Tài liệu Báo cáo " GENERALIED MASON’S THEOREM " ppt

VNU. JOURNAL OF SCIENCE, Mathematics - Physics. T.XXI, N 0 4 - 2005 GENERALIED MASON’S THEOREM Nguyen Thanh Quang, Phan Duc Tuan Department of Mathematics, Vinh University Abstract. The purpose of this pap er is to give a generalization of Mason’s theorem by the Wronskian technique over fields of characteristic 0. Keywords: The Wronskian technicque, Marson’s theorem. 1. Introduction Let F be a fixed algebraically closed field of characteristic 0. L et f(z)beapoly- nomial non - constants which coefficients in F and let n(1/f) be the number of distinct zeros of f. Then we have the following. Marson’s theorem. ([2]). 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)} a n  1 abc  − 1. It is now well known that Mason’s Theorem implies the following corollary. Corollary. (Fermat’s Theorem over polynomials). The equation x n + y n = z n has no solutions in non - constants and relatively prime polynomials in F if n a 3. The main theorem in this paper is as following: Theorem 1.1. Les f 0 ,f 1 , ,f n be relatively primer polynomials and f 0 ,f 1 , ,f n be lin- early independent over F. If f 0 + f 1 + + f n = f n+1 , then max 0aian+1 deg f i a n  n+1    i=0 n  1 f i   − n(n +1) 2 . Remark. Theorem 1.1 is a generalization of Mason’s theorem which was obtained for case n =1. Typeset by A M S-T E X 34 Generalied Mason’s Theorem 35 2. Proof of the main theorem Let ϕ(x)= f(x) g(x) ≡ 0 be a rational function, where f(x),g(x) are non - zero and relatively prime polynomials on F . The degree of ϕ(x), denoted by deg ϕ(x), is defined to be deg f(x) − deg g(x). Here the n otation deg f(x)meansthedegreeofpolynomialf(x). From the properties of polynomial, we ha ve. Proposition 2.1. If ϕ 1 and ϕ 2 are the rational functions on F, then 1) deg(ϕ 1 ϕ 2 )=degϕ 1 +degϕ 2 2) deg  1 ϕ 1  = − deg ϕ 2 3) deg(ϕ 1 + ϕ 2 ) a max(deg ϕ 1 , deg ϕ 2 ). Definition 2.2. Let ϕ(x) ≡ 0 be a rational function on F.Foreverya ∈ F, we write ϕ(x)=(x − α) m f 1 (x) g 1 (x) , (m ∈ Z), where f 1 (x),f 2 (x) are relatively prime polynomials and f 1 (α) =0,g 1 (α) =0. We call m order of ϕ at α. Proposition 2.3. If ϕ 1 , ϕ 2 are rational functions on F and a ∈ F, then 1) ord α (ϕ 1 ϕ 2 )=ord α ϕ 1 + ord α ϕ 2 2) ord α ( 1 ϕ 1 )=−ord α ϕ 1 3) ord α ( ϕ 1 ϕ 2 )=ord α ϕ 1 − ord α ϕ 2 . Proposition 2.4. Let ϕ(x) be a the rational function on F and let the derivatives order k, ϕ (k) ≡ 0. Then ord α  ϕ (k) ϕ   −k. Proof. Let ϕ(x)=(x − α) m f(x) g(x) , whe re f(x),g(x) are relatively prime and f(α)g(α) =0. Then, we ha ve ϕ  (x)=(x − α) m−1 (mf(x)+(x − α)f  (x)) + (x − α)f(x)g  (x) g 2 (x) . Since ord α (g(x)) = 0, we have ord α (ϕ  (x))  m − 1. Therefore ord α  ϕ  ϕ  = ord α (ϕ  ) − ord α (ϕ)  −1. 36 Nguyen Thanh Quang, Phan Duc Tuan Thus, we obtain ord α  ϕ (k) ϕ  = ord α  ϕ  ϕ . ϕ  ϕ  ϕ (k) ϕ (k−1)  = ord α  ϕ  ϕ  + ord α  ϕ  ϕ   + + ord α  ϕ (k) ϕ (k−1)   −k (1) Proposition 2.5. Let ϕ 1 , ϕ 2 be rational functions on F and a ∈ F. Then ord α (ϕ 1 , ϕ 2 )  min{ord α ϕ 1 ,ord α ϕ 2 } . Proof. Let ord α ϕ 1 = m 1 and ord α ϕ 2 = m 2 . Then ϕ 1 (x)=(x − α) m f 1 (x) g 1 (x), (2) ϕ 2 (x)=(x − α) m f 2 (x) g 2 (x), (3) where f 1 ,f 2 ,g 1 ,g 2 are the polynomials o ver F and f 1 (α),f 2 (α),g 1 (α),g 2 (α) =0. We set m = min(m 1 ,m 2 ). Then ϕ 1 (x)+ϕ 2 (x)=(x − α) m  (x − α) m 1 −m f 1 (x)g 2 (x)+(x − α) m 2 −m f 2 (x)g 1 (x)  f 2 (x)g 2 (x) . Since f 2 (α)g 2 (α) =0, we have ord α (ϕ 1 + ϕ 2 )  m = min(ord α ϕ 1 ,ord α ϕ 2 ). Definition 2.6. Let f 1 ,f 2 , , f n be polynomials on F (but to a large extent what we do depends only on formal properties of devivations). We recall that their W ronskian is W (f 1 ,f 2 , ,f n )=         f 1 f 2 ··· f n f  1 f  2 ··· f  n . . . . . . . . . . . . f (n−1) 1 f (n−1) 2 ··· f (n−1) n         Remark. If f 1 ,f 2 , , f n are linearly independent on F, then W(f 1 ,f 2 , ,f n ) =0. Proof of Theorem 1.1. Let {α 0 , α 1 , ,α n } be a subset of I = {0, 1, ,n+1}. Then the equation f 0 + f 1 + + f n = f n+1 implies W (f α 0 , , f α n )=δW (f 0 ,f 1 , , f n ), where δ =1or−1. Because f 0 ,f 1 , f n are linearly i ndependent, we obtain W (f 0 ,f 1 , ,f n ) =0. Generalied Mason’s Theorem 37 Then, we set P (t)= W (f 0 ,f 1 , ,f n ) f 0 f 1 f n , Q(t)= f 0 f 1 f n+1 W (f 0 ,f 1 , ,f n ) . Hence, we ha ve f n+1 = P(t)Q(t). We first prove that degQ(t) a n  n+1    i=0 n  1 f i   . Let α be a zero of the function Q(z). Then α is a zero of some polynomial f i (0 a i a n +1). By the hypothesis that the polynomials are relatively prime, there exists a number v(0 a v a n + 1) s uch that f v (α) =0. Let {i 0 ,i 1 , ,i n } be a subset I|{v},thenwehave Q(t)=δ f i 0 f i 1 f i n W (f 0 ,f 1 , ,f n ) f v . Denote R(t)= W (f i 0 ,f i 1 , , f i n ) f i 0 f i 1 f i n as the logarithmic Wronskian corresponding to {i 0 ,i 1 , , i n }, which is           11··· 1 f  i 0 f i 0 f  i 1 f i 1 ··· f  i n f i n . . . . . . . . . . . . f (n−1) i 0 f i 0 f (n−1) i 1 f i 1 ··· f (n−1) i n f i n           Then f v = R(t)Q(t)andsoord α R(t)=−ord α Q(t). Then the determinant R(t)is asumoffollowingterms δ f  α 0 f  α 1 f (n−1) α n f α 0 f α 1 f α n , where 0 a α 0 , α 1 , ,α n a n +1and δ =1or−1. By applyi ng the propositions 2.3 and 2.4, we get ord α  f  α 0 f  α 1 f (n−1) α n f α 0 f α 1 f α n  = ord α  f  α 0 f α 0  + ord α  f  α 1 f α 1  + + ord α  f (n−1) α n f α n   −n     0aian+1 f i (a)=0 1    . (4) 38 Nguyen Thanh Quang, Phan Duc Tuan Therefore from Proposition 2.5, we have ord α R(t) a −n     0aian+1 f i (a)=0 1    and so ord α Q(t)=−ord α R(t) a −n     0aian+1 f i (a)=0 1    . Since this inequality holds for any zero α of Q(t), we get deg Q(t) a n  n+1    i=0 n  1 f i   . Next,wewillprovethat deg P(t) a − n(n +1) 2 . Here, we have P(t) as the logarithmic Wronskian corresponding to I = {0, 1, ,n} which is          11··· 1 f  0 f 0 f  1 f 1 ··· f  n f n . . . . . . . . . . . . f (n) 0 f 0 f (n) 1 f 1 ··· f (n) n f n          The determinant P (t)isasumoffollowingterms δ f  β 0 f  β 1 f (n−1) β n f β 0 f β 1 f β n . For every term, by Proposition 2.4 we have deg  f  β 0 f  β 1 f (n−1) β n f β 0 f β 1 f β n  =deg  f  β 0 f β 0  deg  f  β 1 f β 1  + +deg  f (n) β n f β n  = −(1 + 2 + + n)=− n(n +1) 2 . (5) Therefore deg P(t) a − n(n +1) 2 so deg f n+1 =degP(t)+degQ(t) a n  n+1    i=0 n  1 f i   − n(n +1) 2 . Generalied Mason’s Theorem 39 By the similar arguments applying to the polynomial f 0 ,f 1 , , f n , we have max 0a⊂an+1 (degf i ) a n  n+1    i=0 n  1 f i   − n(n +1) 2 . Theorem 1.1 is proved. References 1. S. Lang, Introduction to Complex Hyperbolic Spaces, Springer - Verlag, (1987). 2. S. Lang, Old and new conjecture Diophantine inequalitis, Bull.Amer.Math.Soc., 23 (1990), 37 - 75. 3. R. C. Mason, Diophan tine Equations over Function Fields, London Math. Soc., Lecture Notes, Cambridge Univ. Press, Vol. 96 (1984). 4. M. Ru and J. T. - Y. Wang, A second main type inequality for holomorphic curves intersecting hyperplanes, Preprint. . +1) 2 . Remark. Theorem 1.1 is a generalization of Mason’s theorem which was obtained for case n =1. Typeset by A M S-T E X 34 Generalied Mason’s Theorem 35 2 generalization of Mason’s theorem by the Wronskian technique over fields of characteristic 0. Keywords: The Wronskian technicque, Marson’s theorem. 1. Introduction Let