ON POLYNOMIAL REPRESENTATION FUNCTIONS FOR MULTILINEAR FORMS JUANJO RUÉ In Memoriam of Yahya Ould Hamidoune. Abstract. Given an infinite sequence of positive integers A, we prove that for every nonnegative integer k the number of solutions of the equation n a 1 + +a k, a 1,..., a k A, is not constant for n large enough. This result is a corollary of our main theorem, which partially answers a question of Sárköy and Sós on representation functions for multilinear forms. The main tool used in the argument is an application of the Transfer Theorem for asymptotic enumeration by Flajolet and Odlyko. 1. Introduction Let A be an infinite sequence of positive integers. Denote by rn) the number of solutions of the equation n a 1 + a 2, where a 1, a 2 A. In [3] the authors found, by means of analytic arguments, that rn) cannot be constant for n large enough. As it is shown in [2], and elementary argument also exists: it is obvious that rn) is odd when n 2a, a A, and even otherwise. So it is not possible that rn) is constant for n large enough. This idea can be easily generalied when we consider the number of solutions of the equation n a 1 + +a p, where a 1,..., a p A and p is a prime number: if a A, the number of representations of pa is congruent to 1 modulo p, while the number of representations of pa + 1 is congruent to 0 modulo p. As a can be chosen as big as desired, a contradiction is obtained if we suppose that the representation function is constant for n large enough. However, the argument fails when we consider a composite modulo, and it does not seem that the argument could be extended in the general case using elementary tools. These problems are particular cases of a question posted by Sárköy and Sós [7]: given a multilinear form x 1 + + x r, consider the number of solutions of the equation n a 1 + + a r, where a 1,..., a r A. For which multilinear forms the number of solutions could be constant for n large enough? For bilinear forms the problem is completely solved: when we deal with the bilinear form x 1 + kx 2, k > 1, Moser [6] showed that there exists a set A such that the number of solutions of the equation n a 1 + ka 2 where a 1, a 2 A is constant and equal to 1 see also [8] for additional properties of these sequences of numbers). Recently, Cilleruelo and Rué [1] proved that for and k 2 satisfying 1 < < k 2 and gcd, k 2 ) 1 the number of solutions of the equation n a 1 + k 2 a 2 where a 1, a 2 A is not constant for n large enough. In this paper we found an answer to the question posed by Sárköy and Sós for multilinear forms in several cases: let 0 < < k 2 < < be a finite sequence of positive integers and consider the set m {, n 1 ),...,, n r )}. We say that n i is the multiplicity of k i, and that the degree of m is gcdn 1,..., n r ). Given a set m, we consider the associated multilinear form x 1,1 + + x 1,n1 ) + + x r,1 + + x r,nr ). Given a sequence of positive integers A, the representation function of n with respect to m is the number of different solutions of the equation 1.1) n a 1,1 + + a 1,n1 ) + + a r,1 + + a r,nr ) The author is supported by a JAE-DOC grant from the JAE program in CSIC, Spain. 1
2 JUANJO RUÉ where a i,j A. We denote this value by r m n, A). Our main theorem deals with representation functions which are polynomials: Theorem 1.1. Let m be a set of degree s. Then r m n, A) is not a polynomial of degree smaller than s 1 for n large enough. As a trivial consequence, Theorem 1.1 solves the problem posted by Sárköy and Sós for multilinear forms associated to sets whose degree is greater than 1. Observe also that Theorem 1.1 gives the best possible exponent for a polynomial: let A N, and consider the multilinear form x 1,1 + + x 1,s. Then, the associated set is {1, s)} and the number of representations of n is equal to n+1 s 1), which is a polynomial of degree s 1. The proof of Theorem 1.1 is based on two steps: firstly we translate the combinatorial problem into equations between generating functions. Later, we apply an analytic study of the coefficients of the counting series, using the powerful machinery arising from analytic combinatorics. Plan of the paper: In Section 2 we introduce the necessary background in order to deal with the problem, namely the use of generating functions in order to codify the problem and the singularity analysis for generating functions. In Section 3 we apply these techniques to prove Theorem 1.1. 2. Tools In this section we present the background used in order to prove the main theorem of this paper. We introduce the notion of generating function associated to a sequence of integers, and we show the analytic tools needed in order to deal with the singularities of these counting series. Generating functions. We codify all the enumerative information of the problem using generating functions: for every set A of non negative integers we define the formal power series f A ) a A a. This series is called the generating function associated to the sequence A. A generating function defines an analytic function around 0, as its coefficients are bounded. This function is either an entire function or has at least a singularity at 1, depending on whether A is finite or not. In the second case, the Taylor expansion of f A ) around 0 has radius of convergence equals to 1, hence all its singularities have modulo 1. The combinatorial problem can be translated in the language of generating functions in the following way. Let A be a sequence of nonnegative integers and let m {, n 1 ),...,, n r )}. Then, )) n1 )) nr 2.1) a i,j A a 1,1 + +a 1,n1 )+ + a r,1 + +a r,nr ) r m n, A) n. n0 Singularity analysis. As we have shown, the generating function associated to a sequence A defines an analytic function in a neighbourhood of the origin. Hence we can apply complex analytic techniques in order to study the equations defined by these counting series. Denote the nth Taylor coefficient of f) by [ n ]f). The nature [ n ]f) can be studied by considering the counting series f) as a complex analytic function in a neighborhood of the origin. The growth behavior of coefficients is related to the singularities of the generating function with smallest modulo. In our approach we show that we will need to deal with a finite number of these singularities arising from the roots of a polynomial): their location provides the exponential growth of the coefficients, and their behavior provides the subexponential growth of the coefficients.the seminal results in this area are the so-called Transfer Theorems for singularity analysis [4]. These
ON POLYNOMIAL REPRESENTATION FUNCTIONS FOR MULTILINEAR FORMS 3 theorems allows us to deduce asymptotic estimates of an analytic function using its asymptotic expansion near their dominant singularities. We call domain dented at a value R a domain of the complex plane C of the form { C : < R and arg R) / [ θ, θ]} for some real number R > R and some positive angle 0 < θ < π/2. An example of a dented domain is shown in Figure 1. 0 R Figure 1. A domain of C dented at R When we have a single dominant singularity, then the growth of the coefficients can be obtained by means of integrals on the complex plane. This result is resumed in the following version of Corollary VI.1 of [5]: Theorem 2.1 Transfer for a single singularity). Let α be an arbitrary complex number in C\N. Let f) be a function analytic in a domain dented at ζ, such that f) C1 /ζ) α 1 + o1)) when ζ in the corresponding dented domain. Then, the coefficients [ n ] f) admit, for large n, an asymptotic expansion of the form where Γx) is the classical Gamma function. [ n ]f) C n α 1 Γ α) ζn 1 + o1)), The extension of Theorem 2.1 to functions that have finitely many by necessity isolated) singularities on their circle of convergence follows along entirely similar lines. As it is quoted in [5], in the case of multiple singularities, the separate contributions from each of the singularities, as given by the basic singularity analysis process, are to be added up. More precisely, we have the following simplified version of Theorem VI.5 from [5]: Theorem 2.2 Transfer theorem for multiple singularities). Let f) be analytic in < ζ and have a finite number of singularities on the circle ζ at points ζ j, for j 1,..., r. Assume that there exists a domain, dented at ζ, such that f) is analytic in the indented disc r D ζ j ), j0 where ζ j is the image of by the mapping ζ. Suppose that the expansion of f) around each of these points is of the form f) f j 1 /ζ j ) α j 1 + o1)),
4 JUANJO RUÉ as ζ j in the region where f is analytic, where α j C \ N for j 1,..., r. Then the Taylor coefficients of f) satisfies the asymptotic estimate [ n ]f) r j1 where Γx) is the classical Gamma function. f j n αj 1 Γ α j ) ζn j 1 + o1)), 3. Proof of the main theorem Without loss of generaliation, we may assume that 0 A. We suppose that such a sequence A exists, and we argue by contradiction. The case s 1 is trivial, so we may assume that s > 1. We assume that r m n, A) is equal to a polynomial of degree at most s 2, namely Qn) d i0 n i, with d < s 1. Using the generating function terminology, 3.1) r m n, A) n T ) + n0 Qn) n T ) + n>n i0 n>n n i n T 0 ) + i0 n0 n i n, where T ), T 0 ) are polynomials with degree N. Each term of the form n0 ni n can be Q written as i) 1 ), where Q i+1 i ) is a polynomial in such that Q i 1) 0. Hence, we can write Expression 3.1) in the form T 0 ) + i0 Q i ) 1 ) i+1 Q) 1 ) d+1, where Q) is a polynomial which satisfies that Q1) 0. generating function f A ) satisfies the relation Using now Equation 1.1), the 3.2) )) n1... )) nr Q) 1 ) d+1. Observe that Q0) T 0) r m 0, A) 1. As the degree of m is equal to s, Equation 3.2) can be written in the form 3.3) )) n1/s... )) nr/s Q) 1/s 1 ) d+1)/s, where each quotient n i /s is a nonnegative integer. Let us study the location and the type of the singularities of the rightmost term in Equation 3.3). Firstly, observe that if ζ is a ero of Q) such that ζ < 1, then its order is multiple of s: consider the factoriation over C of Q) 1 /ζ) r Q ζ ), where Q ζ ζ) 0. If r 0 s), then 1 /ζ) r/s is not analytic at ζ. But ζ < 1 and the leftmost term in Equation 3.3) is analytic for < 1, so we get a contradiction. Q) 1/s 1 ) d+1)/s Hence, we need to study singularities with modulo greater or equal to 1. As has a singularity at 1, the main contribution to the asymptotic of the coefficients of )) n1/s )) nr/s will arise from singularities in the circle { C : 1}. Denote by ζ 1..., ζ r the eros of Q) with modulo 1, and let α j be the order of the ero ζ j. We may assume that α j is not divisible by s: if this is the case, Q) 1/s is analytic at ζ j, and it does not contribute to the asymptotic of the coefficients. For small ϵ, θ > 0, let { C : < 1 + ϵ and arg R) / [ θ, θ]} be a domain dented at 1. As the function under study is a root of a rational function, it is analytic in the indented domain D r j0 ζ j ). Hence, in order to study the singularities of the rightmost term in Equation 3.3) we just need to study the behaviour of the eros of Q) with modulo 1 and the singularity at 1, respectively.
ON POLYNOMIAL REPRESENTATION FUNCTIONS FOR MULTILINEAR FORMS 5 Under these assumptions Theorem 2.2 asserts that, for certain constants C, C 1,..., C r [ n ] f A )) n1/s )) nr/s r C nd+1)/s 1 Γ n ) αj/s 1 1 + o1)) + C d+1 j Γ α ) 1 + o1)) j s j1 s O n 1/s), because the orders α j are positive and d + 1)/s 1 < s 1)/s 1 1/s. Hence, there exists a nonnegative integer M and a constant c such that, for all n > M [ n ] f A )) n1 /s )) nr /s c < < 1, n1/s but this is impossible because A is infinite and consequently infinitely many Taylor coefficients of f A ) and also of )) f A k n1 1 /s )) k nr r /s ) are 1. Acknowlegments: Javier Cilleruelo is greatly thanked for inspiring discussions and useful comments in order to improve the presentation of this paper. References [1] Cilleruelo, J., and Rué, J. On a question of sárköy and sós on bilinear forms. Bulletin of the London Mathematical Society 4, 2 2009), 274 280. [2] Dirac, G. Note on a problem in additive number theory. Journal of the London Mathematical Society 26 1951), 312 313. [3] Erdös, P., and Turán, P. On a problem of sidon in additive number theory, and on some related problems. Journal of the London Mathematical Society 16 1941), 212 215. [4] Flajolet, P., and Odlyko, A. Singularity analysis of generating functions. SIAM Journal on Discrete Mathematics 3, 2 1990), 216 240. [5] Flajolet, P., and Sedgewick, R. Analytic Combinatorics. Cambridge Univ. Press, 2008. [6] Moser, L. An application of generating series. Mathematics Magaine 1, 35 1962), 37 38. [7] Sárköy, A., and Sós, V. On additive representation functions, in The mathematics of Paul Erdös I. Springer-Verlag, 2008. [8] Shevelev, V. On unique additive representations of positive integers and some close problems. Available on-line at arxiv:0811.0290. Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM), 28049 Madrid, Spain E-mail address: juanjo.rue@icmat.es