On the Conjectures Regarding the 4-Point Atiyah Determinant

For the case of 4 points in Euclidean space, we present a computer aided proof of Conjectures II and III made by Atiyah and Sutcliffe regarding Atiyah's determinant along with an elegant factorization of the square of the imaginary part of Atiyah's determinant.


Introduction
The Atiyah determinant is a complex-valued determinant function At(P 1 , ..., P n ) associated with n distinct points P 1 , ..., P n of R 3 . It was constructed by M. F. Atiyah in [1] in his attempt at answering a natural geometric question posed in [3] and arising from the study of the spin statistics theorem using classical quantum theory. The original conjecture of Atiyah was that At does not vanish for all configurations of distinct points P 1 , ..., P n ∈ R 3 . The conjecture was verified in the linear case (all points lie on a straight line) and in the case n = 3 by Atiyah in [1]. However, the case n ≥ 4 turned out to be notoriously difficult. In a subsequent paper [2], Atiyah and Sutcliffe studied the function At and added two new conjectures (after normalizing At) which imply the original conjecture of Atiyah. They provided compelling numerical evidence of the validity of all three conjectures. The three conjectures can be stated as follows: For all distinct points P 1 , ..., P n of R 3 (and all n ≥ 1) we have: (I) At(P 1 , ..., P n ) = 0.
From the statement of these conjectures we can see that (III) =⇒ (II) =⇒ (I). The three conjectures have been very resistant since their inauguration time and only a few attempts on some special configurations were proved successfully. For example, Eastwood and Norbury [5] were able to prove the case n = 4 only for the first conjecture. Other proofs on special configurations include [4] and [6]. In this paper, we build on the work of Eastwood and Norbury by presenting a computer aided proof of conjectures (II) and (III) in the case n = 4 and we also give an elegant factorization of the square of the imaginary part of the Atiyah determinant.
The construction of the determinant is as follows: One starts with n distinct points P 1 , ..., P n ∈ R 3 . By considering P j as an observer of the other n − 1 points we obtain We lift each of these vectors from R 3 to C 2 using the Hopf map h : C 2 → R 3 given by h(z, w) = ((|z| 2 − |w| 2 )/2, zw) to obtain n − 1 points of C 2 . Note that the lifts are not unique and are defined up to phase because h(λz, λw) = |λ| 2 h(z, w). Consequently, our lifts can be considered as points of CP 1 . Taking the symmetric product of these lifts gives a vector V j in C n because ⊙ n CP 1 = C n . Atiyah's first conjecture was that {V 1 , ..., V n } is a linearly independent set. In other words, the determinant of the matrix having the vector V j as its jth column is nonzero. This determinant (when properly normalized) is called the Atiyah determinant and is denoted by At.
It is immediate from the above construction that At is coordinate free and is independent of solid motion. In other words, the determinant function At is invariant under translations and rotations in R 3 . Moreover, At gets conjugated under a plane reflection (see [1]). One consequence of this last property is that At must be real-valued when the points are planar since a reflection in their plane leaves them fixed. In addition to these properties, the Atiyah determinant is built so that it is independent of the order of the points. In other words, if (j 1 , ..., j n ) is a permutation of (1, ..., n) then At(P j 1 , ..., P jn ) = At(P 1 , ..., P n ).
Let us start computing At in the cases n = 2 and n = 3. We will work throughout this paper with the normalization imposed by Atiyah on At (to get rid of the phase factors) which requires that (−w, z) must be the lift of − − → P j P i whenever i < j and (z, w) is the chosen lift of − − → P i P j . For the case n = 2, we have two distinct points A and B. We can identify R 3 with R × C and assume (possibly after a solid motion) that A and B have coordinates (0, 0) and (0, x) respectively, where x > 0 is the distance from A to B. By choosing ( √ x, √ x) as a lift of −→ AB, we are forced to take (− √ x, √ x) as a lift of −→ BA. Consequently, Atiyah's determinant is: Let us now consider the case n The symmetric tensor product of the vectors are then , respectively. Consequently, we obtain the Atiyah determinant for three points as This determinant expands to xyz[6 + 2(cos α + cos β + cos γ)], which can be written as xyz[8 + 8 sin α 2 sin β 2 sin γ 2 ]. Using the identity sin α 2 = 1 2 (a+b−c)(a+c−b) bc and similar identities for sin β 2 and sin γ 2 , we can rewrite the Atiyah determinant for three points as From the triangle inequality it follows that d 3 (x, y, z) is nonnegative, and so Conjecture III is verified for three points.
Eastwood and Norbury use the notation 144V 2 in place of z 4 (u). If u = U(A, B, C, D), the value z 4 (u) equals 144V 2 , where V denotes the volume of the tetrahedron formed by the points A, B, C, D, and it therefore follows that z 4 (u) ≥ 0. It would be erroneous to infer from this that the polynomial z 4 is nonnegative on all of R 6 ; the above statement implies only that z 4 is nonnegative on feasible vectors. Having expressed ℜAt(A, B, C, D) = d 4 (u) as in (2.1), Eastwood and Norbury then invoke the inequalities z 4 (u) ≥ 0, (b + c) 2 ≥ y 2 , d 3 (x, y, z) ≥ 0 and abcxyz ≥ d 3 (xc, ay, bz) (ie n 4 (u) ≥ 0) to conclude that Note that since w 4 is skew-symmetric it follows that w 2 4 is symmetric. As mentioned in the introduction, the imaginary part of At(A, B, C, D) vanishes whenever the four points A, B, C, D are coplanar and this is born out in the above factorization since  If (3.2) is feasible and if the optimal objective value is α = 64 (we will see later that α > 64 is impossible), then we immediately obtain (3.1). The remaining difficulty is that of finding suitable polynomials {f j }. One means of generating a large collection of such polynomials, which we now describe, stems from the triangle inequality. The four points A, B, C, D contain four (possibly degenerate) triangles and each triangle, by means of the triangle inequality, gives rise to three linear polynomials which are nonnegative when u = (a, b, c, x, y, z) is feasible. For example, the triangle A, B, C yields −x + y + z, x − y + z and x + y − z. In all, there are twelve such linear polynomials which we refer to as triangular variables and use the notation t = (t 1 , t 2 , . . . , t 12 ), where A vector α ∈ Z 12 + is called a multi-index with order |α| = α 1 +α 2 +· · ·+α 12 . Employing the standard notation t α = t α 1 1 t α 2 2 · · · t α 12 12 , we see that t α represents a homogeneous polynomial of degree |α| in the variables (a, b, c, x, y, z). Applying the symmetric average, we conclude that av[t α ] represents a symmetric homogeneous polynomial of degree |α| which is nonnegative on feasible vectors. For integers ℓ ≥ 0, we define T ℓ to be the set of all polynomials av[t α ] with |α| = ℓ: Numerically, we have found that if one chooses {f j } equal to T 6 , then the linear program (3.2) is feasible and has optimal objective value α = 32. The formulation (2.1) of Eastwood and Norbury can be understood in the context of (3.2) as the result of including, in addition to T 6 , the two symmetric polynomials z 4 and n 4 which are nonnegative on feasible vectors. Numerically solving (3.2) with {f j } equal to {z 4 , n 4 } ∪ T 6 , we have found that the optimal objective value is α = 60, and (2.1) is indeed an optimal solution of (3.2) as the term av[a((b + c) 2 − y 2 )d 3 (x, y, z)] can be written as a nonnegative linear combination of polynomials in T 6 .
In order to further increase the optimal objective value α in (3.2), we need other symmetric polynomials which are nonnegative on feasible vectors. In pursuit of this, we have identified the following twenty-one feasible vectors u where d 4 (u) = 64 p 4 (u) (all are obtained as u = U(A, B, C, D) with A, B, C, D collinear or non-distinct).  (3.4), whenever λ j > 0. It has been verified that z 4 vanishes on all of these vectors, but n 4 does not. Therefore, the coefficient of n 4 will be 0 if (3.2) has been solved with α = 64. We have considered numerous symmetric homogeneous polynomials of degree 6 which vanish on the vectors in (3.4), but only one of these has resulted in an improvement. Let v 4 denote the skew-symmetric homogeneous polynomial of degree 3 defined by Then v 4 vanishes on the vectors in (3.4), and numerically solving (3.2) with {f j } equal to {z 4 , n 4 , v 2 4 } ∪ T 6 , we have found that the optimal objective value is α = 188/3. Our obtained identity, which has been verified in Maple, is the following: where the six nonzero coefficients λ α and corresponding multi-indices α are given by We will show that m 4 is nonnegative on feasible vectors, but unfortunately, we have been unable to formulate a proof using only polynomials of degree 6. Rather, we have had to multiply m 4 by p 4 and then work with polynomials of degree 12. where the sixty-four nonzero coefficients {λ α } are all positive integers as given in the following The 114 nonzero coefficients ν α and corresponding monomials α are given in the fol-