A Note on Browkin-Brzezinski s Conjecture

Transcription

1 Int J Contemp Math Sciences, Vol, 007, no 7, A Note on Browkin-Brzezinski s Conjecture Nguyen Thanh Quang Department of Mathematics,Vinh University Vinh, Nghe An, Vietnam Phan Duc Tuan Department of Mathematics,Vinh University Vinh, Nghe An, Vietnam tuanphanduc@yahoocom Abstract In this paper we prove Browkin-Brzezinski s conjecture for a class of polynomials Keywords: abc-theorem, Browkin-Brzezinski s conjecture, Wronskian 1 Introduction Let F be a fixed algebraically closed field of characteristic 0 Let f(z) bea polynomial non - constants with coefficients in F and let r(f) be the number of distinct zeros of f Then we have the following Mason s theorem ([3]) Let a(z),b(z),c(z) be relatively prime polynomials in F and not all constants such that a + b = c Then max {deg(a), deg(b), deg(c)} r(abc) 1 For different purposes some generalizations of the theorem are given (see [, 4]) In [1], Browkin and Brzezinski suggested the following conjecture Conjecture 11 Let f 0,, f n+1 be n + polynomials not all constants in F [x] and have no common zeros such that f f n + f n+1 =0 (1) Then max deg(f j) (n 1) (r(f 0 f n+1 ) 1) 0 j n+1 In this paper we prove the following theorem Theorem 11 Browkin-Brzezinski s conjecture is true for polynomials f 0,, f n+1 such that gdc(f i,f j,f k )=1for all distinct i, j, k {0,, n +1}

2 1336 Nguyen Thanh Quang and Phan Duc Tuan Proof of the main theorem Let f is a rational function, we write f in the form: f = f 1 f, Where f 1,f are polynomial functions are non-zero and relatively prime on F [x] The degree of f, denoted by deg f, is define to be deg f 1 deg f Let a F, we write f in the form: f =(x a) l g 1 g, and g 1 (a)g (a) 0, then l is called the order of f at a and is denoted by μ a f We have the following easily proved properties of μ a f Lemma 1 Let f,g be two polynomials and a F, we have a) μ a f+g min(μa f,μa g ), b) μ a fg = μa f + μa g, c) μ a f = μ a f μa g, g Lemma Let ϕ be a the rational function on F and let the derivatives order k of ϕ satisfy the following ϕ (k) 0 Then μ α ϕ (k) k + μ α ϕ Proof Let ϕ(x) =(x α) m f(x), where f(x),g(x) are relatively prime and g(x) f(α)g(α) 0 Then, we have ϕ (x) =(x α) m 1 mf(x)g(x)+(x α)(f (x)g(x) f(x)g (x)) g (x) By μ α g =0, we have μ Therefore From this we obtain α ϕ m 1 μ α ϕ 1+μα ϕ μ α ϕ (k) k + μ α ϕ Lemma 3 Browkin-Brzezinski s conjecture is true for polynomials f 0,, f n+1 such that gdc(f i,f j,f k )=1for all distinct i, j, k {0,, n +1} and f 0,, f n are linearly independent

3 Browkin-Brzezinski s conjecture 1337 Proof By the hypothesis f 0,, f n are linearly independent, we have the Wronskian W of f 0,, f n does not vanish We set P = W (f 0,, f n ) f 0 f n, Hence we have We first prove that Q = f 0f n+1 W (f 0,, f n ) f n+1 = PQ () deg Q (n 1)r(f 0 f n+1 ) Suppose that α is a zero of f 0 f 1 f n+1, by the hypothesis there exists ν, 0 ν n + 1 such that f ν 0 By the hypothesis f f n = f n+1 we have μ α f 0 f n+1 W (f 0,,fn) = μ α f 0 f ν 1 f ν+1 f n+1 W (f 0,,f ν 1,f ν+1,,f n+1 ) = n+1 j=0 μ α f j μ α W (f 0,,f ν 1,f ν+1,,f n+1 ) W (f 0,, f ν 1,f ν+1,, f n+1 ) is the sum of follow terms δf α0 (f α1 ) (f αn ) (n), Where α i {0, n +1}\{ν},δ = ±1 We suppose there exists k functions f j such that f j (α) =0, By the hypothesis gdc(f i,f j,f k ) = 1 for all distinct i, j, k {0,, n +1} we have k From this and Lemma 1 and we have μ α f α0 (f α1 ) (f αn ) (n) f j (α)=0 μ α f αj (n +(n 1)) = μ α n+1 (n 1) j=0 fα j By Lemma we have Hence μ α W (f 0,,f ν 1,f ν+1,f n+1 ) μα n+1 (n 1) j=0 fα j μ α f 0 f n+1 W (f 0,,fn) n 1,

4 1338 Nguyen Thanh Quang and Phan Duc Tuan By the definition of degree of a rational function, we have: Next, we will prove that deg Q (n 1)r(f 0 f n+1 ) (3) n(n +1) deg P At here, we have P as the logarithmic Wronskian corresponding to I = {0, 1,, n} which is f 0 f 1 f n f 0 f 1 f n 1 f 1 0 f 0 n f n The determinant P is a summa of following terms For every term, we have deg f β 1 f β β n f β1 f β f βn Therefore From (), (3), (4) we have δ f β 1 f β β n f β1 f β f βn = deg ( f β1 f β1 ) + deg ( ) ( f β f β + + deg = ( n) = n(n+1) deg P βn f βn n(n +1) (4) deg f n+1 = deg P + deg Q (n 1)r(f 0 f n+1 ) (n 1) (r(f 0 f n+1 ) 1) Similar arguments apply to the polynomial f 0,f 1,, f n, we have max (degf i) (n 1) (r(f 0 f n+1 ) 1) 0 ı n+1 ) n(n +1) Proof of theorem 11 The proof proceed by induction on n Forn =1, it is true by Mason s theorem Suppose that the theorem is true for all case m, 1 m n 1 If f 0,, f n are linearly independent, then this is Lemma 3 If f 0,, f n are linearly dependent, rewriting (1) as f n+1 = f f n, (5)

5 Browkin-Brzezinski s conjecture 1339 Let f i1,, f iq,q <n+1 be a maximal linearly independent subset of the f j,j = 0,, n, since n 1 and gdc(f i,f j,f k ) = 1 for all distinct i, j, k {0,, n +1} it follows that q Then each f j, 0 j n, j not one of i k, is a linear combination of the f ik, of the form f j = λ 1 f j1 + + λ q f iq (6) where the λ k F, and at least two of these λ k are not zero Using our inductive hypothesis we apply the theorem to (6) This yields that if λ k 0 then So that q deg(f ik ) (q 1)(r(f j f ik ) 1) (7) k=1 n+1 deg(f ik ) (n 1)(r( f k ) 1) (8) From (6) the same estimate as in (8) follows for deg f j Thus the theorem is proved for such f j and f ik Inserting all the relations of the form (6) into the right side of (5) yields an equation of the form k=0 f n+1 = κ 1 f j1 + + κ q f iq, (9) where the κ j F Moreover, if one of these κ υ = 0 then the corresponding f iυ must have appeared in one of the equations (6) with a non-zero λ υ Hence (7) has been established for this f iυ Finally, for those κ υ 0, we treat (9) exactly as we did (6), (note that q +1<n+ 1), and obtain the estimate (8) for deg f iυ, and for deg f n+1 This completes the induction References [1] Browkin, J and Brzezinski, J, Some remarks on the abc conjecture, Mathematics of Computation, 6, (1994) [] PC Hu and CCYang, Notes on a generalized abc-conjecture over function fields, Ann Math Blaise Pascal 8 (001), no 1, [3] Lang, S, Old and new conjectured Diophantine inequalities, Bull Amer Math Soc 3 (1990), 37?75 [4] Leonid N Vaserstein and Ethel R Wheland, Vanishing polynomial sums, Communications in Algebra, 31, No, (003), [5] Mason, R C, Equations over function fields, Lecture Notes in Math 1068 (1984), , Springer

6 1340 Nguyen Thanh Quang and Phan Duc Tuan [6] Mason, RC, Diophantine equations over function fields, London Math Soc Lec-ture Note Ser 96, Cambridge Univ Press, Cambridge, 1984 [7] Nguyen Thanh Quang and Phan Duc Tuan, Siu-Yeng s lemma in the p- adic case, Vietnam Journal of Mathematics, 3:, No (004), 7-34 [8] HN Shapiro and GHSparer, Extension of a Theorem of Mason, Commm Pure and Appl Math, 47 (1994), Received: May 11, 007