A CLASS OF ALGEBRAIC-EXPONENTIAL CONGRUENCES MODULO p. 1. Introduction

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1 Acta Math. Univ. Comenianae Vol. LXXI, (2002), A CLASS OF ALGEBRAIC-EXPONENTIAL CONGRUENCES MODULO C. COBELI, M. VÂJÂITU and A. ZAHARESCU Abstract. Let be a rime number, J a set of consecutive integers, F the algebraic closure of F = Z/Z and C an irreducible curve in an affine sace A r (F ), defined over F. We rovide a lower bound for the number of r tules (x, y,..., y r ) with x J, y,..., y r {0,,, } for which (x, y x,..., yr x ) (mod ) belongs to C(F).. Introduction In Chater F, section F9 of his well known book [4] on unsolved roblems in number theory, Richard Guy collected some questions on rimitive roots. One of them, attributed to Brizolis, asks if for a given rime > 3, there is always a rimitive root g mod, 0 < g <, and an integer x, 0 < x < such that x g x ( mod ). This question was answered ositively in [2], by showing that for any ɛ > 0 there is a ositive integer (ɛ) such that for any rime > (ɛ) the number of airs (x, y) of rimitive roots mod, 0 < x, y < which are solutions of the congruence x y x (mod ), is at least ( ɛ)e 2γ (log log ), 2 where γ denotes Euler s constant. In the resent aer we consider more general congruences, involving x, y x,..., yr, x and look for all the solutions, including those for which y,..., y r are not necessarily rimitive roots mod. We start with a large rime number and a set J of consecutive ositive integers, of cardinality J. Denote by F the algebraic closure of the field F = Z/Z and let C be an irreducible curve of degree D in an affine sace A r (F ). We assume in the following that C is not contained in any hyerlane and that it is defined over F. Denote as usually by C(F ) the set of oints z = (z,..., z r ) on C with all the comonents z,..., z r in F. The roblem is to find integers x J and y,..., y r {0,,, } such that () (x, y x,..., y x r ) (mod ) C(F ). The method emloyed in [2] may be adated to the resent context. The first idea is to look for oints (x, z,..., z r ) on the curve C for which x is relatively rime to. For any such oint (x, z,..., z r ) we find a solution (x, y,..., y r ) of () by arranging y,..., y r such that yj x z j (mod ), Received Setember 29, Mathematics Subject Classification. Primary T99.

2 4 C. COBELI, M. VÂJÂITU and A. ZAHARESCU j r. To be recise, we choose a ositive integer w such that xw (mod ), then set y j = zj w and from Fermat s Little Theorem one gets yj x = zj xw z j mod. We combine this idea with a Fourier inversion technique, similar to that used in [3]. Consider the sets and A = { (x, y,..., y r ) J Z r : 0 y,..., y r <, (x, y x,..., y x r ) (mod ) C(F ) } B = { (x, z,..., z r ) J Z r : 0 z,..., z r <, (x, ) =, (x, z,..., z r ) (mod ) C(F ) }. Our goal is to obtain lower bounds for A. By the above remark we know that A B, thus it will be enough to find lower bounds for B. We will actually obtain an asymtotical estimation for B. The result is stated in the following theorem. Theorem. Let be a rime number, J a set of consecutive ositive integers and C an irreducible curve of degree D in A r (F ), defined over F and not contained in any hyerlane. Then ϕ( ) B = J + O D (σ 0 ( ) log Here ϕ( ) is the Euler function and σ 0 ( ) is the number of ositive divisors of. As a consequence of Theorem we note the following corollary. Corollary. Let r 2 and D be integers and ɛ > 0 a fixed real number. Then there is a ositive integer (r, D, ɛ) such that for any rime number > (r, D, ɛ) and any irreducible curve C of degree D in A r (F ), defined over F and not contained in any hyerlane, the number of r tules (x, y,..., y r ) with 0 < x, y,..., y r <, (x, ) = and (x, y x,..., yr ) x (mod ) C(F ) is at least ( ɛ)e 2γ log log. 2. Characteristic Functions and Exonential Sums Our first ste is to get an exact formula for B in terms of exonential sums. For this we introduce the following characteristic function: {, if x J and (x, ) = φ J (x) = 0, else. Without any loss of generality, we may assume in the roof of Theorem that the set of consecutive integers J satisfies J [, ]. Let C be as in the statement of the theorem. Then the number we are interested in, can be written as (2) B = φ J (x). (x,z,...,z r ) C(F ) ).

3 ALGEBRAIC-EXPONENTIAL CONGRUENCES 5 Next, using a finite Fourier transform modulo we write the characteristic function defined above as (3) φ J (x) = u F ˆφJ (u)e (ux) where e (t) = e 2πit for any t. The Fourier coefficients (u) are given by (4) ˆφJ (u) = φ J (x)e ( ux). x F We substitute the exression (3) in (2) to obtain (5) B = u F ˆφJ (u)s C (u), in which S C (u) = (x,z,...,z r ) C(F ) e (ux). The exression (5) is the basic formula that will be used in the roof of Theorem. In order to comlete the roof we first need estimates for (u). 3. Estimates for the Fourier coefficients The Fourier coefficients given by (4) behave differently, deending on whether their argument is or is not zero modulo. We have J ϕ( ) σ0 ( ) + O (6) ˆφJ (u) = 2, if u 0 (mod ) ) O, if u 0 (mod ) ( x J (x, )= d ( ) ud/ where denotes the distance to the nearest integer. In order to rove (6), we use well known roerties of the Möbius function to write (u) = e ( ux) = e ( ux) = d ( ) x J d x x J e ( ux). When u = 0 one has (0) = {x J ; d divides x} = d ( ) = J d + O σ0 ( ). d ( ) d ( ) d x d ( ) J d + O()

4 6 C. COBELI, M. VÂJÂITU and A. ZAHARESCU d Emloying the equality d ( ) (see for examle [5]), the relation (6) is roved for u = 0. Let us assume now that u 0 (mod ). The sum x J, d x e ( ux) is a geometric rogression of ratio e ( ud). It follows easily that (7) x J, d x = ϕ( ) e ( ux) ud/. Using (7) for any divisor d of, we find that (u) ud/, which roves (6). d ( ) 4. Proof of Theorem We slit the sum in the main formula (5) into two ranges according as to whether u = 0 or u 0. We write (8) B = M + E, where M = (0) C(F ) contains the rincial contribution, giving the main term of the estimation for B, while the remainder is E = ˆφJ (u) e (ux). 0 u F (x,z,...,z r ) C(F ) We now turn our attention to the evaluation of M. By the Riemann Hyothesis for curves over finite fields (Weil [6]), we know that Then using (6), we obtains C(F ) = + O D ( ). M = J ϕ( ) + O D ( ). Next, we estimate the remainder E. Since C is not contained in any hyerlane it follows for u 0 that ux is nonconstant along the curve C. Then one may aly the Bombieri Weil inequality (see [], Theorem 6), which gives for u 0. Therefore, by (6) we see that E = 0 u F ˆφJ (u)s C (u) D σ 0 ( ) log. This comletes the roof of Theorem. S C (u) D d ( ) u= ud/

5 ALGEBRAIC-EXPONENTIAL CONGRUENCES 7 References. Bombieri E., On exonential sums in finite fields, Amer. J. Math. 88 (966), Cobeli C. and Zaharescu A., An exonential congruence with solutions in rimitive roots, Rev. Roumaine Math. Pures Al. 44() (999), , Generalization of a roblem of Lehmer, Manuscrita Math. 04 (200), Guy R. K., Unsolved roblems in Number Theory, Sringer-Verlag, New York-Berlin, 98, (second edition 994). 5. Ram Murty M., Problems in Analytic Number Theory, Sringer-Verlag, New York, Weil A., Sur les courbes algébriques et les variétés qui s en déduisent, Paris, Hermann, 948. C. Cobeli, M. Vâjâitu, Institute of Mathematics of the Romanian Academy, P.O.Box -764, Bucharest, Romania, ccobeli@imar.ro, mvajaitu@imar.ro A. Zaharescu, Deartment of Mathematics, University of Illinois at Urbana-Chamaign, Altgeld Hall, 409 W. Green St., Urbana, IL 680, USA, zaharesc@math.uiuc.edu