I ( S ) d − are invariant under ref ◦ conj, so is their intersection. Ittherefore has a basis in C [ t, x, y ] (cid:104) ref ◦ conj (cid:105) d − = R [ t, x + iy, x − iy ] d − . (cid:3) Construction 5.5.
Let d = qn for some q ∈ Z + and Γ = D n .Input: Two plane curves f and g satisfying (A1)–(A4).Output: a matrix A ∈ C R ( n, d ) such that f = F A .(1) Set g = g .(2) Split up the distinct d ( d −
1) points of V C ( f, g ) into two disjoint, conjugate sets S ∪ S of C n -orbits such that rot( S ) = S and ref( S ) = S .(3) Extend g to a linearly independent set { g , g , . . . , g d } ⊂ R [ t, x + iy, x − iy ] d − vanishing on all points of S with g j ∈ Λ( ω − j ) d − and set g j = g j for all j ∈ [ d ].(4) For 1 < i ≤ j , choose g ij ∈ Λ( ω i − j ) d − ∩ R [ t, x + iy, x − iy ] d − so that g g ij − g i g j belongs to (cid:104) f (cid:105) and g ii ∈ R [ t, x, y ].(5) For i < j , set g ji = g ij and define G = ( g ij ) i,j ∈ ( R [ t, x + iy, x − iy ] d − ) d × d .(6) Define M = (1 /f d − ) · adj( G ).(7) For (cid:96) ∈ [ d ], write (cid:96) − an + b for some integers a and b with 0 ≤ b ≤ n − P be the permutation matrix that takes (cid:96) = an + b + 1 to bq + a + 1. Define M (cid:48) = P M P T as a matrix with q × q blocks M (cid:48) = ( M (cid:48) kl ) nk,l =1 ( M (cid:48) kl ) ij ∈ Λ( ω k − l ) for k, l ∈ [ n ] and i, j ∈ [ q ] . (8) For each k compute the Cholesky decomposition of each diagonal block M (cid:48) kk andwrite ( M (cid:48) ) − kk = U k U Tk for some U k ∈ R q × q .(9) Define U = diag( U , U , . . . , U k ) and output A = (cid:0) P T U T M (cid:48) U P (cid:1) (0 , , i ). Proof of Theorem 4.1(b).
Let g = g . Here we follow Construction 4.9, but split the in-tersection points V C ( f, g ) into S ∪ S so that rot( S ) = S and ref( S ) = S . Indeed, byCorollaries 4.3 and 5.1, for an orbit O of a point in V C ( f, g ), we may put O and ref( O ) in S while taking ref( O ) and O in S . By Lemma 5.4, we can extend g to a linearly independentset { g , . . . , g d } so that g j ∈ Λ( ω − j ) d − ∩ I ( S ) d − ∩ R [ t, x + iy, x − iy ]. Now let g j = g j .The polynomials f , g , g i g j ∈ R [ t, x + iy, x − iy ], so by Lemma 4.8, we are also able to find g ij such that g ij ∈ Λ( ω i − j ) d − ∩ R [ t, x + iy, x − iy ] d for 1 < i < j . Moreover, g ii ∈ R [ t, x, y ]since f, g , g i g i ∈ R [ t, x, y ]. Let g ji = g ij for i < j and define G = ( g ij ) i,j . Notice that G ∈ R [ t, x + iy, x − iy ] d × dd − . We then complete the construction as in the proof of Theorem 4.1.The matrix M = (1 /f d − ) · adj( G ) satisfies M ∈ R [ t, x + iy, x − iy ] d × d so M (1 , , ∈ R d × d .Next we will show by permuting rows and columns of M we may get the identity matrixas the coefficient of t in our representation. Consider M as a matrix of n × n blocks. Eachblock is a cyclic weighted shift matrix and there are q blocks in total. For (cid:96) ∈ [ d ], write (cid:96) − an + b for some integers a and b with 0 ≤ b ≤ n −
1. Let P be the permutationmatrix that takes (cid:96) = an + b + 1 to bq + a + 1. Define M (cid:48) = P M P T as a matrix with q × q blocks M (cid:48) = ( M (cid:48) kl ) nk,l =1 with( M (cid:48) kl ) ij ∈ Λ( ω k − l ) for k, l ∈ [ n ] and i, j ∈ [ q ]and M (cid:48) (1 , ,
0) is a real block diagonal matrix. By Theorem 3.3 of [36], we know M (1 , , M (cid:48) (1 , ,
0) is definite. For each k ∈ [ n ] write M (cid:48) kk (1 , , − = U Tk U k for some U k ∈ R d × d . Define U = diag( U , U , . . . , U n ). Then M (cid:48)(cid:48) = U M (cid:48) U T is a representation of f since M (cid:48)(cid:48) (1 , ,
0) = I d and f = (1 /λ ) · det ( M (cid:48)(cid:48) ) for λ = det( U ) · det( U T ). Lastly, apply theinverse permutation so f = (1 /λ ) · det( P T M (cid:48)(cid:48) P ). Evaluating ( P T M (cid:48)(cid:48) P )(0 , , i ) gives a cyclicweighted shift matrix of order n and it is real because P T M (cid:48)(cid:48) P ∈ R [ t, x + iy, x − iy ] d × d . (cid:3) Example 5.6 ( d = 6, n = 3, q = 2) . For n = 3 and d = 6, we see that V C ( f, g ) consistsof d ( d −
1) = 30 points, which split into 10 orbits, each of size 3. Each orbit O is eitherfixed by ref ◦ conj, in which case ref( O ) = O , or not, in which case ref( O ) (cid:54) = O . Indeed,since there are 10 orbits total, we see that there must be at least one orbit with ref( O ) = O .Regardless, we can split up the 10 orbits under C n into two conjugate sets of five. The unionof each collection is a set S of size 15 satisfying ref( S ) = S . The counts and constructionsthen continue as in Example 4.10, where the assumption ref( S ) = S lets us take the forms g j in R [ t, x + iy, x − iy ]. See [39, Example 3.2.6] for a detailed example of this construction.6. The Degenerate Case
Theorem 6.1.
Let d = qn for some q ∈ Z + and suppose f ∈ H Γ d . (a) If Γ = C n , then there exists A ∈ C C ( n, d ) so that f = F A . (b) If Γ = D n , then there exists A ∈ C R ( n, d ) so that f = F A . Here we deal with assumptions (A1)–(A4) posed in Section 4. To start, we show that thealgebraic assumptions hold generically.
Proposition 6.2.
For d = qn and Γ = C n or D n , a generic invariant form f ∈ C [ t, x, y ] Γ d defines a smooth plane curve V C ( f ) ⊂ P ( C ) . NVARIANT HYPERBOLIC PLANE CURVES 15
Proof.
Consider the subvariety X of P ( C [ t, x, y ] Γ d ) × P ( C ) given by X = (cid:8) ( f, p ) ∈ P ( C [ t, x, y ] Γ d ) × P ( C ) : ∇ f ( p ) = (0 , , (cid:9) By the Projective Elimination Theorem (e.g. [26, Theorem 10.6]), its image under theprojection π ( f, p ) = f is a subvariety of P ( C [ t, x, y ] Γ d ). Therefore the image is either thewhole space, meaning that either every polynomial in C [ t, x, y ] Γ d defines a singular curve, orbelongs to a proper subvariety of C [ t, x, y ] Γ d , meaning that a generic polynomial in C [ t, x, y ] Γ d defines a smooth curve. To finish the proof, we note that t d + ( x + iy ) d + ( x − iy ) d belongsto C [ t, x, y ] Γ d and defines a smooth plane curve. (cid:3) Proposition 6.3.
For d = qn and Γ = C n or D n and any e ∈ Z + , the plane curvesdefined by generic invariant forms f, g ∈ C [ t, x, y ] Γ with deg( f ) = d and deg( g ) = e intersecttransversely.Proof. First, we argue that it suffices to produce one example of a pair of forms f, g in C [ t, x, y ] Γ with deg( f ) = d , deg( g ) = e whose plane curves intersect transversely. This isbecause the intersecting transversely is a Zariski-open condition on f, g . More precisely,consider the subvariety Y ⊂ P ( C [ t, x, y ] Γ d ) × P ( C [ t, x, y ] Γ e ) × P ( C ) defined by Y = (cid:26) ( f, g, p ) : f ( p ) = 0 , g ( p ) = 0 , rank (cid:18) ∇ f ( p ) ∇ g ( p ) (cid:19) ≤ (cid:27) . Again, by the Projective Elimination Theorem [26, Theorem 10.6], the image of Y under theprojection π ( f, g, p ) = ( f, g ) is a Zariski-closed set. By construction, it is the set of pairs( f, g ) for which the intersection V C ( f ) ∩ V C ( g ) is non-transverse. We need to show that thisdoes not occur for all pairs.First we consider the special case d = n and e = 1 ,
2. Note that since f is invariant underthe action of C n , it has the form f ( t, x, y ) = a ( x + iy ) n + b ( x − iy ) n + (cid:98) n/ (cid:99) (cid:88) j =0 c j t n − j ( x + y ) j where a, b, c , . . . , c (cid:98) n/ (cid:99) ∈ C and a = b if Γ = D n . Note that when a, b are non-zero, theintersection of V C ( f ) with V C ( t ) is transverse for non-zero a, b . Also if a, b are nonzero, then V C ( f ) and V C ( x + y ) have no common points with t = 0. Then by Bertini’s theorem, forgeneric λ, µ ∈ C , the intersection of V C ( f ) and V C ( λ ( x + y ) + µt ) is transverse [2].Now we construct the desired pair f, g ∈ C [ t, x, y ] Γ with deg( f ) = d and deg( g ) = e . Let f be the product of q generic forms in C [ t, x, y ] Γ n of degree n , and let g be the product of (cid:98) e (cid:99) generic quadratic forms C [ t, x, y ] Γ2 and t δ where δ = 2( e − (cid:98) e (cid:99) ). Then by the argumentabove, V C ( f ) and V C ( g ) intersect transversely. (cid:3) Proposition 6.4.
Let d = qn . For Γ = C n or D n and generic invariant forms f, g in C [ t, x, y ] Γ with deg( f ) = d and deg( g ) = d − , the number of intersection points on the line t = 0 is given by |V C ( f, g, t ) | = (cid:40) if n is odd, d if n is even.Proof. We first prove something slightly different. Let e ∈ Z + be an integer satisfying e ∈ N + n N where q · e is even. We claim that generic invariant forms f, g ∈ C [ t, x, y ] Γ withdeg( f ) = d and deg( g ) = e satisfy V C ( f, g, t ) = ∅ . By the Projective Elimination Theorem [26, Theorem 10.6], the set of ( f, g ) ∈ C [ t, x, y ] Γ d × C [ t, x, y ] Γ e for which V C ( f, g, t ) is non-empty is Zariski closed. Therefore it suffices to showthat it is not the whole space.Let ( a, b ) ∈ N so that 2 a + nb = e . Note that if e is even, then we may take b to be even.To see this, note that e = 2 a + nb implies that at least one of b and n is even. If n = 2 k iseven and b is odd, then b ≥ a, b ) with ( a + k, b − m ∈ Z + , let χ ( m ) be 0 if m is even and 1 if m is odd. Then considerpolynomials f = ( u n + v n ) χ ( q ) (cid:98) q/ (cid:99) (cid:89) j =1 ( u n + r j v n )( r j u n + v n ) and g = ( uv ) a ( u n + v n ) χ ( b ) (cid:98) b/ (cid:99) (cid:89) k =1 ( u n + s k v n )( s k u n + v n ) . where r , . . . , r (cid:98) q/ (cid:99) , s , . . . , s (cid:98) b/ (cid:99) ∈ C \{ , } are all distinct and u = x + iy , v = x − iy . Weclaim that both f, g are invariant under the dihedral group and have no common roots with t = 0, so long as χ ( q ) · χ ( b ) = 0. Let ω = e πi/n . For invariance, note that both f , g areinvariant under the change of coordinates ( t, u, v ) (cid:55)→ ( t, ωu, ωv ), which is the action of rotin coordinates ( t, u, v ), as well as the map ( t, u, v ) (cid:55)→ ( t, v, u ), which is the action of ref.The zeros of f with t = 0 consist of the points [ t : u : v ] = [0 : 1 : λω ] where ω isan n th root of unity and λ = 1 , r k , /r k for k = 1 , . . . , (cid:98) q/ (cid:99) . Moreover there is only sucha root with λ = 1 if q is odd. Similarly, the zeros of g with t = 0 consist of the points[ t : u : v ] = [0 : 1 : 0] , [0 : 0 : 1] if a ≥ t : u : v ] = [0 : 1 : λω ] where λ = 1 , s k , /s k for k = 1 , . . . , (cid:98) b/ (cid:99) , where λ = 1 gives a root only if b is odd. Therefore so long as at least oneof q or b is even, V C ( f, g, t ) is empty.Now suppose that e = d − qn − n is odd. Then qn has the same parity as q ,which is different than the parity of e . Furthermore, e = ( n −
1) + ( q − n . Since n − N + n N . The argument from above then shows that V C ( f, g, t ) = ∅ .If n is even, then so is d , meaning that d − g ∈ C [ t, x, y ] Γ of degree d − t , meaning that it can be written as g = t · h where h ∈ C [ t, x, y ] Γ d − . Taking e = d − V C ( f, h, t ) = ∅ . Therefore V C ( f, g, t ) = V C ( f, t ). Since f has degree d , this consists of d points generically, as isachieved by the explicit example f above. (cid:3) Having dealt with the algebraic conditions of non-singularity, now we address the semi-algebraic conditions of hyperbolicity and interlacing.
Theorem 6.5.
For d = qn and Γ = C n or D n , every polynomial in H Γ d is a limit ofpolynomials f ∈ ( H ◦ ) Γ d for which there exists g ∈ H Γ d − such that (i) V C ( f ) is smooth, (ii) g interlaces f , (iii) V C ( f ) ∩ V C ( g ) is transverse, and (iv) |V C ( f, g, t ) | = (cid:40) if n is odd ,d if n is even.Proof. For any strictly hyperbolic f ∈ R [ t, x, y ] d the set of polynomials g ∈ R [ t, x, y ] d − thatinterlace f with respect to (1 , ,
0) is a full-dimensional convex cone, whose interior consistsof those g which strictly interlace f . See e.g. [30, Section 6]. Then by Theorem 2.2, the set I = (cid:8) ( f, g ) ∈ ( H ◦ d ) Γ × R [ t, x, y ] Γ d − : g strictly interlaces f with respect to (1 , , (cid:9) NVARIANT HYPERBOLIC PLANE CURVES 17 is an open, full-dimensional subset of the affine subspace { ( f, g ) : coeff( f, t d ) = 1 } in R [ t, x, y ] Γ d × R [ t, x, y ] Γ d − . Moreover, the image of I under the projection π ( f, g ) = f isall of ( H ◦ d ) Γ . By Propositions 6.2, 6.3, and 6.4, U = (cid:8) ( f, g ) ∈ R [ t, x, y ] Γ d × R [ t, x, y ] Γ d − : conditions (i),(iii), (iv) are satisfied (cid:9) is open and dense in the Euclidean topology on R [ t, x, y ] Γ d × R [ t, x, y ] Γ d − . Furthermore, wenote that U is invariant under diagonal scaling ( f, g ) (cid:55)→ ( λf, g ) where λ ∈ R ∗ . An element( f, g ) can be rescaled to have coeff( f, t d ) = 1 if and only if the coefficient coeff( f, t d ) isnonzero, showing that U is also dense in the subspace given by coeff( f, t d ) = 1. It followsthat I ∩U is dense in I . Since the projection π ( I ) equals ( H ◦ d ) Γ , this gives that the projectionof I ∩ U is dense in ( H ◦ d ) Γ . Then, by Theorem 2.2, we see that π ( I ∩ U ) = π ( I ) = ( H ◦ d ) Γ = H Γ d . Therefore every polynomial in H Γ d belongs to the closure of the set of polynomials f forwhich there exists g ∈ R [ t, x, y ] Γ d − with ( f, g ) ∈ I ∩ U . (cid:3) Proof of Theorem 6.1.
Let f ∈ H Γ d . By Theorem 6.5, f is the limit of some sequence ( f ε ) ε in ( H ◦ ) Γ d satisfying (A1)–(A4). By Theorem 4.1, for each ε , there exists some matrix A ε in C F ( n, d ) such that f ε = F A ε , where F = C for Γ = C n and F = R for Γ = D n . Now f ε ( t, − , f ε ( t, , − i ) are the characteristic polynomials of (cid:60) ( A ε ) = ( A ε + A ∗ ε ) / (cid:61) ( A ε ) =( A ε − A ∗ ε ) / i . These must converge to the roots of f ( t, − ,
0) or f ( t, , − i ) respectively.Therefore, the eigenvalues of (cid:60) ( A ε ) and (cid:61) ( A ε ) are bounded, which bounds the sequences( (cid:60) ( A ε )) ε and ( (cid:61) ( A ε )) ε . Then( (cid:60) ( A ε )) ε + i ( (cid:61) ( A ε )) ε = ( (cid:60) ( A ε ) + i (cid:61) ( A ε )) ε = ( A ε ) ε which is also bounded. Passing to a convergent subsequence gives that lim ε → ( A ε ) ε = A and f = det (cid:18) lim ε → (cid:18) tI d + x + iy A ∗ ε + x − iy A ε (cid:19)(cid:19) = det (cid:18) tI d + x + iy A ∗ + x − iy A (cid:19) = F A . (cid:3) Results on the Classical and k -Higher Rank Numerical Range The authors of [12, 13, 42] were particularly interested in the relationship between thenumerical range and the curve dual to its boundary generating curve. Using this relationshipand Theorem 6.1, we characterize matrices whose numerical range is invariant under rotation.In this section, we describe the interaction between invariance of the numerical range,its boundary generating curve, and the dual variety. We also discuss applications to ageneralization of the numerical range. In the special case d = n , these results appear in [31].Invariance of F A under rotation implies the invariance of W ( A ) under multiplication by ω , as discussed in Proposition 3.4. However, the converse does not hold. That is, there areexamples for which W ( A ) is invariant under multiplication by ω , but F A is not invariantunder the action of C n . See Example 7.2. As discussed below, the invariance of W ( A ) onlyimplies that the invariance of some factor of F A , namely the product of irreducible factorswhose dual varieties contribute to the boundary of W ( A ). The invariance of the boundaryof W ( A ) still gives us information about the dual curve V C ( F A ). - - - - - - Figure 3.
The hypersurface V R ( F B ) in the plane t = 1 for B ∈ C × fromExample 7.2 (right) and W ( B ) (left). Although the plane curve and its dualare not invariant under rotation, the numerical range W ( B ) is. Theorem 7.1.
Let B ∈ C d × d . If W ( B ) is invariant under multiplication by ω = e πin , thenthere exists A ∈ C C ( n, n (cid:100) d/n (cid:101) ) such that W ( B ) = W ( A ) . If in addition, W ( B ) is invariantunder conjugation, then A can be taken to have real entries (i.e. A ∈ C R ( n, n (cid:100) d/n (cid:101) ) .Proof. Kippenhahn’s Theorem [29, Theorem 10] states that W ( B ) equals the convex hull of { x + iy : [1 : x : y ] ∈ V R ( F B ) ∗ } . See also [10, 35]. Recall that every compact, convex set isthe convex hull of its extreme points. Let E denote the extreme points of W ( B ) and let Y denote the Zarski-closure of { [1 : a : b ] : a + ib ∈ E } in P ( R ). By extremality of E , we seethat { [1 : a : b ] : a + ib ∈ E } is contained in V R ( F B ) ∗ and so Y ⊆ V R ( F B ) ∗ . In particular, Y is an algebraic variety of dimension ≤ Y contains no lines, since the intersection of E with any line consists of at mosttwo points. Therefore all irreducible components of the dual variety Y ∗ have dimension one.Since Y ⊆ V R ( F B ) ∗ , we see that Y ∗ ⊆ V R ( F B ).Let f denote the minimal polynomial in R [ t, x, y ] e vanishing on Y ∗ . Note that since f must be a factor of F B , e ≤ d , f is hyperbolic with respect to (1 , , f (1 , , (cid:54) = 0, sowe can take coeff( f, t e ) = 1.Since W ( B ) is invariant under multiplication by ω , so is E . It follows that Y and hence Y ∗ are invariant under the action of C n . Therefore f ∈ R [ t, x, y ] C n e . Similarly, if in addition W ( B ) is invariant under conjugation, then so is E . The curves Y and Y ∗ are then invariantunder D n and so f ∈ R [ t, x, y ] D n e .Let δ = n (cid:100) d/n (cid:101) − e ≥ t δ f ∈ R [ t, x, y ] C n n (cid:100) d/n (cid:101) . By Theorem 6.1, there exists A ∈ C C ( n, n (cid:100) d/n (cid:101) ) such that F A = f . Then V R ( F A ) ∗ = V R ( f ) ∗ and so W ( A ) = W ( B ).Moreover, if f is invariant under D n , then we can take A to be real. (cid:3) Example 7.2.
Take B = − − i − − i
00 0 0 0 0 − − i −
12 0 0 0 0 00 0 0 0 0 − − i − − i . Even though B (cid:54)∈ C C (3 , W ( B ) is invariant under rotation by the angle 2 π/
3. Both V R ( F B ) and W ( B ) are shown in Figure 3. For brevity we use u , v to denote the linear forms u = x + iy NVARIANT HYPERBOLIC PLANE CURVES 19 and v = x − iy . Then F B factors as f f where f = (1 / (cid:0) t − t uv + 1050 t ( u + v ) + 425 it ( u − v ) + 3860( uv ) (cid:1) ,f = (1 / (cid:0) t + 12 u − uv + 12 v (cid:1) , and V C ( f ) ∗ contains the boundary of W ( B ). Notice that F B is not invariant under rotationby 2 π/
3, but the quartic factor f is. By Theorem 6.1, we can find a matrix A ∈ C C (3 , t f = F A and W ( A ) = W ( B ). One such matrix is given by(11) A = − i −
10 + 5 i − i − i − i . Theorem 7.1 shows that any invariant numerical range is the numerical range of a blockcyclic weighted shift matrix, of possibly larger size. One possible strengthening of this is torestrict the the size of this structured matrix.
Conjecture 7.3. If B ∈ C d × d and W ( B ) is invariant multiplication by e πi/n , then thereexists A ∈ C C ( n, d ) with W ( A ) = W ( B ) . Moreover, if W ( B ) is also invariant under conju-gation, the entries of A can be taken to be real. Theorem 6.1 also has implications for the following generalization of the numerical range.
Definition 7.4.
For k ∈ [ d ] the k -higher rank numerical range of A ∈ C d × d is W k ( A ) := (cid:40) k (cid:88) j =1 z ∗ j Az j ∈ C | { z , z , . . . , z k } is orthonormal in C d (cid:41) . This set is compact and invariant under unitary transformation. Building off of the workof Choi et al. [14], Woerdeman [44] showed that W k ( A ) is convex. The classical numericalrange is defined by k = 1. Like before, there is a relationship between the geometry of W k ( A )and the hyperbolic plane curve F A . Chien and Nakazato [11] describe how to compute W k ( A )using F A and the boundary generating curve. They also give conditions for which the k -higher rank numerical range is not uniquely determined by the numerical range when k > Remark 7.5. If A is a complex matrix for which F A is irreducible in C [ t, x, y ], then F A is uniquely determined by its numerical range. That is, if A and B are complex matricesfor which the polynomials F A and F B are irreducible, then W ( A ) = W ( B ) if and only if F A = F B . By the results of Gau and Wu [21], it follows that W ( A ) = W ( B ) if and only if W k ( A ) = W k ( B ) for all 1 ≤ k ≤ (cid:98) d/ (cid:99) + 1. To see this, note that the W ( A ) is uniquelydetermined by its extreme points E . As in the proof of Theorem 7.1, if Y is the Zariskiclosure of points { [1 : a : b ] : a + ib ∈ E } , then the minimal polynomial f vanishing onthe dual variety Y ∗ is a factor of F A . If F A is irreducible, then this gives f = F A , which isuniquely determined by E and thus W ( A ).Examples from [9, 15, 25] show there exist matrices A for which F A ∈ C [ t, x, y ] C n d , but A is not unitarily equivalent to any cyclic weighted shift matrix (with positive weights).Theorem 6.1 proves there must exist some matrix in C C ( n, n (cid:100) d/n (cid:101) ) with the same k -higherrank numerical range of A ∈ C d × d , even if the two matrices are not unitarily equivalent. Corollary 7.6. If F B ∈ C [ t, x, y ] C n d for some B ∈ C d × d , then there exists A ∈ C C ( n, n (cid:100) d/n (cid:101) ) with W k ( A ) = W k ( B ) for all ≤ k ≤ (cid:98) d/ (cid:99) + 1 .Proof. Let m = n (cid:100) d/n (cid:101) − d . By Theorem 6.1, there exists cyclic weighted shift matrix A ∈ C C ( n, n (cid:100) d/n (cid:101) ) so that t m F B = F A . Then [21, Theorem 1] implies that W k ( A ) = W k ( B )for all 1 ≤ k ≤ (cid:98) d/ (cid:99) + 1. (cid:3) An extension of Theorem 6.1 to arbitrary d would yield the following. Conjecture 7.7. If F B ∈ C [ t, x, y ] C n d for some B ∈ C d × d , then there exists A ∈ C C ( n, d ) with W k ( A ) = W k ( B ) for all ≤ k ≤ (cid:98) d/ (cid:99) + 1 . Open Questions and Further Directions
Generalizing to Any Degree.
One could hope to generalize Construction 4.9 for ahyperbolic plane curve of any degree. The main obstruction here is with assumption (A1),specifically the requirement that V C ( f ) is smooth. For curves with d mod n ≥
3, it seemsthere are always multiple singularities at infinity meaning most of these curves do not satisfy(A1). More specifically, there are complex singularities at the points [ t : x : y ] = [0 : 1 : ± i ].We conjecture they each have multiplicity (cid:0) d mod n (cid:1) .To see this recall that monomials in t , x + iy and x − iy form a basis for C [ t, x, y ] C n d , namely(12) C [ t, x, y ] C n d = span C (cid:8) t d − j − k ( x + iy ) j ( x − iy ) k : j ≡ k mod n, and j + k ≤ d (cid:9) . The exponent vectors ( j, k ) are pictured in Figure 4.
Figure 4.
The set of ( j, k ) for which t d − j − k ( x + iy ) j ( x − iy ) k ∈ C [ t, x, y ] C n d for ( n, d ) = (4 , , ,
9) (left to right).
Example 8.1 ( d = 7 , n = 4) . Consider f ∈ C [ t, x, y ] C . Then f is a sum of terms of theform t − j − k ( x + iy ) j ( x − iy ) k where j ≤ k ≤
5, shown on the left in Figure 4. Thisconfirms that [0 : 1 : ± i ] are singular points of V C ( f ).Another way to try to construct a determinantal representation is to use Theorem 6.1 andhope to further specialize its structure. Question 8.2.
Let d = qn + m for some q > and m ∈ [ n − and suppose f ∈ H Γ d . Canwe always write t n − m f = F A for some matrix A ∈ C C ( n, ( q + 1) n ) of the form A = (cid:18) A (cid:48)
00 0 (cid:19) ,where A (cid:48) ∈ C C ( n, d ) ? Example 8.3.
Take f = f from Example 7.2. This is quartic and invariant under actionof the group C . Then t f = F A for the matrix A ∈ C C (3 ,
6) shown in equation (11). Sincethe last two rows and columns of A are zero, f has a determinantal representation f = F A (cid:48) where A (cid:48) is the leading 4 × A . NVARIANT HYPERBOLIC PLANE CURVES 21
Figure 1.
Nodal septic surface with octahedral symmetry bounding a spectrahedron
Figure 2.
Cayley cubic overlaid with ⇥ minors of a ⇥ matrix Figure 5.
A hyperbolic surface in P ( R ) invariant under the octahedral groupwith an invariant definite determinantal representation given in Example 8.5.8.2. Higher Dimensions.
One can also consider invariant hyperbolic polynomials and de-terminantal representations in more than three variables. Suppose that Γ ⊂ GL( R n ) isa finite group that fixes a point e ∈ R n and let H Γ d denote the set of polynomials in f ∈ R [ x , . . . , x n ] d invariant under Γ, hyperbolic with respect to e , and with f ( e ) = 1,as in Section 2. Question 8.4.
Is the analogue of Theorem 2.2 true in higher dimensions? That is, are both H Γ d and its interior contractible? As shown in Theorem 6.1, hyperbolic polynomials invariant under C n and D n have deter-minantal representations that certify their invariance. More generally, let ρ : Γ → GL( C d )be a representation of the group Γ. This defines an action of Γ on the set of d × d Her-mitian matrices by conjugation, γ · A = ρ ( γ ) Aρ ( γ ) ∗ . We say that a d × d linear matrix A ( x ) = (cid:80) i x i A i is invariant with respect to Γ and ρ if for every γ ∈ Γ,(13) A ( γ · x ) = ρ ( γ ) A ( x ) ρ ( γ ) ∗ . The determinant f = det( A ( x )) is then invariant under the action of Γ. Indeed, since Γ isfinite, the determinant of ρ ( γ ) is a root of unity and so the determinants of ρ ( γ ) and ρ ( γ ) ∗ multiply to 1. This shows that the determinants of A ( γ · x ) and A ( x ) are equal for all γ ∈ Γ. Example 8.5.
The elementary symmetric function e n − ( x , . . . , x n ) = (cid:80) nk =1 (cid:81) j (cid:54) = k x j is hy-perbolic with respect to the vector e = (1 , . . . ,
1) and invariant under the natural action ofthe symmetric group S n . Sanyal [38] shows that the form e n − ( x ) has a determinantal repre-sentation A ( x ) = diag( x , . . . , x n − ) + x n J , where J is the all-ones matrix of size n −
1. Thisrepresentation is invariant with respect to S n and the representation ρ : S n → GL( C n − )obtained by restricting S n to the hyperplane of points with coordinate sum one. Specifically,we take the representation ρ ( π ) = ( v π (1) , . . . , v π ( n − ) where for j = 1 , . . . , n , v j is the j th unitcoordinate vector in R n − and v n is the constant vector −
1. Specializing e ( x , . . . , x ) andits determinantal representation to the eight linear forms x j = t ± x ± y ± z gives a surfacein P ( R ) that is hyperbolic and invariant under the octahedral group, shown in Figure 5.For n >
3, most forms in R [ x , . . . , x n ] d do not have d × d determinantal representations,so a verbatim generalization of Theorem 6.1 is false. However, there are two other naturalways of generalizing to more variables. One is to restrict to polynomials that are alreadydeterminantal: Question 8.6.
For every finite group Γ ⊂ GL( R n ) , is there a representation ρ : Γ → GL( C d ) so that every determinantal hyperbolic polynomial f ∈ H Γ d has a definite determinantal rep-resentation f = det( A ( x )) that is invariant with respect to Γ and ρ , as in (13) ? A more ambitious goal would be to show that every invariant hyperbolic polynomial hasdeterminantal representation certifying its hyperbolicity and invariance. For this, we usethe terminology of hyperbolicity cones . If a polynomial f ∈ R [ x , . . . , x n ] is hyperbolic withrespect to e ∈ R n , its hyperbolicity cone, C ( f, e ) is defined to be the connected componentof R n \V R ( f ) containing e . The Generalized Lax Conjecture states that for every hyperbolicpolynomial f , there is some multiple f · g with a definite determinantal representation sothat the hyperbolicity cones of f and f · g agree. This is still open. The discussion abovesuggests the following invariant version: Question 8.7 (Invariant Generalized Lax Conjecture) . Is every invariant hyperbolic poly-nomial a factor of an invariant determinant? That is, for f ∈ H Γ d , does there exist e ∈ N , g ∈ H Γ e , and a representation ρ : Γ → GL( C d + e ) so that the product f · g has an invariant,definite determinantal representation f · g = det( A ( x )) with C ( f, e ) = C ( f · g, e ) ? References [1] O. Axelsson, H. Lu, and B. Polman. On the numerical radius of matrices and its application to iterativesolution methods.
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