NEERAJ KUMAR AND K. SENTHIL KUMAR. 1. Introduction. Theorem 1 motivate us to ask the following:

Transcription

1 NOTE ON VANISHING POWER SUMS OF ROOTS OF UNITY NEERAJ KUMAR AND K. SENTHIL KUMAR Abstract. For xe positive integers m an l, we give a complete list of integers n for which their exist mth complex roots of unity x 1,..., x n such that x l xl n = 0. This extens the earlier result of Lam an Leung on vanishing sums of roots of unity. Furthermore, we characterize all positive integers n with 2 n m, for which there are istinct mth complex roots of unity x 1,..., x n such that x l xl n = Introuction Let m be a positive integer. By an mth root of unity, we mean a complex number ζ such that ζ m = 1. That is, a root of the polynomial X m 1. One can easily see that the roots of X m 1 are istinct, in fact there are exactly m, mth roots of unity. Using the relationship between the roots an the coecients of a polynomial, we see that the sum of all mth roots of unity, which is the coecient of X m 1 in X m 1, is zero. A natural question is: What are all the positive integers n for which there exist mth roots of unity x 1,..., x n (repetition is allowe) such that x x n = 0. A beautiful result of T. Y. Lam an K. H. Leung [1] gives a complete classication of all such integers. Suppose m has prime factorization p a par r, where a i > 0, then we have the following theorem ue to Lam an Leung: Theorem 1. Let n be a positive integer. Then there are mth roots of unity x 1,..., x n such that x 1 + +x n = 0 if an only if n is of the form n 1 p 1 + +n r p r where each n i is a non-negative integer for 1 i r. Theorem 1 motivate us to ask the following: Question 1. Let m an l be positive integers. What are all the positive integers n for which there exist mth roots of unity x 1,..., x n such that x l x l n = 0? Note that when l = 1, the complete answer to Question 1 is given by Theorem 1. However, for l 2, we o not n any results in this irection in the literature. Our objective here is to stuy the case when l 2. First, we x some notations. Date: Submitte: March 30, 2015, Revision submitte: May 04, Key wors: roots of unity, power sum Mathematics Subject Classication: Primary 11L03 Seconary 11R18. 1

2 Let m be a positive integer, an let Ω m enotes the set of all mth roots of unity. For a positive integer l, W l (m) enotes the set of all positive integers n for which there exist n-elements x 1,..., x n Ω m such that x l x l n = 0. When l = 1, we simply enote W l (m) by W (m). With this notation, Question 1 can be reformulate as follows: Let m an l be positive integers. What are all the positive integers in the set W l (m)? It is clear that if m ivies l then W l (m) is an empty set. Suppose that there are mth complex roots of unity, say, x 1,..., x n such that x l x l n = 0. Since the lth power of an mth root of unity is still an mth root of unity, the equation x l x l n = 0 with x i Ω m can be written in the form y y n = 0 with y i Ω m. This shows that for any positive integer m an l, W l (m) is a subset of W (m). It follows from Theorem 1 that any positive integers in the set W l (m) must be of the form n 1 p n r p r where each n i is a non-negative integer for 1 i r. In Section 2, we give a complete list of integers in the set W l (m) ( see Theorem 2). Moreover, in Section 3 we n all positive integers n W l (m) for which there are istinct mth complex roots of unity x 1,..., x n such that x l x l n = 0 ( see Theorem 3). There are algebraic aspects why Question 1 is important. For instance, for a positive integer a, enote by p a the power sum polynomial X1 a + +Xn a of egree a. Let l < k be two positive integers. In commutative algebra, one encounters the following situation: To show that the ieal p l, p k generate by the polynomials p l an p k is a prime ieal in C[X 1,..., X n ], one nees to show that the power sum polynomial X1 l + + Xn l oes not vanish when one allows the X i 's to take values among the (l k)th roots of unity [2, see proof of Theorem 3.8]. Acknowlegment: We are grateful to Professor Ram Murty for his valuable suggestions regaring the paper. We also thank the referee for many useful comments for improving the manuscript. This project was fune by the Department of Atomic Energy (DAE), Government of Inia. 2. vanishing of power sums of roots of unity Let m an l be positive integers. In this section, we completely characterize all the positive integers in the set W l (m). More precisely, we prove the following theorem: Theorem 2. Let m an l be positive integers. Let = (m, l) be the greatest common ivisor of m an l. Then W l (m) = W (m/). In other wors, Theorem 2 says that: For any positive integer n, x l 1+ +x l n = 0 with x i Ω m if an only if y y n = 0 with y i Ω m/. 2

3 Proof. It is well known that Ω m, that is, the set of all mth roots of unity, form a group with respect to the multiplication of complex numbers. In fact, it is a cyclic group of orer m, generate by the complex number ζ m = cos 2π/m + i sin 2π/m. There is a remarkable property about nite cyclic groups. Namely, if G is a nite cyclic group an l is a positive integer relatively prime to the orer of G, then the map x x l (x G) (1) is an automorphism of G (In fact, all the automorphisms of G are of the form (1) for some integer l which is relatively prime to the orer of G). It follows that, if l is a positive integer which is relatively prime to m then every element of Ω m is a lth power of some element of Ω m. Thus, for an integer l which is relatively prime to m, the equation x l x l n = 0 with x i Ω m can be replace by y y n = 0 with y i Ω m, an vice versa. This iscussion proves Theorem 2 for the case when l is relatively prime to m. Now assume that > 1. Consier the map ψ : Ω m Ω m/ (2) ene by x x for x Ω m. This map is clearly onto, an the kernel is exactly Ω. Thus, Ω m /Ω = Ωm/. Now suppose that there are elements x 1,..., x n Ω m ( ) such that x l x l n = 0. Then this sum can be rewritten as x l/ ( ) = 0. Since l/ an m are relatively prime, by the above iscussion, the x l/ n latter equation can be rewritten in the form y1 + + yn = 0 with y i Ω m. Finally, using the map ψ, the latter sum can be realize as z z n = 0 where z i Ω m/ for 1 i n. In fact, all these steps can be reverse. This completes the proof of Theorem 2. Combining Theorems 1 an 2, we have the following corollary: Corollary 1. Let m, n an l be positive integers. Let = (m, l) be the greatest common ivisor of m an l. Then there are mth roots of unity x 1,..., x n such that x l x l n = 0 if an only if n is of the form n 1 q n s q s where each n i is a non-negative integer for 1 i s an q 1,..., q s are istinct prime ivisors of m/. Example. Let m = 60, an let l be an integer with 1 l < 60. By Theorem 2, W l (m) = W (m/) where is the greatest common ivisor of m an l. When varies over the ivisors of m, m/ also varies over the ivisors of m. Thus W l (m) coincies with W () for some ivisor of m. On the other han, by Theorem 1, W () = s i=1 q in where = q b qbs s is the prime factorization of. Here N 3

4 enotes the set of non-negative integers. We thus have the following table which escribe W () for all positive ivisors of m = 60. W () 1 2 2N 3 3N 4 2N 5 5N 6 2N + 3N = N \ {1} 10 2N + 5N = N \ {1, 3} 12 2N + 3N = N \ {1} 15 3N + 5N = N \ {1, 2, 4, 7} 20 2N + 5N = N \ {1, 3} 30 2N + 3N + 5N = N \ {1} 60 2N + 3N + 5N = N \ {1} 3. vanishing of power sums of istinct roots of unity Let m an l be two positive integers. For an integer n W l (m), the height H(n; l, m) of n is ene to be the smallest positive integer h for which there are mth roots of unity x 1,..., x n such that x l 1 + +x l n = 0 an the maximum among the repetition of x i 's is h, that is, h is the maximum among the h i, where h i is the number of times x i appears in the list x 1,..., x n. When l = 1, we enote H(n; l, m) by H(n; m). Note that H(n; m) = 1 provie 2 n m. Gary Sivek [3] rene the work of Lam an Leung by proving that for any integers m 2 an 2 n m, H(n; m) = 1 if an only if both n an m n are expressible as a linear combination of the prime factors of m with non-negative integer coecients. Here we exten Sivek's result to vanishing of power sums of istinct roots of unity: Theorem 3. Let m an l be positive integers, an let n be an integer such that 2 n m. Let be the greatest common ivisor of m an l. Then H(n; l, m) = 1 if an only if H(n; m/). Proof. Let Ω m/ = {z 1,..., z m/ }. Suppose that there are istinct mth roots of unity x 1,..., x n such that x l x l n = 0. Since is the greatest common ivisor of l an m, this equation can be rewritten in the form y1 + + yn = 0 with y 1,..., y n are mth roots of unity. Using the map ψ, the latter equation can be written as m/ ψ 1 i=1 a iz i = 0 where a i is the carinality of the set {y 1,..., y n } (z i) for 1 i m/. On the other han, ψ 1 (z) has exactly elements for each z Ω m/. It follows that H(n; m/) max{a 1,..., a m/ }. This proves that if H(n; l, m) = 1 then H(n; m/). 4

5 Conversely, suppose that H(n; m/). Then there is a partition (a 1,..., a m/ ) of n into non-negative integers a i with a i for 1 i m/ an m/ i=1 a iz i = 0. Let y i be any element of ψ 1 (z i) for 1 i m/. Then ψ 1 (z i) = y i Ω = { y i x x Ω }. Since a i, one can replace a i z i by yi (x x a i ) where x 1,..., x ai are istinct elements of Ω. Hence H(n; l, m) = H(n;, m) = 1 since m/ i=1 a i = n. This completes the proof of Theorem 3. References [1] T. Y. Lam an K. H. Leung: On Vanishing sums of roots of unity. J. Algebra 224 (2000), no. 1, [2] Neeraj Kumar: Prime ieals an regular sequences of symmetric polynomials. arxiv: [math.ac]. [3] G. Sivek: On vanishing sums of istinct roots of unity Integers, 10 (2010), pp. A31, The Institute of Mathematical Sciences, 4th cross roa, CIT Campus, Taramani, Chennai , Inia aress: neerajkr@imsc.res.in, senthilkk@imsc.res.in 5