Counting Polynomials with Distinct Zeros in Finite Fields
aa r X i v : . [ m a t h . N T ] F e b Counting Polynomials with Distinct Zeros in FiniteFields ∗ Haiyan Zhou a Li-Ping Wang b Weiqiong Wang c † a. School of Mathematics, Nanjing Normal University, Nanjing 210023, ChinaEmail: [email protected] b. Institute of Information Engineering, Chinese Academy of Sciences Beijing 100093, ChinaEmail: [email protected] c. School of Science, Chan’an University, Xi’an 710064, ChinaEmail:[email protected]
Abstract
Let F q be a finite field with q = p e elements, where p is a prime and e ≥ ℓ < n be two positive integers. Fix a monic polynomial u ( x ) = x n + u n − x n − + · · · + u ℓ +1 x ℓ +1 ∈ F q [ x ]of degree n and consider all degree n monic polynomials of the form f ( x ) = u ( x ) + v ℓ ( x ) , v ℓ ( x ) = a ℓ x ℓ + a ℓ − x ℓ − + · · · + a x + a ∈ F q [ x ] . For integer 0 ≤ k ≤ min { n, q } , let N k ( u ( x ) , ℓ ) denote the total number of v ℓ ( x ) such that u ( x ) + v ℓ ( x )has exactly k distinct roots in F q , i.e. N k ( u ( x ) , ℓ ) = |{ f ( x ) = u ( x ) + v l ( x ) | f ( x ) has exactly k distinct zeros in F q }| . In this paper, we obtain explicit combinatorial formulae for N k ( u ( x ) , ℓ ) when n − ℓ is small, namelywhen n − ℓ = 1 , ,
3. As an application, we define two kinds of Wenger graphs called jumped Wengergraphs and obtain their explicit spectrum.
Key words
Polynomials, Inclusion-Exclusion Principal, Moments Subset-Sum, Distinct CoordinateSieve, Spectrum of Graphs
Let F q be a finite field with q = p e elements, where p is a prime and e ≥ ℓ < n be two positive integers. Fix a monic polynomial u ( x ) = x n + u n − x n − + · · · + u ℓ +1 x ℓ +1 ∈ F q [ x ] of degree n and consider all degree n monicpolynomials of the form f ( x ) = u ( x ) + v ℓ ( x ) , v ℓ ( x ) = a ℓ x ℓ + a ℓ − x ℓ − + · · · + a x + a ∈ F q [ x ] . ∗ Research is supported in part by 973 Program (2013CB834203), National Natural Science Foun-dation of China under Grant No.61202437 and 11471162, in part by Natural Science Basic ResearchPlan in Shaanxi Province of China under Grant No.2015JM1022 and Natural Science Foundation ofthe Jiangsu Higher Education Institutes of China under Grant No.13KJB110016.
1e are interesting in the number of distinct roots in F q of f ( x ) as the lower degree part v ℓ ( x ) varies. Since a q = a for all a ∈ F q , we can reduce the polynomial u ( x ) modulo x q − x . In this way, we can and will assume that n < q .It is clear that f ( x ) has at most n distinct zeros in F q . For integer 0 ≤ k ≤ n ,let N k ( u ( x ) , ℓ ) denote the total number of v ℓ ( x ) such that u ( x ) + v ℓ ( x ) has exactly k distinct roots in F q , i.e. N k ( u ( x ) , ℓ ) = |{ f ( x ) = u ( x ) + v l ( x ) | f ( x ) has exactly k distinct zeros in F q }| . Understanding N k ( u ( x ) , ℓ ) is an important number theoretical problem with a widerange of applications. For example, in coding theory, v ℓ ( x ) represents a code word inthe [ q, ℓ + 1] q Reed-Solomon codes and u ( x ) represents a received word. The number N k ( u ( x ) , ℓ ) is then the number of code words whose distance to the received word u ( x )is precisely q − k . Determining the largest k such that N k ( u ( x ) , ℓ ) > u ( x ) to the code, which is themost important problem in decoding Reed-Solomon codes, see [11][14][23] for variouspartial results. In the special case k = n (the polynomial f ( x ) splits as a product of n distinct linear factors), the possible large size for N n ( u ( x ) , ℓ ) is the key to prove severalcomplexity results in decoding primitive Reed-Solomon codes [3] and in approximat-ing the minimum distance of linear codes [4]. For another example, in graph theory, N k ( u ( x ) , ℓ ) represents the multiplicity of certain eigenvalue in an important class of al-gebraic graphs extending the classical Wenger graph, see [2]. Deciding this multiplicityis a difficult problem in general.Mathematically, the number N k ( u ( x ) , ℓ ) becomes increasingly more complicated as n − ℓ grows. Thus, we cannot expect an explicit formula for N k ( u ( x ) , ℓ ) if n − ℓ islarge. In the simplest case n − ℓ = 1, A. Knopfmacher and J. Knopfmacher derived anexplicit combinatorial formula for N k ( u ( x ) , ℓ ) in [7]. Using this formula, S.M. Cioab˘a,F. Lazebnik and W. Li obtained the explicit spectrum of Wenger graphs in [5]. In thispaper, we obtain explicit combinatorial formulae for N k ( u ( x ) , ℓ ) when n − ℓ is small,namely when n − ℓ = 1 , ,
3. In the case n − ℓ = 1, we give a simple proof as a simpleapplication of the inclusion-exclusion principal. In the case n − ℓ = 2, we use theinclusion-exclusion principe together with the subset sum result in [12]. In the case n − ℓ = 3, it is more complicated. When k = n , this is an extension of the subset sumproblem up to 2 moments(called Moments Subset-Sum with parameter 2). For a fixed d ≥
1, the Moments Subset-Sum with parameter d is formally defined as follows, see[6]: Moments Subset-Sum( MSS(d)):
Given a set A = { a , · · · , a n } , a i ∈ F q , integer t , and m , · · · , m d ∈ F q , decide if there exists a subset S ⊆ A of size t , satisfying P a ∈ S a i = m i for all 1 ≤ i ≤ d .Note that MSS(1) is the usual subset sum problem and it is well-known for theNP-hardness of subset sum problem. However, it turns out to be much more difficultto prove NP-hardness for MSS(d) for d ≥
2. In 2015, V. Gandikota, B. Ghazi and E.Grigorescu proved the NP-hardness for MSS(d) for d = 2 ,
3, see [6]. Surprisingly, whenthe degree of the extension F q / F p is even, A = F q , t = n, m i = 0 and d = 2, we obtainn explicit combinatorial formula for the number of S , i.e., Theorem 4.1, employing themore advanced sieving formula from [13] together with results on quadratic equationsover finite fields. Finally, we get explicit combinatorial formulae for N k ( x n , n − n − ℓ = 2 ,
3, the smallest k such that N k ( u ( x ) , ℓ ) > N k ( u ( x ) , ℓ ). n − ℓ = 1 In this simplest case, using the generating function over an additive arithmeticalsemigroup, A. Knopfmacher and J. Knopfmacher obtained an explicit combinatorialformula for N k ( u ( x ) , ℓ ) in [7]. Here, we would give the simple proof according to theclassical inclusion-exclusion principal. We recall it briefly.Let S be a finite set of objects and let P , P , · · · , P m be m properties refer-ring to the objects in S . Let I ⊆ { P , · · · , P m } . Define S ∅ = S and S I = { x ∈ S | x satifies all properties in I } for I = ∅ . For any non-negative integer j , we put S j = { x ∈ S | x satifies exactly j properts of { P , · · · , P m }} . It is well-known thatthe classical inclusion-exclusion principal implies | S j | = X | I | = j | S I | − X | I | = j +1 | S I | − · · · + ( − n − j X | I | = n | S I | , where j = 0 . It is worth mentioning that the above formula doesn’t work for j ≥
1. For example,let S be the set consisting of all monic polynomials over F with degree 3, P theproperty that the monic polynomial in S has a zero 0, and P the property thatthe monic polynomial in S has a zero 1. Then it is easy to compute | S | = 4 and P | I | =1 | S I | − P | I | =2 | S I | = 6. Theorem 2.1. N k ( x n , n −
1) = q n − k (cid:18) qk (cid:19) n − k X i =0 ( − i (cid:18) q − ki (cid:19) q − i . Proof.
Let f ( x ) = x n + a x n − + a x n − + · · · + a n and c , c , · · · , c k be k distinctroots of f ( x ) in F q . Then there exists a polynomial g ( x ) ∈ F q [ x ] such that f ( x ) =( x − c )( x − c ) · · · ( x − c k ) g ( x ). For a fixed k -subset J = { c , · · · , c k } ⊂ F q and asubset I ⊆ F q − J , we define the set S I ( J ) = { g ( x ) ∈ F q [ x ] | g ( α ) = 0 for all α ∈ I } . For 0 ≤ | I | < n − k , it is obvious to obtain that the cardinality of the set S I ( J ) is q n − k −| I | . It follows that for 0 ≤ i < n − k , X I ⊆ F q − J, | I | = i | S I ( J ) | = (cid:18) q − ki (cid:19) q n − k − i . or | I | > n − k , it is clear that | S I ( J ) | = 0. By the inclusion-exclusion principle, wededuce that N k ( x n , n −
1) = P | J | = k P I ⊂ F q − J ( − | I | | S I ( J ) | = n − k P i =0 ( − i P | J | = k P I ⊂ F q − J, | I | = i | S I ( J ) | = q n − k (cid:18) qk (cid:19) n − k P i =0 ( − i (cid:18) q − ki (cid:19) q − i . Remark 2.1.
If the degree of f ( x ) = x n + a x n − + a x n − + · · · + a n is greater than q − , i.e., n ≥ q , then by the Euclidean division, there exist g ( x ) , h ( x ) ∈ F q [ x ] suchthat f ( x ) = ( x q − x ) g ( x ) + h ( x ) , where g ( x ) is the monic polynomial with degree n − q and the degree of h ( x ) is less than q or h ( x ) = 0 . If f ( x ) has exactly k ≤ q − distinctroots in F q , then h ( x ) has also exactly k distinct roots in F q . So we obtain N k ( x n , n −
1) = q n − q ( q − q − X r = k q r − k (cid:18) qk (cid:19) r − k X i =0 ( − i (cid:18) q − ki (cid:19) q − i . Since q − P r = k q r − k r − k P i =0 ( − i (cid:18) q − ki (cid:19) q − i = 1 + q P i =0 ( − i (cid:0) q − ki (cid:1) q − i + · · · + q q − − k q − − k P i =0 ( − i (cid:0) q − ki (cid:1) q − i = − q q − k − P i =0 ( − i (cid:0) q − ki (cid:1) (1 − q q − k − i )= ( q − q − k − , we have N k ( x n , n −
1) = (cid:18) qk (cid:19) q n − q ( q − q − k . If k = q , then h ( x ) = 0 . Then N k ( x n , n −
1) = q n − q . Hence for n ≥ q , N k ( x n , n −
1) = (cid:18) qk (cid:19) q n − q ( q − q − k . n − ℓ = 2 In the special case that k = n , this is the counting version for the n -subset sumproblem over F q , which is already handled in [12]. We state this result as a lemma andwill use it in our proof. emma 3.1. (See [ ] ) For b ∈ F q , let M ( n, b ) be the number of n -subsets of F q whoseelements sum to b. If p ∤ n , then M ( n, b ) = 1 q (cid:18) qn (cid:19) . If p | n , then M ( n, b ) = 1 q (cid:18) qn (cid:19) + ( − n + np v ( b ) q (cid:18) q/pn/p (cid:19) , where v ( b ) = − if b = 0 , and v ( b ) = q − if b = 0 . In terms of our earlier notations, we have M ( n, b ) = N n ( x n − bx n − , n − Theorem 3.1. (i) If p ∤ n , then N k ( x n − bx n − , n −
2) = q n − k − (cid:18) qk (cid:19) n − k X i =0 ( − i (cid:18) q − ki (cid:19) q − i . (ii) If p | n , then N k ( x n − bx n − , n −
2) = q n − k − (cid:18) qk (cid:19) n − k P i =0 ( − i (cid:18) q − ki (cid:19) q − i +( − np + n v ( b ) q (cid:18) nk (cid:19) (cid:18) q/pn/p (cid:19) . Proof.
Let c , c , · · · , c k be k distinct roots of f ( x ) in F q . Then there exists a polynomial g ( x ) = x n − k + d x n − k − + · · · + d n − k − x + d n − k ∈ F q [ x ]such that f ( x ) = x n − bx n − + a x n − + · · · + a n = ( x − c )( x − c ) · · · ( x − c k ) g ( x )= ( x k + b x k − + · · · + b k − x + b k )( x n − k + d x n − k − + · · · + d n − k − x + d n − k ) . Comparing the coefficients, we have b + d = − bb + b d + d = a b + b d + b d + d = a · · · · · · · · · b k d n − k = a n For fixed b and fixed k -subset J = { c , · · · , c k } ⊂ F q , the coefficient d = − ( b + c + c + · · · + c k ) is then fixed. The other coefficients { d , · · · , d n − k } of the polynomial g ( x )are free since { a , · · · , a n } are free.or a subset I ⊆ F q − J , define the set S I ( J ) = (cid:8) g ( x ) = x n − k + d x n − k − + · · · + d n − k ∈ F q [ x ] | d = b − ( c + c + · · · + c k ) ,g ( α ) = 0 for all α ∈ I } . For 0 ≤ | I | < n − k , the above argument shows that the cardinality of the set S I ( J ) is q n − k − −| I | . It follows that for 0 ≤ i < n − k , X I ⊆ F q − J, | I | = i | S I ( J ) | = (cid:18) q − ki (cid:19) q n − k − − i . When i = n − k , f ( x ) is forced to have n distinct roots in F q with sum equal to b .Then by Lemma 3.1, we deduce X | J | = k X I ⊆ F q − J, | I | = n − k | S I ( J ) | = (cid:18) nk (cid:19) q (cid:18) qn (cid:19) , p ∤ n, (cid:18) nk (cid:19) (cid:20) q (cid:18) qn (cid:19) + ( − n + np v ( b ) q (cid:18) q/pn/p (cid:19)(cid:21) , p | n, For | I | > n − k , it is clear that | S I ( J ) | = 0. By the inclusion-exclusion principle, wededuce that N k ( x n − bx n − , n −
2) = X | J | = k X I ⊂ F q − J ( − | I | | S I ( J ) | = n − k X i =0 ( − i X | J | = k X I ⊂ F q − J, | I | = i | S I ( J ) | . Hence, N k ( x n − bx n − , n −
2) = (cid:18) qk (cid:19) n − k − P i =0 ( − i (cid:18) q − ki (cid:19) q n − k − − i +( − n − k (cid:18) nk (cid:19) q (cid:18) qn (cid:19) , p ∤ n, (cid:18) qk (cid:19) n − k − P i =0 ( − i (cid:18) q − ki (cid:19) q n − k − − i +( − n − k (cid:18) nk (cid:19) (cid:20) q (cid:18) qn (cid:19) + ( − n + np v ( b ) q (cid:18) q/pn/p (cid:19)(cid:21) , p | n. This Theorem is proved from the fact (cid:18) qk (cid:19) (cid:18) q − kn − k (cid:19) = (cid:18) nk (cid:19) (cid:18) qn (cid:19) . Remark 3.1.
For n ≥ q , we can deduce the formula of N k ( x n − bx n − , n − . If n > q , then n − > q − . Similar arguments to those used in the Remark . show that N k ( x n − bx n − , n −
2) = q n − q − (cid:18) qk (cid:19) ( q − q − k . ) If n = q , then f ( x ) = x q − x − bx q − + a x q − + · · · + a q − x + ( a q − + 1) x + a q .It is easy to get the following conclusions:When b = 0 , N k ( x q − bx q − , q −
2) = ( k = q, q (cid:0) qk (cid:1) (( q − q − k − ( − q − k ) k ≤ q − . When b = 0 , N k ( x q − bx q − , q −
2) = k = q, k = q − , q − q (cid:0) qk (cid:1) (( q − q − k − + ( − q − k ) k ≤ q − . n − ℓ = 3 In this section, we always assume that q is an odd number. Let F nq be the Cartesianproduct of n copies of F q . For convenience, we firstly state some results on the numberof common solutions in F nq of the equations ( a x + · · · + a n x n = a ,b x + · · · + b n x n = b , (4.1)where a , b , b , · · · , b n ∈ F q , a , · · · , a n ∈ F ∗ q , b i = 0 for at least one i , 1 ≤ i ≤ n, seeExercises 6.31-6.34 in [16]. Lemma 4.1.
Denote by N ( n, a , b ) the number of common solutions in F nq of theequations (4.1) . Put a = a a · · · a n , b = b a − + · · · + b n a − n , c = b − a b . Then i ) For b = 0 , c = 0 ,N ( n, a , b ) = { q n − if n even ,q n − + q ( n − / ( q − χ (( − ( n − / ab ) if n odd , where χ is the quadratic character of F q .ii ) For b = 0 , c = 0 ,N ( n, a , b ) = { q n − + q ( n − / χ (( − n/ ac ) if n even ,q n − − q ( n − / χ (( − ( n − / ab ) if n odd .iii ) For b = c = 0 ,N ( n, a , b ) = { q n − + v ( a ) q ( n − / χ (( − n/ a ) if n even ,q n − + q ( n − / χ (( − ( n − / a a ) if n odd , where v is as in Lemma 3.1. iv ) For b = 0 , c = 0 , N ( n, a , b ) = q n − . ow, we begin to recall a sieve for distinct coordinate counting, see [13]. Let X bea subset of F nq . Motivated by diverse applications in coding theory and graph theory,it is very interesting to count the number of elements in the set X = { ( x , · · · , x n ) ∈ X | x i = x j , ∀ i = j } . In [13], J. Li and D. Wan discovered the new sieving formula about | X | . Let S n be thesymmetric group on { , , · · · , n } . For a given permutation τ = ( i i · · · i t ) · · · ( l l · · · l t s )with t i ≥ , ≤ i ≤ s , define X τ = { ( x , · · · , x n ) ∈ X | x i = · · · = x i t , x l = · · · = x l ts } . Now the symmetric group S n acts on F nq by permuting coordinates. That is, for given τ ∈ S n and x = ( x , · · · , x n ) ∈ F nq , we have τ ◦ x = ( x τ (1) , · · · , x τ ( n ) ) ∈ X.X is called symmetric if any x ∈ X and any τ ∈ S n , τ ◦ x ∈ X . Furthermore, If X satisfies the ”strongly symmetric” condition, that is, for any τ and σ in S n , one has | X τ | = | X σ | provided l ( τ ) and l ( σ ), then we call X a strongly symmetric set. Let C n be the set of conjugacy classes of S n . If X is symmetric, then | X | = X τ ∈ C n ( − n − l ( τ ) C ( τ ) | X τ | , where C ( τ ) is the number of permutations conjugate to τ and l ( τ ) is the number ofcycles including the trivial cycle.A permutation τ ∈ S n is said to be of type ( c , c , · · · , c n ) if τ has exactly c i cyclesof length i . We denote by N ( c , c , · · · , c n ) the number of permutations in S k of type( c , c , · · · , c n ) and we have (see [19]), N ( c , c , · · · , c n ) = n !1 c c !2 c c ! · · · n c n c n ! . Since two permutations are conjugate if and only if they have the same type, we have C ( τ ) = N ( c , c , · · · , c n ) . Lemma 4.2.
Put S + ( n ) = X n P i =1 ic i = n, P p ∤ i c i is even N ( c , c , · · · , c n ) Y p | i ( − q ) c i Y p ∤ i ( −√ q ) c i ,S − ( n ) = X n P i =1 ic i = n, P p ∤ i c i is odd N ( c , c , · · · , c n ) Y p | i ( − q ) c i Y p ∤ i ( −√ q ) c i . Then + ( n ) = n !2 (( − n α ( n ) + β ( n )) ,S − ( n ) = n !2 (( − n α ( n ) − β ( n )) , where α ( n ) = X i + pj = n, ≤ i ≤√ q (cid:18) √ qi (cid:19) q −√ qp j ! and β ( n ) = X i + pj = n,i ≥ ( − j (cid:18) √ q − i √ q − (cid:19) q + √ qp j ! . Proof.
Define the generating function C n ( t , t , · · · , t n ) = X n P i =1 ic i = n N ( c , c , · · · , c n ) t c t c · · · t c n n . Then we get the following exponential generating function X n ≥ C n ( t , t , · · · , t n ) u n n ! = e ut + u t + u t + ··· . For given generating function f ( x ), denote by [ x i ] f ( x ) the coefficient of x i in the formalpower series expansion of f ( x ).1) If t i = −√ q for p ∤ i and t i = − q for p | i , then we have C n ( −√ q, · · · , −√ q, − q, −√ q, · · · , −√ q, − q, · · · )= [ u n n ! ] e −√ q ( u + u + u + ··· )+ − q + √ qp ( u p + u p + u p + ··· ) = [ u n n ! ] e √ q ln(1 − u )+ q −√ qp ln(1 − u p ) = [ u n n ! ](1 − u ) √ q (1 − u p ) q −√ qp = [ u n n ! ]( P i ≥ ( − i (cid:18) √ qi (cid:19) u i )( P j ≥ ( − j q −√ qp j ! u pj )= n ! P i + pj = n, ≤ i ≤√ q ( − n (cid:18) √ qi (cid:19) q −√ qp j ! . Similarly, if t i = √ q for p ∤ i and t i = − q for p | i , then we have C n ( √ q, · · · , √ q, − q, √ q, · · · ) = n ! X i + pj = n,i ≥ ( − j (cid:18) √ q − i √ q − (cid:19) q + √ qp j ! . Thus, this lemma is proved from S + ( n ) = C n ( −√ q, · · · , −√ q, − q, −√ q, · · · ) + C n ( √ q, · · · , √ q, − q, √ q, · · · )2 , − ( n ) = C n ( −√ q, · · · , −√ q, − q, −√ q, · · · ) − C n ( √ q, · · · , √ q, − q, √ q, · · · )2 . Lemma 4.3.
Let q = p r with p = 2 and χ be the quadratic character of F q . Then χ | F p is the trivial character of F p if and only if r is even.Proof. Let g be a primitive element of F p . Then χ | F p is the trivial character of F p ifand only if χ ( g ) = 1, that is, x − g = 0 has a root γ in F q . Therefore, F p ( γ ) ⊆ F q ,i.e., r is even. Theorem 4.1.
Let q = p e . Denote by M ( n, , the number of n -subsets of F q whoseelements are the solutions of the equations ( x + · · · + x n = 0 ,x + · · · + x n = 0 . (4.2) i ) For p ∤ n , M ( n, ,
0) = 1 q (cid:18) qn (cid:19) + q − p q ( α ( n ) − ( − n β ( n )) .ii ) For p | n , M ( n, ,
0) = 1 q (cid:18) qn (cid:19) + q − q (cid:18) q/pn/p (cid:19) + q − q ( α ( n ) + ( − n β ( n )) . Proof.
Let X be the set of all solutions of the equations (4.2). Then X is symmetric,so we have n ! M ( n, ,
0) = X τ ∈ C n ( − n − l ( τ ) C ( τ ) | X τ | . For a type ( c , c , · · · , c n ) permutation τ , we have n P i =1 ic i = n and l ( τ ) = n P i =1 c i . Denoteby r the number of the cycles of τ such that the length of it is divisible by p , anddenote by s the number of the cycles of τ such that the length of it is not divisible by p . Note that r + s = l ( τ ) and P p ∤ i ic i ≡ n (mod p ). i ) Since p ∤ n , we have s ≥
1. If s = 1, then | X τ | = q l ( τ ) − . If s ≥
2, then by i ) ofLemma 4 .
1, we have | X τ | = q r q s − if s is even ,q r ( q s − + q ( s − / ( q − χ (( − ( s − / Q p ∤ i i c i P p ∤ i ic i ) if s is odd , Since q = p e , by Lemma 4 . | X τ | = q l ( τ ) − if s is even ,q l ( τ ) − + ( q − q − Q p | i q c i Q p ∤ i √ q c i if s is odd , herefore, we have M ( n, ,
0) = n ! P P i ic i = n,s ≥ ( − n − l ( τ ) C ( τ ) q l ( τ ) − + n ! P P i ic i = n,s =1 ( − n − l ( τ ) C ( τ ) q l ( τ ) − + n ! ( q − q − P P i ic i = n,s> ( − n − l ( τ ) C ( τ ) Q p | i q c i Q p ∤ i √ q c i = n ! n P i =1 ( − n − i c ( n, i ) q i − + n ! ( q − q − P P i ic i = n,s is odd ( − n − l ( τ ) C ( τ ) Q p | i q c i Q p ∤ i √ q c i = q (cid:18) qn (cid:19) + n ! ( − n ( q − q − S − ( n )= q (cid:18) qn (cid:19) + q − √ q ( α ( n ) − ( − n β ( n )) .ii ) Since p | n , we have s = 1. If s = 0, then | X τ | = q l ( τ ) . Denote by CP n theconjugacy classes in C n whose every cycle length is divisible by p , and denote by p ( n, i )the number of permutations in S n of i cycles with the length of its each cycle divisibleby p . If s >
0, then by iii ) of Lemma 4 . . M ( n, ,
0) = n ! P τ / ∈ CP n ( − n − l ( τ ) C ( τ ) | X τ | + n ! P τ ∈ CP n ( − n − l ( τ ) C ( τ ) | X τ | = n ! P τ / ∈ CP n ( − n − l ( τ ) C ( τ ) | X τ | + n ! n P i =1 ( − n − i p ( n, i ) q i = n ! n P i =1 ( − n − i ( c ( n, i ) − p ( n, i )) q i − + n ! n P i =1 ( − n − i p ( n, i ) q i + n ! ( q − q − P s> ( − n − l ( τ ) C ( τ ) Q p | i q c i Q p ∤ i √ q c i = n ! n P i =1 ( − n − i c ( n, i ) q i − + n ! q − q n P i =1 ( − n − i p ( n, i ) q i + n ! ( q − q − P s is even ( − n − l ( τ ) C ( τ ) Q p | i q c i Q p ∤ i √ q c i . Recall that n P i =1 ( − n − i p ( n, i ) q i = ( − n + np n ! (cid:18) q/pn/p (cid:19) (See Lemma 3.1 in [13]) and p is an odd prime number. Therefore, M ( n, ,
0) = q (cid:18) qn (cid:19) + q − q (cid:18) q/pn/p (cid:19) + n ! q − q S + ( n )= q (cid:18) qn (cid:19) + q − q (cid:18) q/pn/p (cid:19) + q − q ( α ( n ) + ( − n β ( n )) . Corollary 4.1.
Let q = p e . Denote by M ′ ( n, , the number of n -subsets of F q whoseelements are the common solutions of the equations P ≤ i 0) = M ( n, , Proof. This result follows from the fact that the equations (4.3) are equivalent to theequations (4.2).Let X be the set of all solutions of the equations (4.3) in F nq . Put X = { ( x , · · · , x n ) ∈ X | x i = x j , ∀ ≤ i = j ≤ n − } . Now the symmetric group S n − acts on F nq by permuting the first n − | X | show that | X | = X τ ∈ C n − ( − n − − l ( τ ) C ( τ ) | X τ | , where C ( τ ) is the number of permutations conjugate to τ ∈ S n − and l ( τ ) is thenumber of cycles including the trivial cycle. Denote by M ( n, , 0) the number of n -subsets { x , x , · · · , x n | x i = x j , ∀ ≤ i = j ≤ n − } of F q whose elements are thecommon solutions of the equations (4.3). The similar proof of Theorem 4 . Theorem 4.2. Let q = p e . Then i ) For p ∤ n , M ( n, , 0) = 1 q (cid:18) qn − (cid:19) + q − q ( α ( n − 1) + ( − n − β ( n − .ii ) For p | n , M ( n, , 0) = 1 q (cid:18) qn − (cid:19) + q − √ q ( α ( n − − ( − n − β ( n − , where α ( n − 1) = X i + pj = n − , ≤ i ≤√ q (cid:18) √ qi (cid:19) q −√ qp j ! and β ( n − 1) = X i + pj = n − ,i ≥ ( − j (cid:18) √ q − i √ q − (cid:19) q + √ qp j ! . heorem 4.3. Let q = p e and k ≤ n − .(i) If p ∤ n , then N k ( x n , n − 3) = q n − k − (cid:18) qk (cid:19) n − k P i =0 ( − i (cid:18) q − ki (cid:19) q − i +( − n − k − (cid:18) n − k (cid:19) q − q D ( n − 1) + ( − n − k (cid:18) nk (cid:19) q − √ q D ( n ) , where D ( n − 1) = α ( n − 1) + ( − n − β ( n − and D ( n ) = α ( n ) − ( − n β ( n ) .(ii) If p | n , then N k ( x n , n − 3) = q n − k − (cid:18) qk (cid:19) n − k P i =0 ( − i (cid:18) q − ki (cid:19) q − i + ( − n − k q − q (cid:18) nk (cid:19) (cid:18) q/pn/p (cid:19) +( − n − k − (cid:18) n − k (cid:19) q − √ q P ( n − 1) + ( − n − k (cid:18) nk (cid:19) q − q P ( n ) , where P ( n − 1) = α ( n − − ( − n − β ( n − and P ( n ) = α ( n ) + ( − n β ( n ) .Proof. Note that (cid:18) n − k (cid:19) (cid:18) qn − (cid:19) = (cid:18) qk (cid:19) (cid:18) q − kn − k − (cid:19) and (cid:18) nk (cid:19) (cid:18) qn (cid:19) = (cid:18) qk (cid:19) (cid:18) q − kn − k (cid:19) . This result follows from the similar proof of Theorem 3 . . Remark 4.1. In the special case k = n , N k ( x n , n − 3) = M ( n, , . For n ≥ q , the formulae of N k ( x n , n − can be obtained from the similararguments of Remark . . i ) If n > q + 1 , then N k ( x n , n − 3) = q n − q − (cid:0) qk (cid:1) ( q − q − k .ii ) If n = q + 1 , then N q ( x n , n − 3) = k = q, k = q − , q − q (cid:0) qk (cid:1) (( q − q − k − + ( − q − k ) k ≤ q − .ii ) If n = q , then N q ( x n , n − 3) = k = q, k = q − , q − q − q (cid:0) qk (cid:1) ( ( q − q − k − q + ( − q − k − ( q − k ) + ( − q − k q +1 q ) k ≤ q − . The spectrum of some jumped Wenger graphs In [22], Wenger introduced a family of p -regular bipartite graphs and then Lazebnikand Ustimenko arrived at a family of bipartite graphs using a construction based on acertain Lie algebra for a prime power q in [8]. Later, Lazebnik and Viglione gave anequivalent representation of these graphs in [10]. Then another useful representationof these graphs was given in [20], on which we concentrate in this section. All graphtheory notions can be found in [1].Let m ≥ g k ( x, y ) ∈ F q [ x, y ] for 2 ≤ k ≤ m + 1. Let P = F m +1 q and L = F m +1 q be two copies of the ( m + 1) − dimensional vector space over F q , which are called the point set and the line set respectively. If a ∈ F m +1 q , then wewrite ( a ) ∈ P and [ a ] ∈ L . Denote G = G q ( g , · · · , g m +1 ) = ( V, E ) by the bipartitegraph with vertex set V = B S L and the edge set E is defined as follows: there isan edge from a point P = ( p , p , · · · , p m +1 ) ∈ B to a line L = [ l , l , · · · , l m +1 ] ∈ L ,denoted by P L , if the following m equalities hold: ( l + p = g ( p , l ) ,l + p = g ( p , l ) , ... l m +1 + p m +1 = g m +1 ( p , l ) . (5.4)If g k ( x, y ) = x k − y, k = 2 , · · · , m + 1, then the graph is just the original Wengergraph denoted by W m ( q ) in [5]. The spectrum, the diameter and the automorphismgroup of W m ( q ) were studied in [5] , [8] , [9] and [21]. In [2], a new class of bipartitegraphs called linearized Wenger graphs was introduced. These graphs were defined by(5.4) together with g k ( x, y ) = x p k − , k = 2 , · · · , m + 1, which denoted by L m ( q ). When m ≥ e , the spectrum of L m ( q ) was explicitly determined using results on linearizedpolynomials over finite fields. The diameter and girth of L m ( q ) also were obtained.Furthermore, the spectrum of a general class of graphs, which defined by g k ( x, y ) = f k ( x ) y and the mapping σ : F q → F m +1 q ; u → (1 , f ( u ) , · · · , f m +1 ( u )) is injective, wasstudied. The eigenvalues of such a graph were determined and their multiplicities werereduced to counting certain polynomials with a given number of roots over F q . Thatis, for all prime power q and positive integer m , the eigenvalues of G , counted withmultiplicities, are ± p qN F ω , ( ω , ω , · · · , ω m +1 ) ∈ F m +1 q , where F ω ( u ) = ω + ω f ( u ) + · · · + ω m +1 f m +1 ( u ) and N F ω = |{ u ∈ F q : F ω ( u ) = 0 }| .For 0 ≤ i ≤ q , the multiplicity of ±√ qi is n i = |{ ω ∈ F m +1 q : N F ω = i }| . In this section, we use our previous results to get the spectrum of a general class ofgraphs defined by (5.4) together with polynomials g k ( x, y ) = f k ( x ) y, ∈ F q [ x, y ] , where f k ( x ) = x k − , ≤ k ≤ m, f m +1 ( x ) = x m +1 and f k ( x ) = x k − , ≤ k ≤ m, f m +1 ( x ) = x m +2 . These graphs are denoted by J W m ( q ) and J W m ( q ) respectively. heorem 5.1. For all prime power q and ≤ m + 1 ≤ q − , the distinct eigenvaluesof J W m ( q ) are ± q, ± p ( m + 1) q, ±√ mq, · · · , ± p q, ±√ q, . The multiplicity of the eigenvalue ±√ iq is ( q − (cid:18) qi (cid:19) m − X d = i d − i X k =0 ( − k (cid:18) q − ik (cid:19) q d − i − k + ( q − N i ( x m +2 , m − , ≤ i ≤ m − and ( q − N i ( x m +1 , m − , i = m, m + 1 . Proof. Let ( ω , ω , · · · , ω m +1 ) ∈ F m +1 q and f ( X ) = ω + ω X + · · · ω m X m − + ω m +1 X m +1 .In the case of f = 0, |{ u ∈ F q | f ( u ) = 0 }| = q . Thus, J W m ( q ) has ± q as itseigenvalues.For any 0 ≤ i ≤ m − 1, there exists a polynomial f over F q of degree at most m + 1 ≤ q − 1, which has exactly i distinct roots in F q .For i = m, m + 1, it is easy to compute N ( m + 1 , i ) > . 1. Thenthere exists a polynomial of degree m + 1 which has exactly i distinct roots in F q . Thusby Theorem 2.2 in [2], J W m ( q ) has ±√ iq, ≤ i ≤ m + 1 as its eigenvalues, and byTheorem 2 . . 1, we obtain that the multiplicity of the eigenvalue ±√ iq is( q − (cid:18) qi (cid:19) m − X d = i d − i X k =0 ( − k (cid:18) q − ik (cid:19) q d − i − k + ( q − N i ( x m +1 , m − , ≤ i ≤ m − q − N i ( x m +1 , m − , i = m, m + 1 . Similarly, we have the following theorem about the spectrum of J W m ( q ) by Theorem4 . Theorem 5.2. Suppose that p is an odd prime number and q = p e . If ≤ m + 2 ≤ q − , then we have ± q, ± p ( m − q, ± p ( m − q, · · · , ± p q, ±√ q, are the distinct eigenvalues of J W m ( q ) . The multiplicity of the eigenvalue ±√ iq is ( q − (cid:18) qi (cid:19) m − X d = i d − i X k =0 ( − k (cid:18) q − ik (cid:19) q d − i − k + ( q − N i ( x m +2 , m − , ≤ i ≤ m − . If m ≤ i ≤ m + 2 , then ±√ iq are the distinct eigenvalues of J W m ( q ) if and onlyif N i ( x m +2 , m − > . 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