1. Introduction
Let
A be an
complex matrix. Toeplitz [
1] introduced the numerical range of
A as the set
which contains all eigenvalues of
A. The inverse numerical range aims to determine a vector
satisfying
for a moving boundary point
of
. 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.
The determinantal ternary form associated to an
matrix
A is the homogeneous polynomial
, where the linear matrix pencil
and
,
. The algebraic curve of
A is the set
where
is the equivalence class of points
with
, under the relation that
, for
,
. The form
is hyperbolic with respect to the point
, that is, the equation
has
n real roots counting multiplicities for any
, and
. Note that
the characteristic polynomial of the Hermitian matrix
.
Kippenhahn [
5] proved that the numerical range
is the convex hull of the real affine part of the dual curve of
. Conversely, Lax [
6] conjectured that every hyperbolic ternary form
of degree
n admits a determinantal representation by a linear matrix pencil
of real symmetric matrices
C and
B, i.e.,
Fiedler [
7] proved that the Lax conjecture is true if
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
using the interlacer
of the hyperbolic form
. 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
of
corresponds to an extreme eigenvalue of
for some angle
. A unital eigenvector
of
corresponding to the maximal eigenvalue
assures
, and
From this view point, the inverse numerical range problem can be renamed in a more general setting: For any nonzero point
on the curve
, to find a kernel vector function
satisfying
The kernel vector function method is used in [
12] to deal with the inverse numerical range of a
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 is a quartic elliptic curve. We show that the intersection points of the kernel vector functions and the algebraic curve 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.
2. Quartic Elliptic Curves
Let
be an irreducible hyperbolic form of degree
n. A point
is called a singular point of the curve
if
A singular point
is called an ordinary double point if the Hessian matrix
of
at
is invertible. The curve
is elliptic if it has genus 1. (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
is elliptic, and the
n real intersection points of the curve
and the line
are distinct non-singular points
with coordinates
, where
. Then there is a real birational transformation which transforms
to a non-singular cubic curve
for some real constants
with
. The real affine part
of the curve
is then parametrized by rational functions of Weierstrass function and its derivative over the torus. The Abel–Jacobi map
is the inverse of the parametrization
. Denote
.
In the paper [
11], the Helton–Vinnikov representation for an elliptic curve is formulated by using the Riemann–Jacobi theta functions
on the normalized Abel–Jacobi variety
,
,
. The theta function
is defined by
where
. The relations among
functions are related as follows.
,
. The main result of [
11] reformulates the Helton–Vinnikov determinantal representation as follows.
Theorem 1 ([
11], Theorem 2.4).
Let be an irreducible hyperbolic ternary form of degree n. Assume that the curve intersects the line at n distinct nonsingular points with , , and assume that is an elliptic curve which is parametrized as by two rational functions . Let be the point of the torus corresponding to the point , and . For , the off-diagonal entries of the matrix in the determinantal representation (1) are given bywhere is a half-period of the Weierstrass function .
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
matrix. Assume that the skew Hermitian matrix
is diagonal with distinct diagonals
. We also assume that the ternary form
is irreducible and its algebraic curve
is elliptic. Then there exist
n points
in the normalized Abel–Jacobi variety
of the elliptic curve for which the reduced kernel vector function
is expressed as
for some constants
,
, on the Abel–Jacobi variety. With respect to the abelian group structure of this variety, one has
If the linear matrix pencil is unitarily equivalent to the matrix pencil realized as the -Helton Vinikov representation for or , then . Furthermore, the point coincides with the point (resp. ) of the normalized Abel–Jacobi variety when (resp. ).
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
nilpotent matrix
, is studied which produces one type of numerical ranges for the classification of the numerical ranges of
matrices. Fladt [
15] formulated this quartic curve
as one of the Kepler’s models of planetary orbits. We consider a more general similar form of the matrix (2a):
where
and
are real numbers.
Theorem 2. Let A be the matrix defined in (2b). If then the algebraic curve has a pair of ordinary double points on the line . In this case, the two ordinary double points are and , where .
Proof. Assume
is an ordinary double point of
. Then the resultant of
and
with respect to
x for
is 0. Using “Resultant” function of the “Mathematica” software, the resultant is given by
Clearly, the condition assures that the resultant is 0.
One easy way to find an ordinary double point
is computing the zeros of
at
. One factor of
for
and
is given by
Its zeros are , and the points are also roots of . □
For simplicity, we present a numerical computation which confirms Conjecture 1. This computation method can be used for employing general computation for a generic quartic curve with two ordinary double points. For this purpose, we define the
matrix
A which is the form of (2b) satisfying the assumption
of Theorem 2 with special values
,
and
. Then
The corresponding curve has just two singular points (ordinary double points) lying on the line
. According to the genus formula [
13] of an algebraic curve:
it implies that the quartic curve
is an elliptic curve.
In the following, we show that the quartic form admits two non-unitarily equivalent determinantal representations.
Theorem 3. Let be the quartic form in (2c). Then, there exist two real symmetric matrices and , such that and are not unitarily equivalent, and , , where , and with entries , and the quartet according to the two inequivalent representations: One matrix is given by and another matrix is given by Proof. By comparing the traces,
the unequal traces ascertain that
and
are non-unitarily equivalent. The diagonal matrix
B whose diagonal entries are the roots of
, i.e.,
which gives
.
Next, we determine the symmetric matrix
satisfying
and
. For this, we compute
where the polynomials
are given by the following:
To prove , it suffices to show that the two real quartets satisfy the simultaneous equations . The computations of the Groebner basis of the ideal of the polynomial ring 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 which confirms the Conjecture 1.
Theorem 4. Let B and be matrices defined in Theorem 3, and let the linear matrix pencil . Assume is the 4-th row of the adjugate matrix of . Then there are 10 distinct intersecting real points on the curves and . Furthermore, their corresponding points on the normalized Abel–Jacobi variety of the quartic curve satisfy the elliptic curve theoretic property:and where and on the normalized Abel–Jacobi variety .
Proof. We construct a kernel vector function
by computing the 4-th row of the adjugate matrix of
, which is given by
The intersection points including multiplicities of the curve
and the curves
are represented by the following divisors:
where the 10 points
are mutually distinct real points of the curve
whose
-coordinates are given by
Hence, the reduced divisors
of
by removing the common multiplicities of the zero points are given by
Note that the numerical intersection points of the curve and the curve can be achieved by applying NSolve function of Mathematica. For instance, let , NSolve 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 .
We are now ready to prove that the points and on the normalized Abel–Jacobi variety of the quartic curve correspond to and satisfy the elliptic curve theoretic property (2d) and (2e).
Firstly, we perform successive real projective and real Cremona transformations. The ternary form
is transformed to
under changing the variables
. The points
and
are transformed on the line
in
-coordinates.
The points
in
-coordinates are given by
Then, changing the variables
the quartic form
is expressed as
The corresponding 8 points of
and
on the quartic curve
are respectively expressed as:
and
Finally, using the Cremona transformation:
and its inverse transformation:
the quartic curve
is transformed into a non-singular cubic curve
The 8 points
on
are transformed to
on the cubic curve
according to the Cremona transformation which are given in
-coordinates:
The cubic curve has a point of reflection at which is the neutral element of the elliptic curve group. On the line , the cubic curve has 3 points which satisfy , , with respect to the elliptic curve group structure. Among them, lies on the pseudo-line part of the real cubic curve , lies on the oval part of the real cubic curve corresponding to the point on the normalized Abel-Jacobi variety, and corresponds to on the Abel-Jacobi variety.
To prove (2e) for
, we shall determine the line
passing through
and
on the cubic curve
. It suffices to show that
. In fact, the lines
are given by
and these line pass through the point
. This proves (2e).
Similarly, to prove (2d) for
, we determine the line
passing through
,
, the line
passing through
. These lines are given by
and the two lines pass through the point
. We also determine the line
passing through
,
, the line
passing through
which are given by
and the two lines pass through the point
. This proves (2d). □
Finally, we prove that the kernel vector function of the linear matrix pencil 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 expressed as a vector function on the normalized torus is given by for some constants , , where and are points in the normalized Abel-Jacobi variety corresponding to the point and on the cubic curve for .
Proof. Computing the numerical coordinates of the points
, we find that the points
and
lie on the pseudo line of the cubic curve, and the points
and
lie on the oval. Changing the variables
the cubic curve
= 0 is expressed in the Weierstrass canonical form
where
The numerical values of
in the normalized Abel-Jacobi variety corresponding to the point
on the cubic curve for
are estimated by
By a result in [
16] and the zero points relations in (2f)–(2i), the functions
,
, satisfy
for some constant
, where
v is any point on the normalized Abel–Jacobi variety. As a consequence, the equation of the kernel vector function
is derived. □
In summary, the results of Theorems 4 and 5 prove that the Conjecture 1 is true for an irreducible elliptic quartic curve.