Approximations of Lovász extensions and their induced interaction index

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1 Discrete Applied Mathematics 56 28) 24 Approximations of Lovász extensions and their induced interaction index Jean-Luc Marichal a, Pierre Mathonet b a Applied Mathematics Unit, University of Luxembourg, 62A, avenue de la Faïencerie, L-5 Luxembourg, Luxembourg b Department of Mathematics, University of Liège, Grande Traverse 2, B37, B-4 Liège, Belgium Received 8 October 26; received in revised form 26 June 27; accepted 3 July 27 Available online September 27 Abstract The Lovász extension of a pseudo-boolean function f :{, } n R is defined on each simplex of the standard triangulation of [, ] n as the unique affine function f ˆ :[, ] n R that interpolates f at the n + vertices of the simplex. Its degree is that of the unique multilinear polynomial that expresses f. In this paper we investigate the least squares approximation problem of an arbitrary Lovász extension fˆ by Lovász extensions of at most) a specified degree. We derive explicit expressions of these approximations. The corresponding approximation problem for pseudo-boolean functions was investigated by Hammer and Holzman [Approximations of pseudo-boolean functions; applications to game theory, Z. Oper. Res. 36) 992) 3 2] and then solved explicitly by Grabisch et al. [Equivalent representations of set functions, Math. Oper. Res. 252) 2) 57 78], giving rise to an alternative definition of Banzhaf interaction index. Similarly we introduce a new interaction index from approximations of fˆ and we present some of its properties. It turns out that its corresponding power index identifies with the power index introduced by Grabisch and Labreuche [How to improve acts: an alternative representation of the importance of criteria in MCDM, Internat. J. Uncertain. Fuzziness Knowledge-Based Syst. 92) 2) 45 57]. 27 Elsevier B.V. All rights reserved. Keywords: Pseudo-Boolean function; Lovász extension; Discrete Choquet integral; Least squares approximation; Interaction index. Introduction and problem setting Real valued set functions are extensively used in both cooperative game theory and multicriteria decision-making see for instance [] and the references therein). Indeed, in its characteristic form, a cooperative game on a finite set of players N : {,...,n}, is a set function v : 2 N R such that v ) which assigns to each coalition S of players a real number vs) representing the worth of S. In multicriteria decision-making, when N represents a set of criteria, a capacity also called non-additive measure)onn is a monotone set function v : 2 N R such that v ) and vn) which assigns to each subset S of criteria its weight vs). There is a natural identification of real valued set functions v : 2 N R with pseudo-boolean functions f : {, } n R through the equality vs) f S ), where S denotes the characteristic vector of S in {, } n whose ith component is if and only if i S). Moreover, it is known [5] that any pseudo-boolean function f :{, } n R addresses: jean-luc.marichal@uni.lu J.-L. Marichal), p.mathonet@ulg.ac.be P. Mathonet) X/$ - see front matter 27 Elsevier B.V. All rights reserved. doi:.6/j.dam.27.7.

2 2 J.-L. Marichal, P. Mathonet / Discrete Applied Mathematics 56 28) 24 has a unique expression as a multilinear polynomial in n variables fx) as) x i, ) S N i S where the set function a : 2 N R is simply the Möbius transform see for instance [2])ofv, which can be calculated through as) T S ) S T vt ). 2) Let S n denote the family of all permutations σ on N. The Lovász extension ˆ f :[, ] n R of any pseudo-boolean function f is defined on each n-simplex S σ : {x [, ] n x σ) x σn) } σ S n ), as the unique affine function which interpolates f at the n + vertices of S σ see [7, Section 3; 24, Section 2]). The concept of Lovász extension was initially used in combinatorial optimization [8,7,24] and then in cooperative game theory [2,]. In this paper we are mostly concerned by the interpretation of this concept in decision-making, where it was observed in [8] that the Lovász extension of a monotone pseudo-boolean function is nothing but a discrete Choquet integral [4,5]. Recall that discrete Choquet integrals were recently proposed in decision-making as aggregation functions that generalize the weighted arithmetic means by the taking into consideration of the interaction phenomena among input variables; see for instance [2,9]. As pointed out in Grabisch et al. [], the Lovász extension of any pseudo-boolean function f can be expressed as the following min-polynomial: ˆ fx) S N as) min i S x i. Its degree is that of the multilinear polynomial ) that expresses f. Hammer and Holzman [4] investigated the approximation of a pseudo-boolean function f by a multilinear polynomial of at most) a specified degree. More precisely, fixing k N, with k n, they defined the best kth approximation of f as the multilinear polynomial f k :{, } n R of degree at most k which minimizes x {,} n [fx) gx)] 2 among all multilinear polynomials g of degree at most k. They obtained an explicit form of the solution for k and 2. For a general k, the explicit form of the solution was obtained by Grabisch et al. [, Section 7]. It is written as where f k x) a k S) x i, S N i S S k a k S) as) + ) k+ S T S T >k T S k S 2 T S at ). 3) Hammer and Holzman [4] also noticed that a {i}) identifies with the classical Banzhaf power index [3] related to the ith player. More generally, Grabisch et al. [] noticed that, for any S N, the coefficient a S S) identifies with the Banzhaf interaction index [22] related to the coalition S. In this paper we investigate the least squares approximation of a given Lovász extension fˆ by a min-polynomial of at most) a specified degree. More precisely, fixing k N, with k n, we give explicit formulas see Section 2) for the

3 J.-L. Marichal, P. Mathonet / Discrete Applied Mathematics 56 28) 24 3 best kth approximation of f ˆ, that is, the min-polynomial fˆ k :[, ] n R of degree at most k which minimizes [ fx) ˆ ĝx)] 2 dx [,] n among all min-polynomials ĝ of degree at most k. It is important to emphasize that this least squares problem has a relevant application in multicriteria decision-making. Indeed, consider a multicriteria decision-making problem modeled by a Choquet integral and its underlying capacity on a set of n criteria. It is sometimes useful to approximate this Choquet integral which is, actually, the Lovász extension of the capacity) by simpler aggregation functions such as Lovász extensions of degree k k<n). Indeed this subclass of functions includes the Choquet integrals constructed from k-order capacities introduced by Grabisch [9]. By analogy with the Banzhaf interaction index obtained from the approximations of f, we show see Section 3) that the approximations of fˆ give rise to a new interaction index having appealing properties see for instance [7]). Furthermore, it turns out that its corresponding power index identifies with a power index introduced axiomatically by Grabisch and Labreuche []. In order to avoid a heavy notation, cardinalities of subsets S,T,... are denoted whenever possible by the corresponding lower case letters s,t,...,otherwise by the standard notation S, T,... Also, for any S N we define the function min S :[, ] n R as min S x) : min i S x i, with the convention that min. Finally, we denote by F N the family of set functions v : 2 N R. 2. Approximations of Lovász extensions We first recall the setting of our approximation problem. For any k {,...,n}, we denote by V k the set of all min-polynomials fˆ k :[, ] n R of degree at most k, i.e., of the form fˆ k x) a k S) min x i. i S S N S k Definition. Let fˆ V n and k {,...,n}. The best kth approximation of fˆ is the min-polynomial fˆ k V k that minimizes [ fx) ˆ ĝx)] 2 dx [,] n among all min-polynomials ĝ of V k. In this section we first discuss existence and uniqueness of the best kth approximation of any function ˆ f V n see Proposition 3). Then we provide a closed-form formula for this approximation see Theorem 9). We begin with the following straightforward proposition: Proposition 2. For any k {,...,n}, the set V k is a linear subspace of V n, which in turn is a linear subspace of the space C [, ] n ) of continuous functions on [, ] n. Moreover, the set B k : {min S S N, s k}, 4) forms a basis of V k and hence dimv k ) k n ) s s. Consider the usual inner product, of C [, ] n ) given by f,f 2 : f x)f 2 x) dx. [,] n

4 4 J.-L. Marichal, P. Mathonet / Discrete Applied Mathematics 56 28) 24 This clearly induces an inner product in V n, which in turn allows to define a norm f ˆ : ˆ f, f ˆ /2 and then a distance d fˆ, fˆ 2 ) : fˆ fˆ 2 in V n. According to this terminology, the best kth approximation of fˆ V n is the solution fˆ k of the following least squares problem min{ ˆ f ĝ 2 ĝ V k }, which amounts to a classical orthogonal projection problem. From now on, for any k {,...,n}, we denote by A k the orthogonal projector from V n onto V k. Proposition 3. For any fˆ V n and any k {,...,n}, the best kth approximation of fˆ exists and is uniquely given by fˆ k A k f). ˆ In particular, it is characterized as the unique element fˆ k of V k such that fˆ fˆ k is orthogonal to V k, i.e., to each element of the basis B k defined in 4). This projection problem can be simplified because of the following result, which immediately follows from the fact that the subspaces V k are nested: Proposition 4. The operators A k, k,...,n, are such that k k A k A k ˆ f)) A k ˆ f). 5) We thus observe that A k f)can ˆ be attained from fˆ by successively carrying out the projections A n,a n 2,...,A k. We will therefore search for the relation that links any two consecutive projections. This relation will enable us to compute gradually A k f)from ˆ f ˆ. Remark 5. Fix k {,...,n } and assume that fˆ k+ A k+ f)is ˆ given. By 5), A k f)is ˆ the orthogonal projection of A k+ f)onto ˆ V k and by Proposition 3, it is given as the unique solution fˆ k V k of the following system: f ˆk, min S f ˆk+, min S S N, s k). To solve this projection problem, we need the following two lemmas. The first one yields an explicit expression of the inner product min S, min T of any two elements of B n. The second lemma, which makes use of the first one, yields an explicit expression of the projections onto V k of the elements of B k+ \B k. Lemma 6. For any S,T N, there holds [,] n min S x)min T x) dx Proof. See Appendix A. Lemma 7. For any S N, with s, we have S T +2 s + + ). 6) t + k + t + A k min S ) T S ) k+t min T. 2k + 2 Proof. See Appendix B.

5 J.-L. Marichal, P. Mathonet / Discrete Applied Mathematics 56 28) 24 5 Proposition 8. Let k {,...,n }. Given the coefficients a k+ S) of A k+ f), ˆ the coefficients a k S) of A k f)are ˆ given by k + s + a k S) a k+ S) + ) k+s 2k + 2 T S tk+ a k+ T ). 7) Proof. Using formula 5) and then Lemma 7, we obtain A k f) ˆ A k A k+ f)) ˆ a k+ S) A k min S ) S N s k+ S N s k S N s k a k+ S) min S + S N sk+ a k+ S) min S + S N sk+ a k+ S) A k min S ) a k+ S) ) k+t T S k + t + min T, 2k + 2 which proves the result. We are now ready to derive the explicit form of the coefficients of A k f)in ˆ terms of the coefficients of f ˆ. It is worth comparing this formula with the solution 3) of the Hammer Holzman approximation problem. Theorem 9. Let k {,...,n}. The coefficients of A k ˆ f)are given from those of ˆ f by k + s + t s a k S) as) + ) k+s T S t>k k s k + t + at ). 8) Proof. We only have to prove that the coefficients a k S) given in 8) fulfill the recurrence relation 7). Let k {,...,n} and fix S N with s k. By substituting 8) into 7), we obtain, after removing the common term as), k + s + r s ) k+s R S r>k k s ar) k + r + k + s + 2 r s ) k+s+ R S r>k+ k + s + + ) k+s k + 2 k s + ar) k + r + 2 k + 2 ) 2k + 3 ) at ) + k + 2 ar) T S R T k + r + 2. tk+ r>k+ k + 2 2k + 2

6 6 J.-L. Marichal, P. Mathonet / Discrete Applied Mathematics 56 28) 24 Let us show that this equality holds. Dividing through by ) k+s and then removing the terms corresponding to ar), with r, the equality becomes k + s + r s k s ar) R S k + r + r>k+ k + s + 2 r s k + s + 2k + 3 r s k + 2 k s + ar) + k + 2 k s + ar). R S k + r + 2 R S 2k + 2 k + r + 2 r>k+ k + 2 r>k+ k + 2 Fix R S, with r>k+, and consider the coefficients of ar) in the previous equality. By equating them, we obtain an identity which can be easily checked. Example. Let ˆ f :[, ] 4 R be given by fx) ˆ 3 x + x 2 + x 3 + min{x,x 2 }+min{x,x 3 }+min{x 2,x 3 }) 2 25 min{x,x 2,x 3 }+ 25 min{x,x 2,x 3,x 4 }. The best constant approximation is given by A f)x) ˆ 25 37, the best linear approximation by A f)x) ˆ x + x 2 + x 3 ) + 25 x 4 and the best min-quadratic approximation by A 2 ˆ f)x) x + x 2 + x 3 ) x min{x,x 2 }+min{x,x 3 }+min{x 2,x 3 }) min{x,x 4 }+min{x 2,x 4 }+min{x 3,x 4 }). Before closing this section, we show that every approximation preserves the symmetry properties of the functions. For instance, in the previous example, we observe that the function fˆ and all its approximations are symmetric in the variables x, x 2, and x 3. Let us establish this result in a more general setting. Let X be a non-empty closed convex set in a finite-dimensional inner product space V. For any u V, the distance between u and X is achieved at a unique point in X. We denote this point by A X u) and call it the projection of u onto X see for instance [6, Chapter 3, Section 3.]). Lemma. Let I : V V be an isometry such that IX) X. Then I and A X commute, i.e., I A X A X I. Proof. We clearly have I[A X u)] X for any u V. Furthermore, for any v X, wehave I u) I[A X u)] u A X u) u v I u) Iv), which shows that I[A X u)] is the projection of I u) onto X. Let us now apply this result to our least squares approximation problem. Definition 2. For any σ in S n, we define the linear operator P σ of V n as P σ fx ˆ,...,x n ) : fx ˆ σ),...,x σn) ). We say that σ S n is a symmetry of ˆ f if P σ ˆ f) ˆ f.

7 J.-L. Marichal, P. Mathonet / Discrete Applied Mathematics 56 28) 24 7 Proposition 3. For any σ S n, the operator P σ is an isometry of V n such that P σ V k ) V k. In particular, the operators P σ and A k commute and A k preserves the symmetries of its arguments. Proof. It is clear that P σ V k ) V k and that P σ is an isometry, i.e., it satisfies P σ ˆ f,p σ ĝ ˆ f,ĝ f,ĝ ˆ V n ). Therefore, by Lemma, P σ and A k commute. In particular, if σ S n is a symmetry of f ˆ, then P σ [A k f)]a ˆ k [P σ f)]a ˆ k f) ˆ and hence σ is also a symmetry of A k f). ˆ 3. A new interaction index In cooperative game theory the concept of power index or value) was introduced in the pioneering work of Shapley [23]. Roughly speaking, a power index on N is a function : F N N R that assigns to every player i N in a game v F N his/her prospect v, i) from playing the game. The Shapley power index of a player i N in a game v F N is given by Sh v, i) : n [vt {i}) vt )]. n t T N\{i} Another frequently used power index is the Banzhaf power index [3,6] which, for a player i N in a game v F N, is defined by B v, i) : T N\{i} [vt {i}) vt )]. 2n The concept of interaction index, which is an extension of that of power index, was recently introduced axiomatically to measure the interaction phenomena among players.an interaction index on N is essentially a function I : F N 2 N R that assigns to every coalition S N of players in a game v F N its interaction degree. Various interaction indices have been introduced thus far in the literature: the Shapley interaction index [9], the Banzhaf interaction index [3,22], and the chaining interaction index [2], which all belong to the class of cardinal-probabilistic interaction indices see Definition 4 below) newly axiomatized in [7]. For instance, the Banzhaf interaction index on N, which extends the concept of Banzhaf power index on N, is the mapping I B : F N 2 N R defined by I B v, S) : T N\S 2 n s Δ SvT ), where Δ S vt ) is the S-derivative of v at T defined for any disjoint subsets S,T N by Δ S vt ) : R S ) s r vr T). 9) This index can be easily expressed in terms of the Möbius transform 2) of v as I B v, S) T S at ). 2t s It is noteworthy that, besides the axiomatic approach presented in [3,22], the Banzhaf interaction index can also be defined from the Hammer Holzman approximation problem. Indeed, as pointed out in [4;, Section 7], by considering the leading coefficients in 3) of the best sth approximation, for all s,...,n, we immediately observe that I B v, S) a s S).

8 8 J.-L. Marichal, P. Mathonet / Discrete Applied Mathematics 56 28) 24 In this section we use the same approach to define a new interaction index from our approximation problem of Lovász extensions. In this sense this new index can be seen as an analog of the Banzhaf interaction index. Before going on, we recall the concept of cardinal-probabilistic interaction index [7]. Definition 4. A cardinal-probabilistic interaction index on N is a mapping I p : F N 2 N R such that, for any S N, there is a family of non-negative real numbers {pt sn)} t,...,n s satisfying n s t ps t n), such that I p v, S) pt s n) Δ SvT ). ) T N\S It has been proved [7, Section 4] that a mapping I p : F N 2 N R of form ) is a cardinal-probabilistic interaction index on N if and only if, for any integer s {,...,n}, there exists a uniquely determined cumulative distribution function F s on [, ] such that pt s n) x t x) n s t df s x). ) Moreover, defining q s t : p s t s t) xt s df s x) for all integers s, t such that s t n, wehave I p v, S) qt s at ). 2) T S The following lemma provides conditions on arbitrary coefficients q s t so that a mapping of form 2) can be a cardinalprobabilistic interaction index. Lemma 5. Consider an infinite sequence {qt s} t s. Then the following three conditions are equivalent: i) we have m i ) i m) i q s t+i for all m and all t s, ii) there is a unique cumulative distribution function F s on [, ] such that, for any integer t s, q s t x t s df s x), iii) for any N {,...,n}, the function I p : F N 2 N R, defined in 2) is a cardinal-probabilistic interaction index. Moreover, when these conditions are satisfied, then the mapping I p : F N 2 N R, defined in 2), is of form ) and the coefficients qt s and pt sn) are linked through qs t pt s s t) and n pt s n) ) i s t n s t q s i s t i. is+t Proof. i) ii) Follows immediately from the Hausdorff s moment problem see for instance [, Theorem 2.6.4]). ii) iii) Rewriting 2) in terms of v gives I p v, S) qj s ) j l vl). J S L J Partitioning L J into R S and T J \S and then using 9), we obtain I p v, S) ) j s t q s j T N\S J S T R S ) s r vr T) pt s n)δ SvT ), T N\S

9 where the coefficients pt s n) are given by p s t n) n js+t J.-L. Marichal, P. Mathonet / Discrete Applied Mathematics 56 28) 24 9 ) j s t n s t q sj j s t x t x) n s t df s x), which shows that I p is a cardinal-probabilistic interaction index. iii) ii) According to formula 2), we have I p v, S) qt s at ) [ ] x t s df s x) at ), T S T S which completes the proof by uniqueness of this decomposition. We are now ready to introduce our new interaction index, namely the mapping I M : F N 2 N R : v, S) I M v, S) a s S), where a s S) are the leading coefficients in 8) of the best sth approximation of ˆ f, for all s,...,n. This immediately leads to the following equivalent definition, which makes use of the classical beta function Ba,b) : u a u) b du a,b>). Definition 6. For any N {,...,n}, we define the mapping I M : F N 2 N R as where q s t : I M v, S) qt s at ), T S 2s+ s+ s+t+ s+ ) Bt+,s+) Bs+,s+) for all t s. By using Lemma 5, we can easily show that I M is a cardinal-probabilistic interaction index. Proposition 7. The function I M : F N 2 N R is a cardinal-probabilistic interaction index, given by I M v, S) pt s n)δ SvT ), T N\S where pt s Bn t+,s+t+) n) : Bs+,s+). Proof. We immediately observe that Lemma 5 applies with the beta distribution x F s x) us u) s du Bs +,s+ ). Indeed, we have x t s df s x) x t x) s dx qt s Bs +,s+ ). The expression of the coefficients pt s n) then follows immediately from ). The following proposition shows that the mapping v I M v, S)) S N is invertible and yields the transformation formula from I M v, S)) S N to a as)) S N.

10 2 J.-L. Marichal, P. Mathonet / Discrete Applied Mathematics 56 28) 24 Proposition 8. For any S N, we have as) T S h s t I Mv, T ), with s + t h s t )t s t. 2t t Proof. We only need to show that h t r I Mv, R) I M v, S). qt s T S R T We have qt s T S h t r I Mv, R) I M v, R) R T R S qt s ht r, T :S T R where the inner sum equals if S R andifs R. Indeed, in the latter case, we have 2s + qt s ht r s + s + )! r ) r t r s r + t)! T :S T R 2r r! t s s + t + )! ts r 2s + s + s + )! r [ ) r t r s d r s ] xr+t 2r r! t s dxr s ts x r 2s + [ s + s + )! d r s ] 2r r! dx r s xr+s x ) r s x r. In order to conclude, we focus on the power index associated to I M. The following corollary shows that, incidentally, this power index identifies with the index W i f)previously ˆ introduced axiomatically by Grabisch and Labreuche [, Theorem 2] in the context of multicriteria decision-making. Corollary 9. The restriction of I M v, ) to singletons is given by I M v, {i}) T N\{i} Appendix A. Proof of Lemma 6 6n t)!t + )! [vt {i}) vt )]. n + 2)! Observe first that we can assume that S and T are such that S T n. Moreover, suppose that S and T are non-empty. Remark that [, ] n is equal almost everywhere to the disjoint union of the sets S σ : {x [, ]n x σ) < <x σn) } σ S n ).

11 Then we have where I σ : [,] n min S x) min T x) dx S σ J.-L. Marichal, P. Mathonet / Discrete Applied Mathematics 56 28) 24 2 min S x) min T x) dx σ S n ). Now, define for every R N and p {,..., R } the set σ S n I σ, 3) S p) n R) : {σ S n σ),...,σp) R, σp + ) / R}. It is then clear that there is a decomposition of S n into disjoint subsets where n t S n [S n S T)] p S p) n n s S\T) S n S T): {σ S n σ) S T }. Moreover we have { Sn S T) S T n ) )! S p) n t n S\T) p!tn p )! p S p) n p T \S), 4) Indeed, every element of S n S T)corresponds to a choice of σ) in S T, and a permutation of the remaining elements in N. In the same fashion, every element of S p) n S\T)is determined by the choice of p elements in S\T, a permutation of these elements, the choice of a single element in T and a permutation of the remaining elements. Moreover, for any σ S n S T),wehave and I σ xσn) σ S n S T) I σ xσ2) xσ) 2 dx σ) dx σn) 2 S T n )! n + 2)! In the same way, for any σ S p) n S\T),wehave I σ xσn) p + 2 n + 2)!, 2 n + 2)!, 2s + t n) nn + )n + 2). 5) xσp+2) xσp+) xσ2) x σp+) x σ) dx σ) dx σn)

12 22 J.-L. Marichal, P. Mathonet / Discrete Applied Mathematics 56 28) 24 and that is, n t p n t p σ S p) n S\T) I σ σ S p) n S\T) I σ Similarly, we can write t n t n + 2)! t!n t)! n + 2)! t!n t)! n + 2)! t!n t)! n + 2)! p n t p n t p!n p )!p + 2) p n p p + 2) t n t n p n + 2) t p [ n n + 2) t t ) n t t n t + p )], n p t n t)n + 2t + 2) nn + )n + 2)t + ). 6) n s p σ S p) n T \S) I σ n s)n + 2s + 2) nn + )n + 2)s + ) 7) and by Eqs. 3) 7), we have [,] n min S x) min T x) dx σ S n S T) n t I σ + p σ S p) s + t + 2 n + 2)s + )t + ) n S\T) I σ + as desired. We can easily see that the result still holds if S or T. n s p I σ σ S p) n T \S) Appendix B. Proof of Lemma 7 We only have to prove that the explicit expression we give for A k min S ) fulfills the conditions of Proposition 3, namely, min S A k min S ), min T T N,t k), or, equivalently, J S I J,T ) k+j k + j + T N, t k), 8) where I J,T : min J, min T is given explicitly in 6).

13 J.-L. Marichal, P. Mathonet / Discrete Applied Mathematics 56 28) Partitioning J S into P S\T and Q R with R : S T hence r t k), the left-hand side of 8) becomes I P Q,T ) k+p+q k + p + q + P S\T Q R k r+ p k r+ p r k r + p k r + p q r q p + q + t + 2 p + t + 2)p + q + )t + ) )k+p+q ) ) k r+ p p + t + 2 where g : N R is defined by gz) z + t + 2 z + z + )t + ) z + + z +. k t + Consider the difference operator Δ n f n) : fn+ ) f n) r q r ) r q gp + q), q k + p + q + for functions on N. It is well known that we have k Δ k k n f n) ) k j fn+ j) k N). 9) j j Applying this to g, we obtain r q r ) r q gp + q) Δ r q p gp) and the left-hand side of 8) becomes k r+ p k r + p ) ) k r+ p p + t + 2 p + + k r t + p + + k r t + Applying 9) again, we see that this latter expression can be written as [ Δ k r+ z + z + )] z + z + t + 2 k r t + k r +. z p + k r + ) p +. k r + We now show that the expression in brackets is identically zero. We do this by considering two cases: If t k then z + t + 2 z + k r )z + k + 2) ) + z + k + 2 k r + ) z + t + k r + ) )k r + ) is a polynomial P k r z) of degree k r, and Δ k r+ z P k r z). z + k r

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