The Polynomial Form of the Scattering Equations

The scattering equations, recently proposed by Cachazo, He and Yuan as providing a kinematic basis for describing tree amplitudes for massless particles in arbitrary space-time dimension (including scalars, gauge bosons and gravitons), are reformulated in polynomial form. The scattering equations for $N$ particles are shown to be equivalent to a Moebius invariant system of $N-3$ equations, $\tilde h_m=0$, $2 \leq m \leq N-2$, in $N$ variables, where $\tilde h_m$ is a homogeneous polynomial of degree m, with the exceptional property of being linear in each variable taken separately. Fixing the Moebius invariance appropriately, yields polynomial equations $h_m=0$, $1 \leq m \leq N-3$, in $N-3$ variables, where $h_m$ has degree $m$. The linearity of the equations in the individual variables facilitates computation, e.g the elimination of variables to obtain single variable equations determining the solutions. Expressions are given for the tree amplitudes in terms of the $\tilde h_m$ and $h_m$. The extension to the massive case for scalar particles is described and the special case of four dimensional space-time is discussed.


Introduction
In this paper, we study the scattering equations recently proposed by Cachazo, He and Yuan (CHY) [1] as a basis for describing the kinematics of massless particles, and where k a are the momenta of the particles labeled by a ∈ A, with N = |A|, the number of elements in A, and k 2 a = 0. Typically, we shall take A = {1, 2, . . . , N }. Our principal result is that these equations are equivalent to the homogeneous polynomial equations where the sum is over all N !/m!(N − m)! subsets S ⊂ A with m elements, and Note that the coefficients, k 2 S , S ⊂ A, in (1.2) are precisely all the Mandelstam variables, i.e. the denominators of propagators of tree diagrams for processes involving the particles with momenta k a , a ∈ A.
We first recall basic properties of the equations (1.1) [1]- [4]. The equations (1.1) are invariant under Möbius transformations, z a → ζ a = αz a + β γz a + δ , a ∈ A, (1.4) in that, if z = (z a ) is a solution to (1.1), ζ = (ζ a ) also provides a solution, f a (ζ, k) = 0, a ∈ A, provided that momentum is conserved and the particles are massless, for, using momentum conservation, which vanishes if f a (z, k) = 0 and k 2 a = 0. The Möbius invariance implies that only N − 3 of the equations (1.1) are independent. Relations between them are conveniently derived by noting that, if (1. 6) and the Möbius invariance of the system of equations (1.1) can be deduced from that of U (z, k) [4]. Then, considering an infinitesimal Möbius transformation, δz a = ǫ 1 + ǫ 2 z a + ǫ 3 z 2 a , δU = 0 implies the identities: a∈A f a (z, k) = 0; a∈A z a f a (z, k) = 0; a∈A z 2 a f a (z, k) = 0. (1.7) So only N − 3 of the conditions (1.1) are needed to restrict to its set of solutions. This can be achieved by the delta function ′ δ (f a (z, k)) , (1.9) so that the integrand of where the invariant measure on the Möbius group, dω = dz r dz s dz t (z r − z s )(z s − z t )(z t − z r ) , (1.11) is Möbius invariant provided that the function Ψ N (z, k), which may also depend on polarizations, is itself Möbius invariant. [Note that, in (1.10), a cyclic ordering on A has been chosen, with a + 1 following a, say (1, 2, . . . , N ) with N + 1 ≡ 1.] In order to give a precise interpretation of (1.10) that includes a sum over all, possibly complex, solutions of the equations (1.1), CHY rewrote the expression as a contour integral where the contour O encloses all the solutions of (1.1) and ′ is defined as in (1.9).
CHY initially proposed a form of Ψ N (z, k) in (1.10) for tree amplitudes in pure Yang-Mills theory and in gravity in arbitrary space-time dimension [2], subsequently noting that taking Ψ N constant would yield massless scalar φ 3 theory [3]. This result was proved for massless φ 3 and pure Yang-Mills theory in [4], where it was noted that the scattering equations (1.1) could be extended to the massive case, k 2 a = µ 2 , and that (1.12) then produced the tree amplitudes of massive φ 3 for Ψ N constant.
It seems that the scattering equations (1.1) were first written down by Fairlie and Roberts [5], who, before the string-theory interpretation of dual resonance models had been established in detail, were seeking a variation of the Veneziano model that was free of tachyons. They subsequently occurred in the work of Gross and Mende on the high energy behavior of string theory [6]. Fairlie and Roberts replaced the Fubini-Veneziano position field, X µ (z), and the corresponding momentum field, P µ (z), (1.13) expanded in terms of an infinite number of harmonic modes, a µ n , with (1.14) The singularities of x µ (z), p µ (z) on the Riemann sphere are evidently only at z = z a , provided that we have momentum conservation, a k a = 0. [If not, there will be a singularity at Fairlie and Roberts replaced the Virasoro conditions, which can be written : P (z) 2 : = constant, with Given that p(z) 2 vanishes sufficiently fast at infinity, it vanishes everywhere provided it has no singularities at z = z a , for a ∈ A. The absence of double poles requires k 2 a = 0, a ∈ A, and the absence of simple poles implies (1.1). In an approach anticipating in general terms that of [2,3], they sought to define their model as a sum over the solutions of the scattering equations (1.1), rather than as an integral as in the Veneziano model.
An important property of the scattering equations (1.1) is the way they factorize when one of the Mandelstam variables, k 2 S , vanishes. If k 2 S = 0 for all subsets S with between 2 and N − 2 elements, the values of the z a , a ∈ A, are distinct. To demonstrate this, suppose that, as the k a vary (maintaining momentum conservation and the massless conditions), two or more of the z a to tend to the same value, z 0 . Suppose z a = z 0 + ǫx a + O(ǫ 2 ) for a ∈ S, and z a → z 0 for a / ∈ S, as ǫ → 0; then the scattering equations factorize into two sets: implying in the limit that f S a (x, k) = 0, a ∈ S, and In what follows, in general, we shall assume k 2 S = 0 for all subsets S with between 2 and N − 2 elements and, consequently, that z a = z b for a = b, a, b ∈ A.
In section 2, we derive the polynomial form (1.2) of the scattering equations, first directly and then, more elegantly, by using their equivalence to (1.15). We then show show that the polynomial form possesses a factorization property corresponding to (1.16) and (1.17). For the special kinematic configuration in which k 2 S is independent of S ⊂ A if |S| = m, the solutions of (1.2) are given by complex roots of unity, which we describe in section 3, where we also show how the fact that (1.2) is linear in each z a taken separately enables simple forms for single-variable equations to be found which determine the solutions to the scattering equations, at least for N ≤ 6. Expressions for the scattering amplitudes in terms of the polynomials (1.
implies that λ S = k 2 S , for suitable light-like momenta k a , a ∈ A. The extension to the massive case is given in section 6, and special features of the four-dimensional space are discussed in section 7. Some comments and indications of further directions for investigation are given in section 8.

Polynomial Form for the Scattering Equations
The Polynomial Equations. Studies of the scattering equations indicate that, after using Möbius invariance to fix three of the z a , they have (N − 3)! solutions [1,2]. Bézout's Theorem states that number of solutions to a system of polynomial equations, possessing a finite number of solutions, is bounded by the product of their degrees and, suitably counted, the number will attain this bound (see, e.g. [7]). This suggests that the scattering equations (1.1) should be equivalent to N −3 homogeneous equations in the z a , h m = 0, 1 ≤ m ≤ N −3, where h m has degree m.
in (1.7), we noted that Möbius invariance implies that g −1 , g 0 and g 1 vanish identically. For other integral values of m, (2.1) defines an homogeneous polynomial of degree m in the z a , Thus the N − 3 equations are equivalent to the equations f a = 0, a ∈ A, N − 3 of which are independent. Since g m = 0, for all integers m, follows from f a = 0, a ∈ A, and these equations themselves follow from the set (2.4), it follows that this set implies g n = 0 for all integers n.
The set of N − 3 homogeneous polynomial equations (2.4) is equivalent to the scattering equations (1.1) but this form is not convenient, e.g. for taking advantage of the Möbius invariance to fix z 1 at ∞. We can find a form of these polynomials that is more convenient as follows. Write k a 1 a 2 ...an = k a 1 + k a 2 + · · · + k an , (2.9) So a 0 ,a 2 ,...,am∈A a i uneq.
where |S| denotes the number of elements of S, and k S , z S are defined by (1.3).
The homogeneous polynomials are quite simply related to the g m defined by (2.1) because a 0 ,a 2 ,...,am∈A a i uneq.
so that, since g 1 vanishes identically, where Σ A r is the elementary symmetric function (2.14) Since the equations f a = 0, a ∈ A, are equivalent to the equations g m = 0, 2 ≤ m ≤ N − 2, they are also equivalent to the equations,h m = 0, 2 ≤ m ≤ N − 2.h m is a homogeneous polynomial of degree m with the special property that, although it is of degree m in the z a , it is linear in each one of them taken individually, i.e. the monomials that it comprises are square-free. This considerably simplifies various calculations and constructions.
The coefficients, k 2 a 1 a 2 ...am , in theh m are all determined in terms of the 1 2 Of course, these are not independent but must satisfy the constraints of momentum conservation. This leaves 1 2 ). An Alternative Derivation. We can give a more elegant and succinct, if less direct, derivation of the polynomial form of the scattering equations as follows. As we noted in section 1, the scattering equations (1.1) are equivalent to p(z) 2 , as given by (1.15), vanishing everywhere as a function of z for z a , a ∈ A, satisfying these equations, and this statement is equivalent to the vanishing of the polynomial of degree N − 2, so that the coefficients of z N −2 and z N −3 vanish and establishing the equivalence of the polynomial form (1.2) to the original scattering equations (1.1).

Partially Fixing the Möbius Invariance.
Because the set is equivalent to the equations f a = 0, a ∈ A, the Möbius group acts on the solutions of the set of equationsh m = 0, 2 ≤ m ≤ N − 2 (which define an algebraic variety in C N ). As usual, it is convenient to fix this invariance, at least partially. We take z 1 → ∞, z N → 0. This is facilitated by the fact that h m is an homogeneous polynomial of degree m in the variables z 2 , . . . , z N −1 , linear in each of them separately. Fixing z 1 = ∞, z N = 0 breaks the Möbius group down to scalings These equations define a (presumably) zero-dimensional projective variety in CP N −3 , a set consisting typically of points by Bézout's Theorem.
Factorization. To see how the factorization, evident in (1.16) and (1.17), when z a → z 0 , a ∈ S, implying k 2 S = 0, manifests itself in terms of the polynomial equations, consider the equationsh m = 0, 2 ≤ m ≤ N − 2, without specializing to z 1 = ∞, z N = 0, and use the translation invariance that then holds to take z 0 = 0, without loss of generality. Let n = |S|, h n vanishes because k 2 S = 0, and, for n < m < N , hS 0 m is the appropriate form ofh m for the polynomial scattering equations for n + 1 particles with variables z a , a ∈S and 0, associated with momenta k a , a ∈S, and k S , whileh S∞ m is the appropriate form of h m for the polynomial scattering equations for N − n + 1 particles with variables z b , b ∈ S, and ∞, associated with momenta k b , b ∈ S, and kS, thus demonstrating that the polynomial equations factorize as they should.

Solutions to the Scattering Equations
The configuration describes massless particles with momenta k 1 , k N annihilating in their center of mass frame to produce symmetrically N − 2 massless particles with momenta k a , 2 ≤ a ≤ N − 1, the spatial components of whose momenta correspond to the vertices of a regular (N − 3)-simplex.] If its coefficients k 1a 1 ...am are all equal, h m is proportional to the elementary symmetric function, Σ A ′ m , of (2.14). The conditions Thus each z = z a , 2 ≤ a ≤ N − 1, satisfies the equation z N −2 = λ N −2 and, taking λ = 1, because only ratios of the z a matter, the solutions of the scattering equations for these momenta are and ω is a complex (N − 2)-th root of unity. The ratios z a /z 2 , 3 ≤ a ≤ N − 1, say, are thus given by the N − 3 distinct complex (N − 2)-th roots of unity, taken in some order, giving the (N − 3)! possible solutions. For this particular symmetric solution there is a symmetry under permuting the coordinates, which takes one solution into another, which is not present in general.
For further discussion of special configurations see [8], where Kalousios discusses kinematical configurations in which the solutions to the scattering equations can be identified with the zeros of Jacobi polynomials, and [3].
Solutions for N = 4 and N = 5. For N = 4, we have a simple linear equation determining For N = 5, writing (x, y, z) = (z 2 , z 3 , z 4 ), the equations yield a quadratic equation for z/y by elimination of x. This can be conveniently achieved by considering 12 is quadratic in y and z (as can be seen, e.g., by noting, using the approach of (3.7), that any third derivative of h Solution for N = 6. This approach can be extended to N = 6 without much difficulty, where we expect a sextic equation to determine the ratios z a /z b . Write (x, y, z, u) = (z 2 , z 3 , z 4 , z 5 ) and (3.8) Then the technique of (3.7) shows that ∂ x h is indeed independent of both x and y because ∂ x h which is a form that can be obtained from classical elimination theory [9].

Amplitudes in Terms of Polynomial Constraints
We want to rewrite (1.12) in terms of h m , 1 ≤ m ≤ N − 3, rather than f a , a ∈ A, a = i, j, k. We can invert (2.1) to give where Z ma = z m+1 a , a ∈ A, −1 ≤ m ≤ N − 2, noting that g −1 = g 0 = g 1 = 0. Since Z −1 = (det Z) −1 adjZ, the Jacobian of {f a : a ∈ A, a = i, j, k} with respect to {g m : 2 ≤ m ≤ N − 2} is given by Thus, up to a sign and factors of 2, Taking z 1 → ∞, z 2 fixed, z N → 0, (4.5)

Möbius Transformations
Möbius Invariance of the Integrand for the Amplitude. It is straightforward to demonstrate directly the Möbius invariance of (4.4). The Möbius group is generated by translations, scaling and inversion, and, in this instance, demonstrating invariance under translation, z a → z a + ǫ, involves a little more than the others. We havẽ Given U with |U | = n, and it may easily be checked that (4.4) is invariant under these as well, so that it is invariant under the full Möbius group.
Representations of the Möbius Algebra. Combining translation, T ǫ , (5.5), with inversion, I, (5.7), gives the special conformal transformation, S ǫ = IT −ǫ I, If L −1 denotes the generator of translations (5.5), the generator of special conformal transformations (5.8), With these definitions, the appropriate generator of scale transformations is takes the value ( 1 2 N − 2)( 1 2 N − 1). More generally, the ring, R N , of polynomials in z 1 , z 2 , . . . , z N , provides a graded infinitedimensional representation space for (5.12). This is the N -fold tensor product of the representation acting on the ring of single-variable polynomials, R 1 , in z. It is easy to see that the only invariant subspace of R 1 for (5.14) is the two-dimensional subspace, F 1 , consisting of linear polynomials, i.e. that spanned by {1, z}, which carries a representation of 'Möbius spin' 1 2 , for which L 2 = 3 4 . [The quotient space R 1 /F 1 provides an irreducible infinite-dimensional representation of the Möbius algebra with 'Möbius spin' 1 2 and L 0 ≤ − 3 2 . Correspondingly, R N has a Möbius invariant subspace F N , which is the tensor product of N copies of F 1 , one for each z a , a ∈ A, and consists of polynomials that are linear in each of the z a taken separately. F N has dimension 2 N and has a basis consisting of the square-free monomials, It is straightforward to show that, if the polynomial ϕ(z) ∈ R N involves a monomial term with a factor z n a , with n ≥ 2, then (L 1 ) M ϕ involves a monomial term with a factor z M +n  We can verify directly that the dimension of the space of tensors ϕ i 1 ...in satisfying these conditions is dim H N n as given by (5.15). The action of L 1 on ϕ ∈ H N n , defined by (5.17), is given by (5.19) where, in general, we define Then, if m ≥ n, and So L r 1 ϕ = r!ϕ n+r and, using the algebra (5.12) and L −1 ϕ = 0, we can calculate L −1 L r 1 ϕ and deduce that where ϕ n−1 = 0, and, as we see below, ϕ N −n+1 = 0, and It follows that Iϕ m = (−1) n ϕ N −m , from which it follows that ϕ m = 0 if m > N − n, consistent with the fact that the N − 2n + 1 polynomials ϕ n , ϕ n+1 , . . . , ϕ N −n are a basis for the invariant subspace generated from the highest weight polynomial ϕ ≡ ϕ n .
A basis for F N 0 is provided by ϕ ≡ ϕ 0 = 1, and then ϕ m = Σ A m , 0 ≤ m ≤ N , the elementary symmetric polynomial defined by (2.14). An element of F N 1 is of the form ϕ ≡ ϕ 1 = a∈A λ a z a , where a∈A λ a = 0, and then Uniqueness. If ϕ ∈ H N n , the invariant linear subspace generated from ϕ has dimension N − 2n + 1, and so it is the case n = 2 that is of particular interest to us because it is only in this case that the conditions ϕ m = 0 provide the right number of constraints to determine a finite number of points in CP N −1 , up to Möbius transformations. Further, from (5.15), dim H N 2 = 1 2 N (N − 3), which, if the dimension of space-time is sufficiently high, is the number of independent degrees of freedom, k a · k b , of light-like momenta k a , a ∈ A, satisfying momentum conservation (up to Lorentz transformation), with ϕ U determined by To see more directly that, if ϕ ∈ H N 2 , then ϕ m is of the form (1.2), take vectors, k a , a ∈ A, summing to zero, with {k 1 , . . . , k N −1 } linearly independent, define a scalar product by Then, taking ϕ a,b = (k a + k b ) 2 , ϕ U , as defined by (5.20), is given by (5.27) and ϕ m =h m , defined as in (2.11).

Massive Particles
In the discussion of massless particles, the order of the particles is not relevant. To discuss massive particles [4], we need to select a cyclic ordering on A, which we take to be the ordering 1, 2, . . . , N, 1. We replace the definition of f a in (1.1) by , a ∈ A. (6.1) Using (6.1) in (2.1) to define g m , it is again the case that Möbius invariance of this system of equations implies that g −1 , g 0 , g 1 all vanish identically, provided that k 2 a = µ 2 for each a ∈ A.
Then, following section 2, we obtain, as in (2.5), where n a,a±1 = 1, n ab = 2, b = a ± 1, provided that k 2 a = µ 2 , a ∈ A, and, as in (2.6), (k 2 abc − n abc µ 2 )z a z b z c ; (6.5) where n abc = n ab + n ac + n bc − 3, provided that k 2 b = µ 2 for b ∈ A. [n 2 abc = 3, if none of a, b, c are adjacent, n abc = 2, if just two them are, and n abc = 1, if a, b, c are sequential (in some order).] Thus, as in (2.8), For S ⊂ A, let n S denote the number of disjoint strings of adjacent elements of S with the given cyclic ordering on A. Then 1 ≤ n S ≤ |S|, with n S = 1 if and only if the elements of S are sequential. Then, as in (2.10), a 0 ,a 2 ,...,am∈A a i uneq.
The equations f a = 0, a ∈ A, are equivalent to the equationsh m = 0, 2 ≤ m ≤ N − 2, wherẽ Theh m are related to the g m , defined by (2.1) with f a given by (6.1), by (2.13). The Möbius invariance of this system can be demonstrated directly as in section 4.
As in section 2, it is convenient to fix the Möbius invariance partially by letting z 1 → ∞ and z N → 0. Again, writing A ′ = {a ∈ A : a = 1, N } and S 1 = S ∪ {1},

Four-Dimensional Space-Time
Fairlie and Roberts [5], who were concerned with four-dimensional space-time, noted that a particular solution of the scattering equations (1.1) is given by for D = 4 and for all N . The demonstration of this can conveniently be expressed using twistor variables, π a ,π a , defined by In terms of these, z a = π 1 a /π 2 a , k a · k b = π a , π b [π a ,π b ], (7 .3) with π a , π b = π 1 a π 2 b − π 2 a π 1 b , [π a ,π b ] =π 1 Another solution to the scattering equations in D = 4, not equivalent under Möbius transformations to that of [5] (except for N = 4), is given by z a =π 1 a /π 2 a . These are two rational explicit solutions which exist for all N . As noted in [10], these solutions are can be associated with the MHV and anti-MHV amplitudes, and the others can be associated other combinations of helicities. [The scattering equations in four-dimensional space have also been discussed recently by Weinzierl [11].] Although CHY introduced the scattering equations in order to describe tree amplitudes for Yang-Mills and gravity [2,10], perhaps more basically, they also describe scalar particles [3,4]. It seems that the other solutions can be associated with the division of the particles into n, say, positive helicities and N − m negative helicities. However, perhaps strangely, the solutions only depend on how many positive and negative helicities there are, i.e. they depend on n, not on which particles are assigned positive helicity and which negative.
To see this, as in [12] (see also [13]), we divide the N indices a ∈ A into m 'positive' indices i ∈ P and n 'negative' indices r ∈ N . Then the link variables c ir , introduced by Arkani-Hamed, Cachazo, Cheung and Kaplan [14,15], satisfy and these are related to the parameters of twistor string theory [16] - [18] by where it is understood that r, s are summed over N and j over P, with j = i. Thus the scattering equations (1.1) hold for a = i ∈ P and a similar argument shows that it holds if a ∈ N .
Spradlin and Volovich [13] conjectured that the number of solutions of the equations (7.7) and (7.8) is given by the Eulerian number N −3 m−2 , 2 ≤ m ≤ N − 2, and CHY have given a demonstration of this [10]. These numbers sum to (N − 3)!, the number of solutions of (1.1). This implies that the solutions depend on an assignment of a number of positive and negative helicities, but must be independent of which particular momenta are associated with positive helicities and which negative, otherwise there would be too many solutions. To demonstrate this independence, consider moving i into N and r into P, So, definingP = {r; j ∈ P, j = i} andÑ = {i; s ∈ N , s = r}, we have ; (7.14) and, similarly, we can showπ t = ℓ∈Pπ ℓcℓt , t ∈Ñ .

Comments
A general understanding of the polynomial form of the scattering equations was provided by the discussion of Möbius invariance in section 5. If the Möbius algebra acts as in (5.9)-(5.11), the polynomial scattering equations are the only sets of homogeneous polynomials, related by Möbius transformations, determining a finite set of points.
In general, for a given highest weight polynomial ϕ, satisfying L −1 ϕ = 0, L 0 ϕ = ( 1 2 N − n)ϕ, the equations, ϕ m = 0, n ≤ m ≤ N − n, defined as in (5.21), determine an algebraic variety in the complex projective space CP N −1 on which the Möbius group acts. [We shall assume ϕ S = 0, S ⊂ A, n ≤ |S| ≤ N − n.] For n = 0, where, up to a constant, ϕ = 1, this is empty. For 1 ≤ n ≤ 1 2 N , we can use the Möbius invariance to send z 1 → ∞, z N → 0, as in (2.20) and (6.11), where, again, S 1 = S ∪ {1} and A ′ = {a ∈ A : a = 1, N }. The homogeneous equations h ϕ m = 0, n − 1 ≤ m ≤ N − n − 1, define a complex projective variety, V ϕ , in CP N −3 , which we expect typically to have dimension 2n − 4. For n = 1, h ϕ 0 is constant and there are no solutions, V ϕ = ∅. As we have discussed, n = 2 provides precisely the scattering equations, for all possible light-like momenta satisfying momentum conservation, with typically V ϕ containing (N − 3)! points. It would be interesting to determine conditions on ϕ so that, for n = 2, V ϕ contains precisely this number of distinct points, and, for n > 2, dim V ϕ = 2n − 4. It would also be interesting to know whether the varieties V ϕ , for n > 2, have any physical significance.
We may realize differently the Möbius group by changing the action of inversion from (5.7) to φ(z 1 , . . . , z N ) → (z A ) K φ(1/z 1 , . . . , 1/z N ), (8.2) where K is a positive integer, and then, in place of (5.9)-(5.11), which correspond to K = 1, we have defining the action of translation, scaling and the special conformal transformation for n = −1, 0, 1, respectively. In this case, for N = 1, a single variable, the finite-dimensional invariant subspace has 'Möbius spin' 1 2 K. For general N it will be the tensor product of N copies of this space and so will have dimension (K + 1) N ; it consists of all polynomials involving no power of z a higher than z K a for each a ∈ A. Clearly, we may consider Möbius invariant systems of polynomial equations in this more general context but we have not yet analyzed these.
The CHY expressions for the tree amplitudes in scalar, gauge and gravity theories involve, in effect, summing a rational function (their integrand) over the finite projective variety V ϕ , where ϕ is the degree 2 highest weight polynomial whose coefficients are the scalar products of the external (light-like) momenta. Although the points of V ϕ are, in general, irrational, the result is a rational function of the coefficients of ϕ and the rational integrand. It would be desirable to have formula for this result, and an understanding of it, in terms of the algebro-geometric properties of V ϕ , perhaps along the lines of elimination theory [19]. The understanding of scattering equations in terms of (1.15) and (2.16) may provide insight into possible extensions of the scattering equations from the Riemann sphere to the torus.