Elimination and recursions in the scattering equations

We use the elimination theory to explicitly construct the (n-3)! order polynomial in one of the variables of the scattering equations. The answer can be given either in terms of a determinant of Sylvester type of dimension (n-3)! or a determinant of B\'ezout type of dimension (n-4)!. We present a recursive formula for the Sylvester determinant. Expansion of the determinants yields expressions in terms of Pl\"ucker coordinates. Elimination of the rest of the variables of the scattering equations is also presented.


Introduction
The last decade a great amount of progress has been made in our understanding of scattering amplitudes for massless theories starting with the revolutionary work of Witten [1] on four dimensional scattering amplitudes in twistor space. In 2013 a new formulation for the computation of the tree level S-matrix of massless particles in any dimension was proposed in a series of papers by Cachazo, He and Yuan (CHY) [2][3][4][5][6]. The formalism was proven for Yang-Mills and scalars in [7]. By defining a map from the space of kinematic invariants of the scattering of n massless particles to the space of punctures over a Riemann sphere, CHY found the so-called scattering equations, where k a denotes the momentum of the a th particle and σ b the position of the b th puncture on the sphere. Those equations have appeared previously in different contexts, particularly in the work of Fairlie and Roberts [8][9][10] and in the study of the high energy behavior of scattering of strings by Gross and Mende [11] (see also [12][13][14][15][16]). In this work we use the polynomial form of (1) that was presented in [17] and which can be written as where A = 1, 2, . . . , n and the sum runs over all subsets of S with m elements. Furthermore, According to CHY, massless scattering at tree level in arbitrary dimensions can in general be described by the contour integral A n = d n σ vol SL (2, C) σ ij σ jk σ ki a =i,j,k δ(f a ) I n (k, ǫ, σ), which is completely localized over the solutions of the scattering equations, and I n (k, ǫ, σ) depends on the theory. So far I n (k, ǫ, σ) has been proposed for several different theories in various dimensions, including Einstein gravity, pure Yang-Mills [3], φ 3 [4], Einstein-Yang-Mills [5], massive scalar-gravity [18] and lately for Dirac-Born-Infeld and U(N) non-linear sigma model [6].
Although at first sight the CHY formalism seems quite simple and straightforward, when it comes to the study of solutions of the scattering equations they rapidly become difficult to manage since the number of solutions grows factorially as a function of the number of particles. However the amplitudes involve a sum over the solutions which can be found without even solving the equations. With that in mind, several interesting methods have been developed lately. In [35] the elimination method was used in order to write the scattering amplitude in terms of the solutions of a polynomial in a single variable, σ a , and then use the Vieta formulas to get the sums over roots from the coefficients of the one variable polynomial in σ a . This idea was enhanced in [36] by using the companion matrix method, which allows to encode the single variable polynomial in a matrix T σa such that its eigenvalues correspond to the roots of the polynomial and hence the sum over the roots is reduced to taking the trace of the given matrix. In [37] we further elaborated on the equivalence of [35] and [36]. A related approach has been recently considered in [38] where the authors use the Bézoutian matrix to compute the amplitudes with emphasis to numerics. A parallel method was developed in [39] where known results for φ 3 theory are used as building blocks for the computation of other amplitudes. A different approach was considered in [40,41], where based on some results from [33] a direct matching between Feynman diagrams and integration measures in the scattering equation formalism of CHY was found.
This work is partly motivated by some ideas discussed in [35] and then in [36] and [37]. If one can eliminate the variables in the scattering equations, then using simple algebra one can evaluate any rational expression of the variables of the scattering equations summed over roots. Of most importance are of course the scattering amplitudes. The elimination theory was first used in the n = 5 and n = 6 case in the work of [17]. Here we are concerned with the general case.
We know from Bézout theorem that the number of solutions of (2) is (n − 3)!. This means that in the general case each of the variables appearing in the scattering equations can be expressed as one variable polynomials of order (n − 3)!. In the following we will show how this can be done for the chosen variable σ n−1 . We will then express the rest of the variables of the scattering equations as functions of σ n−1 only. In the mathematical literature this goes by the name of elimination theory. We will study the elimination theory of (2) following two separate approaches, by Sylvester and Bézout. The results obtained are the same and can be expressed in various determinantal forms. On our way we will also discover an on-shell recursion for the scattering equations.
In the whole body of this paper we think of (2) as a special case of the multilinear system of equations where b are arbitrary constants and we have fixed the SL(2, C) invariance of (1) by specifying arbitrary values to σ 1 , σ 2 , σ n . It is notationally convenient for what follows to consider σ n−1 as a parameter and rewrite (5) as where c now depends linearly on σ n−1 . This defines a system of n − 3 equations in n − 4 unknowns. We now turn to the elimination via Sylvester type determinants.
2 Elimination in the scattering equations via Sylvester type determinants In this part we will express the advertised relations in terms of Sylvester type determinants. Before focusing on the general case, we will first discuss the n = 6, n = 7 and n = 8 case. The n = 5 case is trivial.
The number of scattering equations is three in this case. Let S(a 1 , a 2 , . . . , a i ) denote the vector space of all polynomials of degree less than or equal to a i in x i . Then the dimension of S(a 1 , a 2 , . . . , a i ) is i j=1 (a j + 1). We consider the linear map where g i are given by (6) for n = 6. Both the range and image of φ 3 have dimension 6. We choose to use the canonical order for the monomial bases for both vector spaces. We choose σ 3 and σ 4 to be the variables of the scattering equations and we consider σ 6 to be a parameter. Then φ 3 is given by a 6 × 6 matrix whose determinant is the resultant of the three scattering equations. By setting the resultant to zero we get the desired sixth order polynomial in σ 5 . We proceed with the explicit construction of the resultant. From (6) the three scattering equations are The monomial basis of We multiply each of the three scattering equations with the elements of the monomial basis S(0, 1) and we consider the following system of six equations Then (9) is a linear system in the variables of the monomial basis S(1, 2). The desired sixth order polynomial in σ 5 is then given by the determinant of (9) and it is where . We can also rewrite (10) compactly as There are many ways to expand (10), for example we can consider the Laplace expansion of the first three columns to get (10) where we have used the n-bracket notation In order to complete the elimination we need to express σ 3 and σ 4 as function of σ 5 . This can be achieved in various ways. We can consider any five of the six equations in (9), for example the first five, and use Cramer's rule to solve the corresponding linear system for σ 3 and σ 4 . We respectively find The two determinants in (14) came from (10) by removing the sixth column which corresponds to the sixth scattering equation in (9) and by multiplying the second (third) row by σ 3 (σ 4 ) and adding that to the first row, finally eliminating the second (third) row.
In the case of n = 6 it is easy to simplify the problem even further by noticing that the first three equations in (9) only depend on three variables (σ 3 , σ 4 , σ 3 σ 4 ). Therefore, it is sufficient to only consider the top left 4 × 3 block of (10) and apply the same operations that led to (14). Equivalently we can only consider the top left 3 × 3 blocks of (14). More specifically we find We consider the map In this case the range and image of φ 4 have dimension 24. The monomial basis of (6) the four scattering equations are We consider the following set of 24 equations and we expand them in the monomial basis of S(1, 2, 3). Then the sought 24 degree polynomial in σ 6 is given by the 24 × 24 determinant The range of j i is 0,1.
We now need to express σ 3 , σ 4 , σ 5 as functions of σ 6 . The construction is identical to the one in the n = 6 case. We can consider any of the 24 equations in (17) and solve the corresponding linear system for the desired σ i . We arbitrarily choose the first 23 equations. Then the desired 23 × 23 determinant in σ 3 is constructed from the 24 × 24 determinant in (19) by removing the last column, multiplying the second row by σ 3 and adding that to the first row and finally by deleting the second row. For the σ 4 polynomial we do the same but now instead of the second row we consider the third row and for the σ 5 polynomial we consider the seventh row according to the canonical order of our basis.
It is possible to expand (19) in terms of 4-brackets using its Laplace expansion. We consider the first four columns of (19) and view this as a 24 × 4 matrix. We then form all possible 4-brackets made out of any non-zero four rows of the aforementioned 24 × 4 matrix. There are 70 possibilities. We next form all possible non-zero 4-brackets made out of the columns 5,6,7,8 of (19). We repeat this until we exhaust all columns of (19). We end up with six groups of 70 4-brackets each. We then form all the products that contain one 4-bracket of each group such that each row of any 4-bracket appears only once. Of course in doing the Laplace expansion an appropriate sign needs to be considered. We end up with an expression that contains 3274 terms of products of six 4-brackets each. It is possible to express an expansion of (19) in terms of 4-brackets as a 6 × 6 determinant with elements 4-brackets. This construction will be presented in the next section.

n = 8
In order to easier understand the general case it is instructive to study the n = 8 case that requires the construction of a 120 × 120 determinant. We consider the map The construction is similar to the previous cases. We have worked it out and found that the 120 degree polynomial in σ 7 is given by the 120 × 120 determinant where and An expansion in terms of 5-brackets can be constructed by considering groups of five columns of (21), forming all possible 5-brackets and then forming products of the 5brackets. The final answer is a sum and difference of terms each one containing the product of 24 5-brackets.

General case
For general n we consider the map The dimension of the range and image of φ n−3 is (n − 3)!. We introduce the notation A (2) j 1 j 2 ...j n−5 = c 0j 1 j 2 ...j n−5 1 c 0j 1 j 2 ...j n−5 2 · · · c 0j 1 j 2 ...j n−5 (n−3) c 1j 1 j 2 ...j n−5 1 c 1j 1 j 2 ...j n−5 2 · · · c 1j 1 j 2 ...j n−5 (n−3) and A (m) Then the (n − 3)! order polynomial in σ n−1 is given by the determinant It is easy to solve the recursion (25), (26) and explicitly write down A (n− 3) . The final answer is a repeated structure isolated from the non-zero rows of the first (n−3) columns. The elimination of the rest of the variables as well as the expansion on A (n−3) in terms of n-brackets follows from the discussion of the n = 7 case. We should also comment that the determinant of A (n−3) can be associated to the Cayley hyperdeterminant of a hypermatrix of the boundary format with entries the coefficients of (6).

How to evaluate the amplitudes
One of the motivations of this work was to find a way to explicitly evaluate the scattering amplitudes using the scattering equations. We have succeeded in doing something more. In fact one can in principle explicitly evaluate any expression of the form where the sum runs over the solutions of the generalized scattering equations (5) and f (σ i ) is an arbitrary rational function of the variables that appear in the scattering equations. The idea is simple to understand and it was first presented in [35], enhanced in [36] and further elaborated in [37]. One uses the elimination to express the function f (σ i ) in terms of σ n−1 only. We call the resulted function g(σ n−1 ). Then, the sum of roots of (28) is given purely in terms of the coefficients of the (n − 3)! order polynomial in σ n−1 using the Vieta formulae for the roots of a polynomial. Furthermore, it is a rational function. A way to efficiently do this is via the help of the companion matrices ( [36] and [37]). One replaces σ n−1 in g(σ n−1 ) by its companion matrix, T (σ n−1 ), performs the multiplication and summation of the matrices and then takes the trace of the final matrix. Schematically we have For more details and for some examples we refer the reader to the aforementioned literature.
3 Elimination in the scattering equations via Bézout type determinants As we have seen in the previous section, the Sylvester determinant is an elegant way to compute the resultant of a set of multivariate polynomials. In the multilinear case we considered we ended up with determinants of size (n − 3)!. The Sylvester method is by no means the only way to use elimination. In this section we consider an alternative to Sylvester and we discuss the construction of the so-called Bézout formula for the scattering equations. The resultant coming from the determinant of the Bézout formula has dimension (n − 4)! and it is equivalent to the one obtained in the previous section. The elements of the Bézout formula are immediately given in terms of (n − 3)-brackets. Other mixed approaches of intermediate dimensionality can also be considered but we will not discuss them here.
Although the construction of the Bézout formula applies straightforwardly to general n with a factorially increasing difficulty, for the sake of simplicity we are going to explicitly consider only the particular cases n = 6 and n = 7. The general idea is simple and was originally considered in [42] although in this section we will follow [43].
This in turn implies that the affine bézoutian is independent of σ 3 andσ n−2 . In the formulae that follow we subtract σ 3 times the second row from the first row in (30) in order to explicitly eliminate σ 3 . This is possible because det M = 0 upon use of the scattering equations. More importantly, the affine bézoutian can be interpreted as a linear map from the dual vector space S(n − 5, n − 6, . . . , 0) * to S(0, 1, . . . , n − 5), which is represented by a (n − 4)! × (n − 4)! matrix B, i.e, the coefficients of the affine bézoutian are the components of the aforementioned matrix B, which in the mathematical literature is called the Bézout formula. Since the components of the Bézout formula come from the computation of a determinant, namely det M, they can be written in terms of brackets.
Here we sketch one way to obtain them. First choose an order for the monomial basis and think of it as a vector. Take into account that there are two such vectors, one in the basis built out of σs for S(0, 1, . . . , n−5) and the other in the dual basis built out ofσs for the dual space S(n − 5, n − 6, . . . , 0) * .
Let us consider the bracket composition of the (l, m) th element of the matrix B, namely B lm . The l th ⊗ m th component of the basis S(0, 1, . . . , n − 5) ⊗ S(n − 5, n − 6, . . . , 0) * is given by an element in the set (31) with values for α i and β j that depend on the chosen order or our monomial basis. Then we have The notation | (σ,σ)=0 means we set all the σs andσs to zero after taking the derivatives.
Let us now move to some particular examples.

n = 6
In the case of six particles, the polynomial form of the scattering equations is given by (8). For this polynomial set, the i th column of the M matrix is given by, The where the 3-brackets were defined in (13). The determinant of B 6 produces the same resultant as the one given in (10).
The expansion of the determinant above gives us the 24 degree polynomial in σ 6 corresponding to the resultant of the system (17).

Discussion
In this work we applied the classical elimination theory in the context of the scattering equations. We chose to use a generalized set of the scattering equations in their polynomial form which leads to more compact expressions. We have achieved to construct the one variable polynomial of degree (n − 3)! in one of the variables of the scattering equations. The answer is given compactly by the determinant of a (n − 3)! × (n − 3)! matrix of Sylvester type. Then we expressed the rest of the variables of the scattering equations as function of one of the variables. The determinant we found satisfies on-shell recursion relations and it would be interesting to study its relation to BCFW recursions [44,45].
One of the features of our relations is that one can interestingly expand our expressions in terms of n-brackets that can be viewed as Plücker coordinates on the variety of lines in P n . One way to see this is to use a particular Laplace expansion of the Sylvester determinant. Then the Sylvester determinant has the interpretation of the Chow form of the subvariety P 1 × P 1 × · · · × P 1 in the Segre embedding [46].
There is an alternative way to use the elimination, namely the Bézout construction. In this case one ends up with a (n − 4)! × (n − 4)! determinant with elements constructed out of (n − 3)-brackets. The expansion of the Bézout determinant gives the same polynomials as the Sylvester determinant after making use of Plücker relations. Mixed determinantal expressions of intermediate dimensionality should also exist but we did not investigate this here.
Our derivation plays the role of the construction of a Gröbner basis of the scattering equations. One of the applications of our results is the evaluation of the sum over the solutions of the generalized scattering equations of any rational expression of the variables that appear in the scattering equation. This can now be achieved by pure algebraic manipulations. The idea is based on the fact that the sum over roots is given by particular combinations of the coefficients of the scattering equations via simple algebra. Therefore, one does not need to explicitly know the solutions of the equations. In fact one cannot know explicitly the solutions in view of the Abel-Ruffini theorem. In the case of the scattering amplitudes, we hope that specialized versions of the elimination could lead to huge calculational simplifications.
There is further a connection of the Sylvester determinant and the Cayley hyperdeterminant of a particular hypermatrix with elements the coefficients of the scattering equations.
One could hope that our construction can lead to a further investigation of the scattering equations and amplitudes and even perhaps connections with different geometrical and combinatorial structures.