Inverse Numerical Range and Determinantal Quartic Curves

: A hyperbolic ternary form, according to the Helton–Vinnikov theorem, admits a determinantal representation of a linear symmetric matrix pencil. A kernel vector function of the linear symmetric matrix pencil is a solution to the inverse numerical range problem of a matrix. We show that the kernel vector function associated to an irreducible hyperbolic elliptic curve is related to the elliptic group structure of the theta functions used in the Helton–Vinnikov theorem.


Introduction
Let A be an n × n complex matrix. Toeplitz [1] introduced the numerical range of A as the set W(A) = {ξ * Aξ : ξ ∈ C n , ξ * ξ = 1}, which contains all eigenvalues of A. The inverse numerical range aims to determine a vector ξ ∈ C n satisfying ξ * Aξ = t 0 ξ * ξ for a moving boundary point t 0 of W(A). The inverse numerical range problem has been discussed by many authors (see [2][3][4]). Our approach to this inverse problem is based on the algebraic curve theory and the determinantal representation of a hyperbolic ternary form.
Kippenhahn [5] proved that the numerical range W(A) is the convex hull of the real affine part of the dual curve of F A (t, x, y) = 0. Conversely, Lax [6] conjectured that every hyperbolic ternary form F(t, x, y) of degree n admits a determinantal representation by a linear matrix pencil M(t, x, y) = tI n + xC + yB of real symmetric matrices C and B, i.e., F(t, x, y) = det(tI n + xC + yB). (1) Fiedler [7] proved that the Lax conjecture is true if V C (F) is a rational curve, and raised a similar conjecture in a relaxed form where B and C are Hermitian matrices. Plaumann and Vinzant [8] provided a method to construct a linear matrix pencil with Hermitian matrices B, C using the interlacer ∂ ∂t F(t, x, y) of the hyperbolic form F(t, x, y). Recently, Helton and Vinnikov [9] confirmed that the Lax conjecture is true by algebraic curve theory for the construction of real symmetric matrices B and C using Riemann-Jacobi theta functions with characteristics (see also [10,11]).
It is well known that any boundary point t 0 of W(A) corresponds to an extreme eigenvalue of (e −iθ A) for some angle θ. A unital eigenvector ξ θ of (e −iθ A) corresponding to the maximal eigenvalue λ max ( (e −iθ A)) assures ξ * θ Aξ θ ∈ ∂W(A), and From this view point, the inverse numerical range problem can be renamed in a more general setting: For any nonzero point (t, x, y) ∈ C 3 on the curve F A (t, x, y) = 0, to find a kernel vector function The kernel vector function method is used in [12] to deal with the inverse numerical range of a 3 × 3 matrix for which its algebraic curve is a cubic elliptic curve.
In this paper, we continue our work on the inverse numerical range problem using the kernel vector function method in the case that V C (F A ) is a quartic elliptic curve. We show that the intersection points of the kernel vector functions and the algebraic curve V C (F A ) induce an abelian group structure on the Abel-Jacobi variety, and the kernel vector functions can be expressed in terms of the theta functions used in the Helton-Vinnikov theorem.

Quartic Elliptic Curves
Let F(t, x, y) be an irreducible hyperbolic form of degree n. A point (t 0 , x 0 , y 0 ) is called a singular point of the curve V C (F) if (For reference on algebraic curve theory, see, for instance, [13]. ) We recall some previous results on the determinantal representation of elliptic curves in [11]. Assume that the curve V C (F) is elliptic, and the n real intersection points of the curve F(t, x, y) = 0 and the line x = 0 are distinct non-singular points Q 1 , . . . , Q n with coordinates Q j = (b j , 0, −1), where b j = 0. Then there is a real birational transformation which transforms V C (F) to a non-singular cubic curve Y 2 Z = 4X 3 − g 2 X 2 Z − g 3 Z 3 for some real constants g 2 , g 3 with g 3 2 − 27g 2 3 > 0. The real affine part F(1, x, y) = 0 of the curve V C (F) is then parametrized by rational functions of Weierstrass function and its derivative over the torus. The Abel-Jacobi map φ : V C (F) → Jac(X) is the inverse of the parametrization s → (1, x, y). Denote Q j = φ(Q j ).
The main result of [11] reformulates the Helton-Vinnikov determinantal representation as follows.
Theorem 1 ([11], Theorem 2.4). Let F(t, x, y) be an irreducible hyperbolic ternary form of degree n. Assume that the curve V C (F) intersects the line x = 0 at n distinct nonsingular points Q j = (b j , 0, −1) with b j = 0, j = 1, 2, . . . , n, and assume that V C (F) is an elliptic curve which is parametrized as where ω 1 > 0 is a half-period of the Weierstrass function ℘(u).
We raise the following conjecture that the kernel vector functions play a role for inverse numerical range and formulate the determinantal representation as well.
Let Conjecture 1. A be an n × n matrix. Assume that the skew Hermitian matrix (A) is diagonal with distinct diagonals b 1 , b 2 , . . . , b n . We also assume that the ternary form F A (t, x, y) is irreducible and its algebraic curve V C (F A ) is elliptic. Then there exist n points R 1 , R 2 , . . . , R n in the normalized Abel-Jacobi variety C/(Z + τZ) of the elliptic curve for which the reduced kernel vector function (ξ 1 ,ξ 2 , . . . ,ξ n ) T is expressed asξ for some constants C k = 0, k = 1, 2, . . . , n, on the Abel-Jacobi variety. With respect to the abelian group structure of this variety, one has If the linear matrix pencil M(t, x, y) is unitarily equivalent to the matrix pencil realized as the θ δ -Helton Vinikov representation for δ = 2 or δ = 3, then (R 1 − Q 1 ) + (R 1 − Q 1 ) = 0. Furthermore, the point R 1 − Q 1 coincides with the point 1/2 (resp. (1 + τ)/2) of the normalized Abel-Jacobi variety when δ = 2 (resp. δ = 3).
The result of a previous paper [12] shows that the conjecture is true for cubic elliptic curves. In this section, we confirm that the conjecture is also true for some quartic elliptic curves.
In the paper [14], a 4 × 4 nilpotent matrix is studied which produces one type of numerical ranges for the classification of the numerical ranges of 4 × 4 matrices. Fladt [15] formulated this quartic curve F A (1, x, y) = 0 as one of the Kepler's models of planetary orbits. We consider a more general similar form of the matrix (2a): where a > 0 and a 12 , a 13 , a 14 are real numbers.
The corresponding curve has just two singular points (ordinary double points) lying on the line y = x. According to the genus formula [13] of an algebraic curve: it implies that the quartic curve V C (F 2 ) is an elliptic curve.
In the following, we show that the quartic form F 2 (t, x, y) admits two non-unitarily equivalent determinantal representations. Theorem 3. Let F 2 (t, x, y) be the quartic form in (2c). Then, there exist two real symmetric matrices H 1 and H 2 , such that H 1 + iB and H 2 + iB are not unitarily equivalent, and F 2 (t, x, y) = det(tI 4 where the polynomials C j = C j (h 12 , h 13 , h 14 , h 34 ) are given by the following: To prove F 2 (t, x, y) = det(tI 4 + x H j + y B), it suffices to show that the two real quartets (h 12 , h 13 , h 14 , h 34 ) satisfy the simultaneous equations C 1 = C 2 = C 3 = C 4 = 0. The computations of the Groebner basis of the ideal (C 1 , C 2 , C 3 , C 4 ) of the polynomial ring C[h 12 , h 13 , h 14 , h 34 ] is efficient to solve the system, and direct computations show that the two quartets are solutions of the simultaneous equations.
We present a numerical computation for the quartic form F 2 (t, x, y) which confirms the Conjecture 1.
Proof. We construct a kernel vector function ξ = (ξ 1 , ξ 2 , ξ 3 , ξ 4 ) by computing the 4-th row of the adjugate matrix of M(t, x, y), which is given by The intersection points including multiplicities of the curve F 2 (t, x, y) = 0 and the curves ξ j (t, x, y) = 0 are represented by the following divisors: where the 10 points Q 1 , Q 2 , Q 3 , Q 4 , R 1 , R 2 , R 3 , R 4 , S 1 , S 2 are mutually distinct real points of the curve F 2 (t, x, y) = 0 whose (t, x, y)-coordinates are given by , Hence, the reduced divisorsξ j of F 2 · ξ j by removing the common multiplicities of the zero points are given byξ Note that the numerical intersection points of the curve F 2 (t, x, y) = 0 and the curve ξ j (t, x, y) = 0 can be achieved by applying NSolve function of Mathematica. For instance, let t = 1, NSolve[F 2 == 0, ξ j == 0, {x, y}] produces numerical solutions. The performance of symbolic computations of the intersection points needs more delicate treatments. Using the Resultant function, we can get the algebraic equations defining the coordinates of the intersection point (1, x, y).
We are now ready to prove that the points Q j and R k on the normalized Abel-Jacobi variety of the quartic curve F 2 (t, x, y) = 0 correspond to Q j and R k satisfy the elliptic curve theoretic property (2d) and (2e).
The corresponding 8 points of Q j and R k on the quartic curveK(φ, ξ, η) = 0 are respectively expressed as: (1, (1, Finally, using the Cremona transformation: and its inverse transformation: the quartic curveK(φ, ξ, η) = 0 is transformed into a non-singular cubic curve The 8 points Q j , R k onK(φ, ξ, η) = 0 are transformed to Q j , R k on the cubic curve F 3 (u, v, w) = 0 according to the Cremona transformation which are given in (u, v, w)-coordinates: The cubic curve F 3 (u, v, w) = 0 has a point of reflection at (u, v, w) = (0, 1, −1) which is the neutral element of the elliptic curve group. On the line v = w, the cubic curve has 3 points E j which satisfy E j + E j = 0, j = 1, 2, 3, with respect to the elliptic curve group structure. Among them, To prove (2e) for E 1 , we shall determine the line K j : a j v + b j w + c j u = 0 passing through R j and −Q j on the cubic curve F 3 (u, v, w) = 0. It suffices to show that − 3 2 a j − 3 2 b j + c j = 0, j = 1, 2, 3, 4. In fact, the lines K j are given by and these line pass through the point (u, v, w) = (1, −3/2, −3/2). This proves (2e). Similarly, to prove (2d) for E 2 , we determine the line N 1 passing through −Q 1 , −Q 2 , the line N 3 passing through −Q 3 , −Q 4 . These lines are given by and the two lines pass through the point (u, v, w) = (1, −1, −1). This proves (2d).
Finally, we prove that the kernel vector function of the linear matrix pencil M(t, x, y) = tI 4 + x H 2 + y B in Theorem 4 can be expressed in terms of theta functions used in the Helton-Vinnikov theorem.
Theorem 5. Using the notation and terminology of Theorem 4, the kernel vector function ξ = (ξ 1 , ξ 2 , ξ 3 , ξ 4 ) T expressed as a vector function on the normalized torus is given by for some constants C j = 0, j = 1, 2, 3, 4, where Q j and R j are points in the normalized Abel-Jacobi variety corresponding to the point Q j and R j on the cubic curve for j = 1, 2, 3, 4.

Proof.
Computing the numerical coordinates of the points Q j , we find that the points Q 1 and Q 4 lie on the pseudo line of the cubic curve, and the points Q 2 and Q 3 lie on the oval. Changing the variables (v + w), the cubic curve F 3 (u, v, e) = 0 is expressed in the Weierstrass canonical form where g 2 = 7357 76800 , g 3 = − 11023 22118400 .
In summary, the results of Theorems 4 and 5 prove that the Conjecture 1 is true for an irreducible elliptic quartic curve.

Conflicts of Interest:
The authors declare no conflict of interest.