J. Combin. Theory Ser. A 116(2009), no. 8, A NEW EXTENSION OF THE ERDŐS-HEILBRONN CONJECTURE

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1 J. Combin. Theory Ser. A 116(2009), no. 8, A NEW EXTENSION OF THE ERDŐS-HEILBRONN CONJECTURE Hao Pan and Zhi-Wei Sun Department of Mathematics, Naning University Naning , People s Republic of China haopan79@yahoo.com.cn, zwsun@nu.edu.cn Abstract. Let A 1,..., A n be finite subsets of a field F, and let f(x 1,..., x n ) x x n + g(x 1,..., x n ) F x 1,..., x n with deg g <. We obtain a lower bound for the cardinality of f(x 1,..., x n ) : x 1 A 1,..., x n A n, and x i x if i }. The result extends the Erdős-Heilbronn conecture in a new way. 1. Introduction In 1964 P. Erdős and H. Heilbronn EH made the following challenging conecture: If p is a prime, then for any subset A of the finite field Z/pZ we have x 1 + x 2 : x 1, x 2 A and x 1 x 2 } p, 2 A 3}. It had remained open for thirty years until it was confirmed fully by Dias da Silva and Hamidoune DH who actually obtained the following generalization with the help of the representation theory of symmetric groups: For any finite subset A of a field F, we have the inequality x x n : x 1,..., x n A, and x i x if i } p(f ), n( A n) + 1}, Key words and phrases. Erdős-Heilbronn conecture, Combinatorial Nullstellensatz, value sets of polynomials Mathematics Subect Classification. Primary 11B75; Secondary 05A05, 11P99, 11T06, 12E10. The second author is responsible for communications, and supported by the National Natural Science Foundation (grant ) of China. 1

2 2 HAO PAN AND ZHI-WEI SUN where p(f ) p if the characteristic of F is a prime p, and p(f ) + if F is of characteristic zero. Recently P. Balister and J. P. Wheeler BW extended the Erdős-Heilbronn conecture to any finite group; namely they showed that for any nonempty subsets A 1 and A 2 of a finite group G written additively we have x 1 + x 2 : x 1 A 1, x 2 A 2 and x 1 x 2 } p(g), A 1 + A 2 3}, where p(g) is the least positive order of a nonzero element of G, and p(g) is regarded as + if G is torsion-free. In 1996 N. Alon, M. B. Nathanson and I. Z. Ruzsa ANR used the socalled polynomial method (see also Alon A, Nathanson N, pp , and T. Tao and V. H. Vu TV, pp ) to deduce the following result: If A 1,..., A n are finite subsets of a field F with 0 < A 1 < < A n, then x x n : x 1 A 1,..., x n A n, and x i x if i } } p(f ), 1 + ( A i i). i1 Consequently, if A 1,..., A n are finite subsets of a field F with A i i for all i 1,..., n, then x x n : x 1 A 1,..., x n A n, and x i x if i } } p(f ), 1 + ( A ). i n i1 (Choose A i A i with A i i + i n( A ) A i. Then A 1 < < A n.) For other results on restricted sumsets obtained by the polynomial method, the reader may consult HS, PS, S03, SY and S08a. Recently Z. W. Sun S08 obtained the following result on value sets of polynomials. Theorem 1.1 (Z. W. Sun S08). Let A 1,..., A n be finite nonempty subsets of a field F, and let f(x 1,..., x n ) a 1 x a n x n + g(x 1,..., x n ) F x 1,..., x n (1.1) with Z + 1, 2, 3,... }, a 1,..., a n F F \ 0} and deg g <. (1.2)

3 A NEW EXTENSION OF THE ERDŐS-HEILBRONN CONJECTURE 3 (i) We have f(x 1,..., x n ) : x 1 A 1,..., x n A n } } Ai 1 p(f ), 1 +. i1 (ii) If n and A i i for i 1,..., n, then f(x 1,..., x n ) : x 1 A 1,..., x n A n, and x i x if i } } Ai i p(f ), 1 +. i1 (1.3) (1.4) Throughout this paper, for a predicate P we let 1 if P holds, P 0 otherwise. For a Z and Z +, we use a} to denote the least nonnegative residue of a modulo. Let C be the field of complex numbers. By S08, Example 4.1, if Z +, q 0, 1,... }, and A z C : z 1,..., q}} R with R z C : z q + 1} and R r <, then A q + r and x x n : x 1,..., x n A, and x i x if i } n( A n) n} A n} + r n} > r + 1. Motivated by this example, Sun S08 raised the following extension of the Erdős-Heilbronn conecture. Conecture 1.1 (Z. W. Sun S08). Let f(x 1,..., x n ) be a polynomial over a field F given by (1.1) and (1.2). Provided n, for any finite subset A of F we have f(x 1,..., x n ) : x 1,..., x n A, and x i x if i } p(f ) n 2 & a 1 a 2, n( A n) n} A n} } + 1. (1.5) Sun S08 noted that this conecture in the case n 2 follows from PS, Corollary 3, and proved (1.5) with the lower bound replaced by p(f ), A n + 1}. In this paper we establish a similar version of (1.4) for the case n under the condition a 1 a n. It implies Conecture 1.1 in the case a 1 a n. Here is our first result.

4 4 HAO PAN AND ZHI-WEI SUN Theorem 1.2. Let A 1,..., A n be finite subsets of a field F with A i i for i 1,..., n. Let f(x 1,..., x n ) x x n + g(x 1,..., x n ) F x 1,..., x n (1.6) with deg g < n. Then f(x 1,..., x n ) : x 1 A 1,..., x n A n, and x i x if i } p(f ), q q n + 1} (1.7) where q i i n i (mod ) for i 1,..., n. (1.8) Remar 1.1. If n then q i ( A i i)/ for i 1,..., n. So Theorem 1.2 is a complement to Theorem 1.1(ii). In the case 1, Theorem 1.2 yields the main result of ANR. Theorem 1.2, together with Theorem 1.1(ii), implies the following extension of the Erdős-Heilbronn conecture. Theorem 1.3. Let A 1,..., A n be finite subsets of a field F with A 1 A n m n, and let f(x 1,..., x n ) be given by (1.6) with deg g <. Then f(x 1,..., x n ) : x 1 A 1,..., x n A n, and x i x if i } p(f ), n(m n) n} } m n} + n} m} < n} + 1. (1.9) Remar 1.2. If n or m n is divisible by, then the lower bound in (1.9) becomes p(f ), n(m n)/+1}. In the case 1 and A 1 A n, Theorem 1.3 yields the Dias da Silva-Hamidoune extension (cf. DH) of the Erdős-Heilbronn conecture. In the next section we are going to present an auxiliary theorem. Theorems 1.2 and 1.3 will be proved in Section An Auxiliary Theorem For a polynomial P (x 1,..., x n ) over a field, by x 1 1 x n n P (x 1,..., x n ) we mean the coefficient of the monomial x 1 1 x n n in P (x 1,..., x n ). In this section we prove the following auxiliary result.

5 A NEW EXTENSION OF THE ERDŐS-HEILBRONN CONJECTURE 5 Theorem 2.1. Let q 1,..., q n N 0, 1, 2,... } and 1,..., n}. Then n x q + 1 (x x n) N (x x i ) N! 1 s1 where N q q n. 1 i< n 0 i< (n s)/ (q +s + (q i+s + i)) (n s)/, 0 (q +s + )! To prove Theorem 2.1, we need a lemma. (2.1) Lemma 2.1. Let σ be a permutation of a finite nonempty set X. Suppose that A is a subset of X with σ(a) A. Then ε(σ A )ε(σ X\A ), (2.2) where stands for the sign of σ and σ A denotes the restriction of σ on A. Proof. Write σ τ 1 τ 2 τ, where τ 1, τ 2,..., τ are disoint cycles. As σ(a) A, for each i 1,...,, either all elements in the cycle τ i lie in A, or none of the elements in the cycle τ i belongs to A. Set and Then I 1 i : all the elements in the cycle τ i lie in A} Ī 1 i : all the elements in the cycle τ i lie in X \ A} I Ī 1,..., }, σ A i I τ i A and σ X\A i Ī τ i X\A. Therefore i1 ε(τ i ) i I This completes the proof. ε(τ i ) Ī ε(τ ) ε(σ A )ε(σ X\A ). For a finite nonempty set X, we let S(X) denote the symmetric group of all permutations of X. If X n, then the group S(X) is isomorphic to the symmetry group S n S(1,..., n}). Recall that the deterant of a matrix a i, 1 i, n over a field is defined as deta i, 1 i, n σ S n n a i,σ(i) i1 σ S n n a σ(),. 1

6 6 HAO PAN AND ZHI-WEI SUN Proof of Theorem 2.1. By the multinomial theorem, (x x n) N i 1,...,i n N i 1 + +i n N N! i 1! i n! xi 1 1 x i n n. In view of linear algebra, σ S n n 1 x σ() 1 detx i 1 1 i, n 1 i< n (x x i ) (Vandermonde). Thus n 1 n 1 n 1 x q + 1 (x x n) N (x x i ) 1 i< n n x q + 1 N! x q + 1 N! i 1,...,i n N i 1 + +i n N σ S n σ() for 1,...,n i 1,...,i n N σ S n i 1 + +i n N σ(x s )X s for s1,..., 1 n 1 x i +σ() 1 i! x i +σ() 1, i! where X s 1 n : s (mod )}. Set n s X s. Then n s n s q N : s + q n} + 1. If σ S n and σ(x s ) X s for all s 1,...,, then ε(σ X1 )ε(σ X2 X ) ε(σ X1 ) ε(σ X ) by Lemma 2.1.

7 A NEW EXTENSION OF THE ERDŐS-HEILBRONN CONJECTURE 7 Let (x) 0 1 and (x) i i 1 r0 (x r) for i 1, 2, 3,.... By the above, n x q + 1 (x x n) N (x x i ) N! 1 n 1 ( s1 x q + 1 i 1,...,i n N i 1 + +i n N σ S(X s ) q +( σ())/ 0 for X s ( s1 s1 σ S n s n s 1 1 i< n ( s1 σ S(X s ) X s 1 X s x (q + ( σ())/)! ) (q ( 1)+s + 1) σ() 1 (q ( 1)+s + 1)! det(q ( 1)+s + 1) i 1 1 i, ns ns 1 (q ( 1)+s + 1)! It is well nown that y i (y) i + 0 r<i s1 i +σ() 1 ) i! S(i, r)(y) r for i 0, 1, 2,..., ) det(q +s + ) i 0 i, ns 1 ns 1 0 (q. +s + )! where S(i, r) (0 r < i) are Stirling numbers of the second ind. So det(q +s + ) i 0 i,<ns det(q +s + ) i 0 i,<ns (q +s + (q i+s + i)) (Vandermonde), 0 i<<n s and hence (2.1) follows. 3. Proofs of Theorems 1.2 and 1.3 Let us recall the following powerful tool. Combinatorial Nullstellensatz (Alon A). Let A 1,, A n be finite subsets of a field F, and let P (x 1,, x n ) F x 1,, x n. Suppose that deg P n where 0 i < A i for i 1,..., n. If x 1 1 x n n P (x 1,..., x n ) 0, then P (x 1,..., x n ) 0 for some x 1 A 1,..., x n A n. Proof of Theorem 1.2. Let m be the least nonnegative integer not exceeding n such that m<i n q i < p(f ). For each m < i n let A i be a subset of

8 8 HAO PAN AND ZHI-WEI SUN A i with cardinality q i + i A i. In the case m > 0, p p(f ) is a prime and we let A m be a subset of A m with ( A m p 1 m<i n q i ) + m < q m + m A m. If 0 < i < m then we let A i A i with A i i. Clearly q i ( A i i)/ q i. Whether m 0 or not, we have n i1 ( A i i) n i1 q i (N 1), where N p(f ), q q n + 1}. Let s 1,..., }. For any 0 < i < n s (n s)/ + 1 we have q (i 1)+s (i 1)+s n s (mod ) i+s n s (mod ) q i+s and hence q (i 1)+s q i+s. (If (i 1) + s m then q (i 1)+s q (i 1)+s q i+s q i+s.) So Define 0 q s < q +s + 1 < q 2+s + 2 < < q (n s 1)+s + n s 1. P (x 1,..., x n ) (x x n) N 1 In light of Theorem 2.1, n 1 1 i< n x q + 1 P (x 1,..., x n ) he, where e is the identity of the field F, and / h (N 1)! s1 n s r<q +s + +i: 0 i<} r q i+s (x x i ) F x 1,..., x n. (q +s + r) is an integer dividing (N 1)!. Since p(f ) > N 1, we have he 0. Set C f(x 1,..., x n ) : x 1 A 1,..., x n A n, and x i x if i }.

9 A NEW EXTENSION OF THE ERDŐS-HEILBRONN CONJECTURE 9 Suppose that C N 1 and let Q(x 1,..., x n ) denote the polynomial f(x 1,..., x n ) N 1 C c C (f(x 1,..., x n ) c) 1 i< n (x x i ). Then deg Q (N 1) + ( ) n 2 ( A i 1) i1 (q i + i 1) i1 and n x A x A n n 1 Q(x 1,..., x n ) 1 x q + 1 P (x 1,..., x n ) 0. In light of the Combinatorial Nullstellensatz, there are x 1 A 1,..., x n A n such that Q(x 1,..., x n ) 0. This contradicts the fact f(x 1,..., x n ) C. By the above, we have f(x 1,..., x n ) : x 1 A 1,..., x n A n, and x i x if i } C N. This concludes the proof. Proof of Theorem 1.3. Write n q 0 + n 0 with q 0 N and 1 n 0. Then i1 r1 r1 i n i (mod ) q+r n 0 q (n r)/ r (mod ) 0 q (n r)/ m m r (n r)/ ( )( ) n r m r n r + 1 r1 n 0 ( ) m r (q 0 + 1) q 0 + ( ) m r q 0 q r1 n 0 <r m r n 0 m r q 0 + q 0 ((q 0 + 1)n 0 + (q 0 1)( n 0 )). r1 r1

10 10 HAO PAN AND ZHI-WEI SUN Observe that m r r1 So we have ( m r r1 i1 i n i (mod ) m r} ) m r1 1 r s m. s0 n 0 m r + q 0 (m ) q 0 (2n 0 + (q 0 1)) r1 n 0 ( ) m r q 0 + q 0 (m n). r1 To simplify the last sum, we now subtract a term which will be added later. Clearly n 0 ( ) m r m n q 0 n 0 r1 n 0 ( ) m n + n0 r m n r1 0 1 m n} + s. s0 If m} n 0, then 0 1 m n} + s s0 0 1 If m} < n 0, then 0 1 m n} + s s0 s0 m} n} + s s0 m} n s s 0,..., n 0 1} : s n 0 m} } m}. Therefore i n i1 i (mod ) m n q 0 (m n) + n 0 + m} m} < n 0 n m n (m n) + n} + m} m} < n} n(m n) n} m n} + m} m} < n}.

11 A NEW EXTENSION OF THE ERDŐS-HEILBRONN CONJECTURE 11 In view of the above, by applying Theorem 1.2 for n and Theorem 1.1(ii) for n, we immediately get the desired (1.9). References A N. Alon, Combinatorial Nullstellensatz, Combin. Probab. Comput. 8 (1999), ANR N. Alon, M. B. Nathanson and I. Z. Ruzsa, The polynomial method and restricted sums of congruence classes, J. Number Theory 56 (1996), BW P. Balister and J. P. Wheeler, The Erdős-Heilbronn conecture for finite groups, Acta Arith., to appear. DH J. A. Dias da Silva and Y. O. Hamidoune, Cyclic spaces for Grassmann derivatives and additive theory, Bull. London Math. Soc. 26 (1994), EH P. Erdős and H. Heilbronn, On the addition of residue classes modulo p, Acta Arith. 9 (1964), HS Q. H. Hou and Z. W. Sun, Restricted sums in a field, Acta Arith. 102 (2002), N M. B. Nathanson, Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Graduate Texts in Math., Vol. 165, Springer, New Yor, PS H. Pan and Z. W. Sun, A lower bound for a + b: a A, b B, P (a, b) 0}, J. Combin. Theory Ser. A 100 (2002), S03 Z. W. Sun, On Snevily s conecture and restricted sumsets, J. Combin. Theory Ser. A 103 (2003), S08 Z. W. Sun, On value sets of polynomials over a field, Finite Fields Appl. 14 (2008), S08a Z. W. Sun, An additive theorem and restricted sumsets, Math. Res. Lett. 15 (2008), SY Z. W. Sun and Y. N. Yeh, On various restricted sumsets, J. Number Theory 114 (2005), TV T. Tao and V. H. Vu, Additive Combinatorics, Cambridge Univ. Press, Cambridge, 2006.