General Solution of the Scattering Equations

The scattering equations, originally introduced by Fairlie and Roberts in 1972 and more recently shown by Cachazo, He and Yuan to provide a kinematic basis for describing tree amplitudes for massless particles in arbitrary space-time dimension, have been reformulated in polynomial form. The scattering equations for N particles are equivalent to N-3 polynomial equations h_m=0, m=1,...,N-3, in N-3 variables, where h_m has degree m and is linear in the individual variables. Facilitated by this linearity, elimination theory is used to construct a single variable polynomial equation of degree (N-3)! determining the solutions. \Delta_N is the sparse resultant of the system of polynomial scattering equations and it can be identified as the hyperdeterminant of a multidimensional matrix of border format within the terminology of Gel'fand, Kapranov and Zelevinsky. Macaulay's Unmixedness Theorem is used to show that the polynomials of the scattering equations constitute a regular sequence, enabling the Hilbert series of the variety determined by the scattering equations to be calculated, independently showing that they have (N-3)! solutions.


Introduction
In this paper, we describe the solution of the scattering equations originally introduced by Fairlie and Roberts in 1972 [1] in a search for new dual models (string theories) without tachyons and subsequently found again by Gross and Mende while studying the high-energy behavior of string theory [2]. Recently they have been rediscovered by Cachazo, He and Yuan (CHY) [3]- [6], who have shown that they provide a basis for describing the kinematics of massless particles, by proposing remarkable formulae for their tree amplitudes, which have been proved for scalar field theory and gauge theory [7].
We consider N massless particles, labelled by a ∈ A = {1, 2, . . . , N }, with momenta, k a , a ∈ A, with k 2 a = 0 and a∈A k a = 0, and introduce a variable z a ∈ C for each a ∈ A. Then the scattering equations are the N equations f a (z, k) = 0, a ∈ A, where (1.1) This system of N equations is Möbius invariant system, and consequently satisfy three relations, 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, (1.2) so that there are only N − 3 linearly independent equations in the system.
The equations (1.2) can clearly be written in polynomial form by multiplying by suitable factors of z a − z b , but it is desirable to obtain a set of polynomials whose degrees are as small as possible. The equations typically have (N − 3)! solutions [3], up to Möbius transformations, and so Bézout's theorem, which states that the number of solutions of a system of M polynomial equations in M unknowns is the product of the degrees of the polynomials, suggests that there should be a system of N − 3, equations of degrees 1, 2, . . . N − 3, that is equivalent to the scattering equations, up to Möbius transformations, and this has been shown to be the case [8].
The proposals of CHY for tree amplitudes in massless theories were expressed as integrals of rational functions of the z a and the momenta, around poles occurring at the each of the solutions of the scattering equations. These integrals are hence just sums over the solutions and thus, necessarily, rational functions of the coefficients in the scattering equations. Therefore, the prescription is fundamentally one of attaching an algebraic expression to the scattering equations, or, equivalently, the zero-dimensional variety that they describe; the integral is somewhat symbolic. The objective is to understand the amplitudes in terms of natural algebraic objects attached to this variety.
A number of authors have discussed the use of techniques of algebraic geometry and commutative algebra to discuss the polynomial scattering equations [9]- [12]. There are also extensions of the scattering equations to massive particles [7], [13]; discussions of world sheet theories and loop amplitudes [14]- [17], leading to one-loop (or torus) scattering equations that can also be put in polynomial form [18]; and discussions of the loop amplitudes by mapping onto the Riemann sphere [19]- [23], but we shall not discuss any of these further here.
The scattering equations are equivalent [8] to the N − 3 polynomial equationsh m (z, k) = 0, 2 ≤ m ≤ N − 2, whereh (1.6) p(z) 2 has no double poles because k 2 a = 0, a ∈ A, and, since it vanishes sufficiently fast at infinity, it will vanish everywhere provided it has no pole at z = z a , for each a ∈ A, and this condition is just the scattering equations (1.1). However, the vanishing of p(z) 2 is equivalent to the vanishing of the polynomial of degree N − 2, whereS = {a ∈ A : a / ∈ S}. It is straightforward to show that the coefficients of z N −2 and z N −3 in F (z) vanish, and that We can fix the Möbius invariance of the system partially by taking two of the z, z 1 and z N , say, to ∞ and 0, respectively.
. h m is an homogeneous polynomial of degree m in the variables z 2 , . . . , z N −1 , linear in each of them separately. The equations h m (z, k) = 0, 1 ≤ m ≤ N − 3, define a (presumably) zero-dimensional projective variety in CP N −3 , a set consisting typically of points by Bézout's Theorem. Writing u = z N −2 , v = z N −1 , we seek to construct a polynomial equation of degree δ N = (N −3)! in u/v, whose roots determine the solutions of the scattering equations.
As we shall see, while the degree of the h m determine the degree of this polynomial and so the number of solutions, it is the linearity of the h m in each z a taken separately that facilitates the derivation of the degree δ N polynomial in a relatively simple explicit form. As we shall review in section 2, this has already been done for N ≤ 6 [8]. For N = 4 and 5, it is immediate: for N = 4, there is just one linear equation, h 1 = 0, which just relates u and v and so determines their ratio; for N = 5, the two equations, h 1 = h 2 = 0, provide two relations between x, u, v, where x = z 2 , linear in x, which, when x is eliminated between them, provide the desired quadratic equation for u/v.
For N = 6, writing x = z 2 , y = z 3 , the equations h 1 = h 2 = h 3 = 0, although linear in the individual z a , now involve xy as well as x, y, and so the elimination of x and y is more complicated, but can be achieved by a simple application of the elimination theory developed by Sylvester and Cayley in the 19th century [24]- [26]. We supplement h 1 = h 2 = h 3 = 0 with xh 1 = xh 2 = xh 3 = 0 to provide six linear relations between 1, x, y, xy, x 2 and x 2 y, and the condition of their consistency is that the vanishing of the determinant of the matrix of their coefficients provides the desired sextic in u/v, which, as we review in section 2, we can write the form ∆ 6 = 0 [8], where The advantage of this form is that we can see directly that ∆ 6 = 0 provides a sextic in u/v, vanishing when h 1 = h 2 = h 3 = 0, and so determining the solutions of the scattering equations, without relying on its derivation using elimination theory.
The sextic ∆ 6 is the resultant of h 1 , h 2 , h 3 , regarded as polynomials in x, y. The resultant is the polynomial of lowest degree, with integral coefficients, whose variables are the coefficients of h 1 , h 2 , h 3 , (which coefficients are themselves in this case homogeneous polynomials in u, v, at most linear in either) and which vanishes when the corresponding three equations have a common solution for x, y. As we discuss in section 5, from elimination theory, we know that the resultant is irreducible (that is has no factors other than ±1) and is unique up to sign.
We shall see in section 3 that the discussion for N = 6 extends to N = 7, where we seek a polynomial equation of degree δ 7 = 24, ideally obtained as the determinantal condition for the consistency of 24 independent linear relations between 24 monomials in x, y, z, where x = z 2 , y = z 3 , z = z 4 , the equations being obtained by multiplying the polynomial scattering equations h m , 1 ≤ m ≤ 4, by suitable monomials.
The general approach of elimination theory, as outlined in Cayley's classic article of 1848, which prefigured much of the development of commutative algebra [27], is to obtain new equations by multiplying the polynomial equations to be solved, treating these equations as linear relations between monomials, until the equations are overdetermined and so yield a consistency condition. It may be that the polynomial equations are not linearly independent and Cayley describes how to take account of the corresponding linear relations between the equations, and, potentially linear relations between these relations and so on. Of course, it is best if such complications can be avoided and fortunately we are able to do this for the polynomial scattering equations.
For N = 7, we use the 24 equations given by 12) which provide linear relations between the 24 monomials compromising 23 variables together with 1. The determinantal condition for consistency of the equations (1.12), which we can show are linearly independent, provides the desired polynomial equation of degree 24 for u/v, which we can write in a form like (1.11).
As we show in section 4, these results extend to general N . Writing where C M denotes the set of monomials, (1.14) These provide linear relations between the (N − 3)! monomials β ∈ C N −4 , including unity. The resulting determinantal consistency condition is the required polynomial, ∆ N , of order δ N in u/v. Each element of the determinant can be taken to be a partial derivative of one of the h m as in (1.11); this is specified in (4.9).
Because h m is linear in each of the z a , 2 ≤ a ≤ N − 1, it is not a general sum of monomials in the z a , and, for this reason it is not the theory of resultants for general multinomials that is relevant to the solution of the scattering equations but, as we outline in section 5, the theory of sparse resultants. Further, the sparse resultant of multilinear equations of the form of the scattering equations is given by the hyperdeterminant of a multidimensional array, within the theory developed by Gel'fand, Kapranov and Zelevinsky [27]. Multilinear systems of this sort occur, e.g. in discussions of Nash equilibria in game theory (see chapter 6 of [28]); hyperdeterminants have previously occurred in theoretical physics, e.g. in the context of black hole entropy [29].
Information about the number of solutions of the scattering equations is encoded in the Hilbert series of the variety they determine. We calculate the Hilbert series by using Macaulay's Unmixedness Theorem [30] to show that the polynomials of the scattering equations constitute a regular sequence, and this provides an independent determination that they have (N − 3)! solutions.
Some of our resultants overlap with work of Cardona and Kalousios [31], who also obtained the formula for the polynomial of order δ N , using an approach based on elimination theory.
2 Review of the Cases N = 4, 5 and 6 As described in section 1, in order to solve the scattering equations f a (z, k) = 0, a ∈ A, or, equivalently, h m (z, k) = 0, 1 ≤ m ≤ N − 3, we seek to eliminate z a , 2 ≤ a ≤ N − 3, in favor of u = z N −2 and v = z N −1 ; to give a polynomial of order (N − 3)! in u/v. As before, when appropriate, write x = z 2 and y = z 3 .

Solutions for N = 4 and N = 5.
For N = 4, we have one simple linear equation determining u/v, For N = 5, eliminating x between the equations yields a quadratic equation for u/v given by the vanishing of the determinant where h x m = ∂ x h m , as we see by adding x times the second column to the first. Now, we can see directly that ∆ 5 = 0, where gives the required quadratic in u/v because (i) it vanishes when h 1 = h 2 = 0; (ii) it is a homogeneous quadratic in (x, u, v); (iii) it is independent of x. To see that ∆ 5 is independent of x, note that To see that ∆ 5 is quadratic, note that the entry in row r and column c is of degree r − c + 1; since the sum of the row numbers equals the sum of the column numbers for any product contributing to the determinant, ∆ is of degree 2. Note that the term in v 2 comes from the product h 1 h x 2 and its coefficient is σ 4 σ 24 . Another way, which will be useful when we consider larger values of N , to establish the elementary fact that ∆ 5 is independent of x is as follows. We are seeking the condition on u, v (and the coefficients in the h m ) that h 1 (x, u, v) = 0 and h 2 (x, u, v) = 0 have a common solution for some value of x. For any given value of x 0 , this is clearly the same as the condition that h 1 (x 0 + ξ, u, v) = 0 and h 2 (x 0 + ξ, u, v) = 0 have a common solution for some value of ξ. So this condition must be independent of x 0 . But and the condition for these equations to have a common solution for ξ is ∆ 5 = 0 evaluated at (x, u, v) = (x 0 , u, v). So this condition is independent of x 0 , which implies that ∆ 5 , as defined by (2.4), is independent of x.
Since h 1 , h 2 and h x 2 are linear in v and h x 1 is constant, the leading power of v in ∆ 5 , v 2 , comes from the product h 1 h x 2 and its coefficient is The corresponding result for N = 4 is that the coefficient of v in ∆ 4 = h 1 is σ 3 . These leading coefficients are nonzero if all the Mandelstam variables k 2 S , S ⊂ A, are nonzero.

Solution for N = 6.
For N = 6, we seek to eliminate x and y between the equations to obtain a sextic in u, v. We shall see that obtaining a simple formula for the polynomial in u, v depends on each h m being linear in each z a , while the degree of this polynomial depends on the degree of h m equalling m. We can write where in fact d 1 = a 3 = 0. Following the approach of elimination theory, we consider the equations, h m = 0, 1 ≤ m ≤ 3, together with the equations xh m = 0, 1 ≤ m ≤ 3, which provide six homogenous linear equations relating the five quantities y, x, xy, x 2 , x 2 y, the consistency of which leads to the condition ∆ 6 = 0, where (2.10) By row and column operations, we can show that we can rewrite ∆ 6 as where h xy m = ∂ x ∂ y h m . We can establish the condition ∆ 6 = 0 in the form (2.11), following the approach we used for N = 5, by considering the condition for h m (x + ξ, y + η, u, v) = 0, 1 ≤ m ≤ 3, to have a common solution for ξ, η, given values of x, y, u, v, noting that this condition is clearly independent of the values of x, y. Since where h m = h m (x, y, u, v), etc., giving the vanishing of (2.11) as the condition.

2.3
Direct demonstration that ∆ 6 is the required sextic.
Again, we can see directly, without using elimination theory, that ∆ 6 , as defined by (2.11), gives the required sextic in u/v because (i) it vanishes when To see that ∆ 6 is a homogeneous sextic, note again that the entry in row r and column c, if non-zero, is of degree m r −n c +1 in x, y, u, v, where (m 1 , m 2 , m 3 , m 4 , m 5 , m 6 ) = (1, 2, 3, 2, 3, 4) and (n 1 , n 2 , n 3 , n 4 , n 5 , n 6 ) = (1, 2, 2, 3, 3, 4). Since any product contributing to the determinant involves one element from each row and one from each column, the degree of the product is 6 plus the sum of the m r less the sum of n c , i.e. it equals 6, showing that ∆ 6 is a ho-mogeneous sextic. Because no element in the representation (2.11) is more than linear in v, the term in v 6 must be the product of linear factors, and it is then easy to see that it comes only from the product h 1 h y 2 h xy 3 h 1 h x 2 h xy 3 and its coefficient is It is immediately obvious that ∂ y ∆ 6 = 0, because the y-derivative of any column either vanishes by linearity of the entries in y, or equals some other column. To show directly that ∂ x ∆ 6 = 0 in a way that will generalize to N > 6, first write and, subtracting x times column 6 from column 4, and x times column 5 from column 3, and then x times column 4 from column 2, and x times column 3 from column 1, where h o = h − xh x and h y o = h y − xh xy ; then adding x times row 1 to row 2, and then, again, subtracting x times column 6 from column 4, and x times column 5 from column 3, we obtain which is manifestly independent of x.

2.4
Determination of x, y in terms of u, v satisfying ∆ 6 = 0.
Given u, v satisfying ∆ 6 = 0, x, y can be determined from linear relations. Consider Thus Γ 6 is independent of x, linear in y and vanishes when h m = 0, 1 ≤ m ≤ 3, and so provides a linear relation to determine y, given u, v satisfying ∆ 6 = 0. Similarly, h h y h xy = 0 provides a linear relation, independent of y, to determine x.
We can derive the expression (2.18) for Γ 6 as follows. Given u, v satisfying ∆ 6 = 0, and any x, y, we can find ξ, η such that h(x + ξ, y + η, u, v) = 0, implying ∆ 6 = 0. Then (1, η, ξ, ξη, ξ 2 , ξ 2 η) T is a null vector of the matrix obtained by putting h = h(x, y, u, v) in the matrix corresponding to the right hand side of (2.11) and 3 Elimination Theory for N = 7 3.1 Construction of the 24th order polynomial ∆ 7 using elimination theory , we seek to eliminate x, y and z between the equations h m = 0, 1 ≤ m ≤ 4, to obtain a homogeneous polynomial of order 24 in u, v. We can write and the column vector h = (h 1 , h 2 , h 3 , h 4 ) T ; then h = 0 provides 4 linear relations between 7 variables x, y, z, xy, zx, yz, xyz. Adding the 4 equations xh = 0 brings in the additional variables x 2 , x 2 y, zx 2 , x 2 yz; further adding yh = 0 brings in the variables y, xy 2 , y 2 z, xy 2 z, giving a total of 15 variables with 12 linear relations, still leaving 3 more variables than relations. Then adding the 4 equations xyh = 0, just adds two more variables x 2 y 2 , x 2 y 2 z, leaving one more variable than linear relation. Adding x 2 h = 0 leaves this balance unchanged with 4 more variables, x 3 , x 3 y, zx 3 , x 3 yz, to match the 4 new relations. But, finally, adding the 4 relations x 2 yh = 0 adds just the two variables x 3 y 2 , x 3 y 2 z, so giving a total of 23 variables with 24 linear relations, and so a consistency condition ∆ 7 = 0, where ∆ 7 is the 24 In principle, h m and its derivatives should be evaluated at x = y = z = 0, but we need not impose this condition because, as in the N = 5, 6 cases, for general x, y, z, the vanishing of (3.2) provides the condition that h(x + x ′ , y + y ′ , z + z ′ , u, v) = 0 has solutions for some values of x ′ , y ′ , z ′ , and this is independent of x, y, z, and there is no need to impose the conditions x = y = z = 0.

3.2
Direct demonstration that ∆ 7 is the required polynomial of degree 24.
So we can see directly from (3.2) that ∆ 7 vanishes when h = 0 and is a homogenous polynomial of degree 24 in x, y, z, u, v. As in the N = 5, 6, we can show directly that ∆ 7 is independent of x, y, z. For this purpose, it is notationally convenient to write h for the 4 × 2 matrix (h, h z ), and h γ = ∂ γ h, γ = x, y or xy.
It follows immediately that ∂ z ∆ 7 = 0. We can show ∂ y ∆ 7 = ∂ x ∆ 7 = 0 using the same method that we used to show ∂ x ∆ 6 = 0. To show ∆ 7 is independent of y, first take y times the n-th column from the (n − 1)-th, for n = 3, 6, 9, 12, and then do the same for n = 2, 5, 8, 11, leading to where h • = h − yh y , h x • = h x − yh xy , which are independent of y, by linearity. Now add y times the n-th row to the (n + 1)-th row for n = 1, 3, 5, and then again subtract the n-th column from the (n − 1)-th, for n = 3, 6, 9, 12, leading to which is manifestly independent of y. A similar, but longer, argument shows that ∂ x ∆ 7 = 0. Thus we have shown directly that ∆ 7 , as defined by Following the approach adopted in the N = 6, 7 cases, we consider the (N − 3)! linear relations between the variables that is (N − 3)! − 1 variables other than 1. So, provided that the relations (4.1) are independent, we can use them to eliminate the x a , 2 ≤ a ≤ N − 3, and obtain a polynomial equation for u/v.  The relations (4.1) are labelled by (α, m) ∈ R N −4 ,  Here, in principle, the h γ m are evaluated at x a = 0, 1 ≤ a ≤ N − 4, but, by the argument used for N = 5, 6, 7, the condition ∆ N = 0 is independent of the x a .
From (4.10) it follows that ∆ N is a homogeneous polynomial of degree δ N , where We have already established that it is a homogeneous polynomial of degree δ N and we now wish to show that it is independent of x b , 1 ≤ b ≤ N − 4. We do this by generalizing the argument of section 3.2. Let and Then, the non-zero elements of the column of M (N ) specified by are in the rows specified by (α, m), where:   all of whose elements are independent of x b and whose determinant equals ∆ N , showing that this also is independent of x b . It follows that ∆ N has the desired properties of being a homogeneous polynomial of degree δ N in u, v, independent of x b , 1 ≤ b ≤ N − 4, vanishing when h = 0.
To complete the proof that this polynomial does indeed determine the solutions, we need to show that it does not vanish identically. To do this, we show in section 4.3 that only one product in the expression for the determinant ∆ N contributes to the coefficient of the leading power, v δ N .

The
(1 − x a+1 ). (4.22) Note that putting x = 1 in this relation verifies that [ * ] (4.24) We will show that there is precisely one such map.
It follows from the block structure of M (N ) that any such φ defines bijective maps R j N −4 → C j N −4 , and so our task is equivalent to showing that, for each 0 ≤ j ≤ d N , there is one exactly one bijective map φ : and R j N −4 into subsets C j,k N −4 and R j,k N −4 , respectively. We define so that C j,k n comprises elements β, which are the product of factors x ma n−a+1 , 1 ≤ a ≤ n, where 0 ≤ m a < a if 1 ≤ a ≤ n − k, m n−k+1 = k, and 0 ≤ m a ≤ a if n − k + 2 ≤ a ≤ n. It follows that where the range of k is limited above by both j and N − 4, and below by 0 and the requirement that Then R j,k n = ∅ if k < j−d N −1 or k > N −4; and R j n is the union of R j,k n for max(0, j−d N −1 ) ≤ k ≤ min(j, N − 4). Note |R j,k n | = |C j,k n | and R j,k n = (x k−1 n−k+1 α ′ , m ′ + 1) : (α ′ , m ′ ) ∈ R j−k n−1 (x 1 , . . . , x n−k , x n−k+2 , . . . , x n−1 ) , 1 ≤ k ≤ j. If (α, m) ∈ R j,k n , with k > 0, then x n−a a | α, 1 ≤ a ≤ n − k. Suppose φ(α, m) = γα, where γ ∈ B n . Then x n−a+1 a | γα, 1 ≤ a ≤ n − k, for γ can add at most one factor of x a to α. Thus φ(α, m) / ∈ C j,n−a+1 n , 1 ≤ a ≤ n − k, i.e. φ(α, m) / ∈ C j,ℓ n , k + 1 ≤ ℓ ≤ n. Since φ(R j n ) = C j n and φ(R j,0 n ) = C j,0 n , because we always have φ(α, 1) = α, it follows that In particular φ(R j,1 n ) ⊂ C j,1 n , and since |R j,1 n | = |C j,1 n | and φ is bijective, it follows that φ(R j,1 n ) = C j,1 n , and similarly, by induction, that φ(R j,k n ) = C j,k n , for all k.
, respectively, and subdividing further and so on inductively defines orders on C j N −4 and R j N −4 , with respect to which φ is lower triangular, and, consequently, reordering the rows and columns of M (N ) brings it into a form which is lower triangular.
It follows that det M (N ) , which contains the term in ∆ N involving v δ N , is given, up to sign, by a single product of the (N − 3)! diagonal elements (after reordering as above), which are of the form h γ m , where γ ∈ B N −4 and m = deg γ + 1, with each m occurring (N − 4)! times. It follows immediately from this that the coefficient of v δ N is nonzero provided that all the Mandelstam invariants σ S = 0.
To evaluate this coefficient more explicitly, consider further det M (N ) . Up to sign, it is given by a product of the form where A ′′ = {a : 1 < a < N − 2}.
We can establish (4.34) directly from our iterative construction of the map φ. From this construction, if, inductively, det M (N ) is given by (4.33), up to a sign, det M (N +1) is given by

Sparse Resultants
It is the special feature of the polynomial scattering equations, that the h m are linear and homogeneous in the variables z a , 2 ≤ a ≤ N − 1, and it is this linearity that ensures that we can obtain relatively simple explicit forms for the equation ∆ N = 0, obtained by eliminating 2), that is the polynomial in the coefficients of these polynomials (which are combinations of factors σ S , u and v) whose vanishing is the necessary and sufficient condition for these equations (5.1) to have a common solution for the variables (5.2), taking values on the Riemann sphere. The theory of resultants, in the two variable case, has its origins in the work of Sylvester [24] and Cayley [25], extended to the multivariable case by Macauley [32], and revived and set in a modern context by Jouanolou [33] and Gel'fand, Kapranov and Zelevinsky (GKZ) [27]. For accounts, see, e.g., chapter 4 of [28] or chapter 3 of [34].
The existence of the resultant, of a set of M + 1 polynomial equations in M variables, as an irreducible polynomial in the coefficients of the equations with integral coefficients, unique up to sign, can be established using techniques of algebraic geometry (see [27] and [33]- [34]). However, these results do not provide explicit expressions for the resultant when M > 1. Moreover, these existence theorems are typically formulated for general polynomials of a particular degree, involving all its coefficients, and they can vanish identically when one specializes to polynomials in which certain terms are absent. This is the situation for the scattering equations (5.1), where the condition of linearity in each of the variables (5.2) means that many terms are absent from the general form of a multinomial of the same degree.
Although the theory of resultants for general polynomials can not be directly applied to the scattering equations, the theory of sparse resultants, developed to deal with situations in which some of the monomial terms in the polynomial equations considered are absent, precisely addresses their case (see, e.g., section 4.3 of [28]) While a complete theory of sparse resultants is lacking, the scattering equations fall into a category for which some precise and explicit results exist [27], [35]- [37]. In particular, it follows from the discussion of chapter 13 in [27] that the (sparse) resultant of the multilinear equations (5.1), viewed as polynomials in the variables (5.2), defined as above, exists and, by Proposition 2.1 of that chapter, has degree is (N − 3)! (see also [36]). Because ∆ N , constructed as in section 4.1, vanishes when (5.1) holds, has this degree and has no integer factors, it must indeed be this sparse resultant (which is unique up to sign), and so irreducible as a polynomial. Consequently, there is a solution to (5.1) for (5.2) for each of the δ N roots of ∆ N = 0 for u/v.
The hyperdeterminant, det M, is defined as the (sparse) resultant of the ℓ 0 + 1 homogeneous equations that is the polynomial in the coefficients of the f i 0 (the elements of the multidimensional matrix M) whose vanishing is the necessary and sufficient condition for (5.4) to have a nontrivial solution for the ξ j , which we should regard as elements of CP ℓ j , because of the multihomogeneity of (5.3). det M itself is nontrivial if and only if ℓ 0 ≤ ℓ 1 + · · · + ℓ M , for otherwise the ξ j ∈ CP ℓ j are always overdetermined. This definition of det M agrees with the conventional definition in the two-dimensional case.
Explicit formulae for det M in the general case are not known, but an exception is the borderline case, when ℓ 0 = ℓ 1 + · · · + ℓ M ; in this case M is said to be of boundary format, there is a simple expression for the degree, deg M = (ℓ 0 + 1)!/ℓ 1 ! · · · ℓ M !, and det M can constructed as a (two-dimensional) determinant, using methods similar to those of sections 3 and 4.
∆ N is the hyperdeterminant of a multidimensional matrix of boundary format, (N − 3) × 2 × · · · × 2, where there are M = N − 4 factors of 2. To show this, we homogenize h m as a function of the variables (5.2) by setting x a = ξ 1 a /ξ 0 a , and defining so that the f m are linear in each ξ a . Then, writing We note that the hyperdeterminant corresponding to N = 6 case is discussed by GKZ as example 4.9 of [35]; and, as Cardona and Kalousios [31] have pointed out, Sturmfels discussed the equations of the N = 7 case as an example of sparse resultants in section 4.5 of [28].

Hilbert Series and Regular Sequences
The CHY expressions for tree amplitudes in massless field theories are given as residue functions evaluated at the solutions of the scattering equations and summed over those solutions. Thus such a residue function, Φ, only need be specified up the addition of sums of polynomial multiples of the h m , because the additional terms vanish at the solutions of the scattering equations.
If R denotes the ring of polynomials in z a , 2 ≤ a ≤ N − 1, and I ≡ h 1 , . . . , h N −3 denotes the ideal in R generated by the h m , 1 ≤ m ≤ N − 3, that is the polynomials of the form we are interested in S = R/I, polynomials modulo elements of I. Further, we consider homogeneous polynomials in the z a of a specific degree, n, say. Suppose R n denotes the homogeneous polynomials of degree n, so that R is the direct sum of the R n ; and I n denotes the elements of I which are homogeneous of degree n, so that I n = I ∩ R n . Then if S n = R n /I n , S is the direct sum of the S n and H S n = dim S n , the number of independent homogeneous functions of degree n modulo the h m , which is the number of independent functions we have to consider. These dimensions are encoded into a formal series, the Hilbert series (see, e.g., [38]), If J is a homogeneous ideal in R, T = R/J and h ∈ R s , a homogeneous polynomial of degree s, h / ∈ J , we can consider the ideal K, generated by J and h, i.e. K = {g 1 + hg 2 : g 1 ∈ J , g 2 ∈ R}, and the corresponding quotient ring, U = R/K. Define φ : T → T by φ(g + J ) = gh + J . The kernel of φ consists of those (g + J ) ∈ T , for which gh ∈ J , so ker φ is trivial if gh / ∈ J whenever g / ∈ J . If there is g / ∈ J with gh ∈ J , h is said to be a zero divisor in T . If h is a nonzero divisor in T , ker φ is trivial, and it is easy to calculate the Hilbert series of U in terms of that of T .
Because J ⊂ K, we can define ψ : T → U by ψ(g + J ) = g + K. The kernel of ψ consists of (g + J ) ∈ T such that g ∈ K, implying g + J = g ′ h + J , for some g ′ , implying ker ψ = im φ, while, clearly, im ψ = U . If deg h = d, we have maps φ : T n → T n+d and ψ : T n → U n . So, provided that h is a nonzero divisor in T , we have an exact sequence: Multiplying by t n+d and summing gives where S (m) = R/I (m) , (1 + t + . . . + t m−1 ), (5.10) consistent with results obtained by computer calculation in [9] for low values of N .
That, in the case of the scattering equations, h 1 , . . . , h N −3 is indeed a regular sequence follows from the Unmixedness Theorem of Macaulay [30]. (For an exposition in modern terminology, see, e.g., [39], specially section 5.) This theorem states that, if g 1 , g 2 , . . . g s is a regular sequence of homogeneous polynomials, in the M (complex) variables, x 1 , x 2 , . . . , x M , then the (projective) variety defined by g 1 = g 2 = · · · = g s = 0 has dimension M − s (i.e. projective dimension M − s − 1), and, conversely, if this variety has dimension M − s then g 1 , g 2 , . . . g s is a regular sequence (irrespective of the order). Taking M = N − 2, with x a = z a+1 , 1 ≤ a ≤ N − 2, we have established in section 4 that the variety defined by the scattering equations (5.1) is discrete, i.e. has projective dimension zero, so that, by the Unmixedness Theorem, h 1 , h 2 , . . . , h N −3 is a regular sequence.
In fact, we do not have rely on the analysis of section 4 to show that we have a regular sequence. If we supplement the N − 3 scattering equations (5.1) with the equation z N −1 = 0, there is no solution to this augmented system, other than the trivial solution in which all the z a vanish, because we have already specialized to z N = 0, and we established in [8] that the values of the z a must be distinct for different a if the Mandelstam variables σ S = k 2 S 1 are nonzero. [We can also deduce this directly by first substituting z N −1 = 0 in h N −3 = 0 to obtain σ A ′′ z 2 z 3 · · · z N −2 = 0, (5.11) where A ′′ = {a ∈ A : a = 1, N −1, N }, implying that one of z a , 2 ≤ a ≤ N −2, vanishes, z b say. Then, putting z b = z N −1 = 0 in h N −2 = 0, we can deduce that another z a vanishes and so on to establish that all the z a , 2 ≤ a ≤ N − 1, vanish, showing again that there is no nontrivial solution to (5.1) together with z N −1 = 0.] So the space of solutions has (affine) dimension zero, i.e. is empty projectively, and by the Unmixedness Theorem, h 1 , h 2 , . . . , h N −3 , z N −1 is a regular sequence, and it follows that the subsequence h 1 , h 2 , . . . , h N −3 is regular.
[A somewhat similar argument has been used in [40] to show that the (nonhomogeneous) polynomials obtained by putting z 2 = 1 in h 1 , . . . , h N −3 form a H-basis, in the sense of Macauley, for the (affine) ideal that they generate.] The regularity of the sequence of scattering equations implies that the formula (5.10) for the Hilbert series holds. H S n , as a function of n, is called the Hilbert function. For sufficiently large n, it is always a polynomial whose degree equals the projective dimension of the variety, and so is eventually constant for a zero-dimensional variety, with the constant equalling the number of points in the discrete variety, counted according to multiplicity. In this case, (1 + t + . . . + t m−1 ), (5.12) so that the H S n is a polynomial of degree 2ℓ − 4 for sufficiently large n, corresponding to a variety of this projective dimension. This expression for H S (t) is again consistent with results in [9] for low values of N .