Mason’s theorem and the similarities among numeric fields and coronary polynomials

In this article, we study about some similarities among numeric fields and coronary polynomials. We propose some theorems about polynomials which are similar to theorems and hypotheisises of arithmetic. Also, we use the Mason’s theorem to prove mentioned theorems.

Ha Noi Metroplolitan University 28 MASON’S THEOREM AND THE SIMILARITIES AMONG NUMERIC FIELDS AND CORONARY POLYNOMIALS Pham Xuan Hinh Hanoi Metropolitan University Abstract: In this article, we study about some similarities among numeric fields and coronary polynomials We propose some theorems about polynomials which are similar to theorems and hypotheisises of arithmetic Also, we use the Mason’s theorem to prove mentioned theorems Keywords: Similarities among numeric fields and coronary polynomials Email: pxhinh@hnmu.edu.vn Received November 2018 Accepted for publication 15 December 2018 INTRODUCTION The development of arithmetic, especially in recent decades, is greatly influenced by the similarities among integers and polynomials, among numeric fields and coronary polynomials In other words, when some hypothesises had not been proven for integers, these events were tried to prove for polynomials In this paper, we mention some similarities among integers and polynomials and use Mason’s theorem to prove the theorems of polynomials MASON’S THEOREM 2.1 Definition First of all, it is clear that the set of integers and the set of polynomials have similar properties listed as follows: • The rules of addition, subtraction, multiplication, division are exactly the same in both sets • If for integers we have prime numbers, then for polynomials we have irreducible polynomials Scientific Journal − No27/2018 29 • For two integers as well as for two polynomials, the largest common denominator can be defined Furthermore, in both cases, the largest common denominator is found by the Euclidean algorithm • Each integer can be analysed into prime factors Each polynomial can be analysed into the product of irreducible polynomials We can extend this list Now we study some more difficult similarities We notice the similarity among analysing prime factors and analysing irreducible polynomials If K is a closed algebraic field, then every polynomial f (x) ∈ K [ x ] can be analysed as: f(x)= p1α1 p2α pnα n where pi ( x) = x − ; ∈ K Thus, we can see that, among the analysis of the irreducibles and the analysis of the prime factors, the solutions of the polynomials is corresponding to the prime divisors of the integers Thus the number of distinct solutions of polynomials has the same role as the number of distinct divisors of the integer From that comment we have the following definition: Definition Let a be an integer We define the root of a, denoted by N0(a), is the product of distinct prime divisors of a: N (a ) = ∏ p p/a 2.2 Mason’s theorem Assume that P, Q, R are polynomials of one variable with complex, relatively prime pairwise coefficients satisfying P + Q = R Then if we denote the number of distinct solutions of a polynomial f by n0(f) then we have: max {deg P, deg Q, deg R} ≤ n0 ( PQR) − Proof Let degP = k and P has solutions a1 , a2 , , am whose their multiples are α1 , α , , α m for α1 + α + + α m = k And, deg Q = l and Q has solutions b1 , b2 , , bs whose their multiples are β1 , β , , β s for β1 , β , , β s = l, degR =p; R has solutions c1 , c2 , , cr whose their multiples are γ , γ , , γ r for γ + γ + , γ r = p Since P, Q, R are relatively prime polynomials, then ≠ bt ≠ c j Since P + Q = R ⇒ P / R + Q / R = Put f = P / R; g = Q / R ⇒ f + g = ⇒ f '+ g ' = Ha Noi Metroplolitan University 30 ⇒ f f '/ f + g g '/ g = ⇒ − ⇒ We have: f g '/ g = g f '/ f − P g '/ g = and : f ' : f = (log f ) '; g '/ g = (log g ) ' Q f '/ f P = ∏ ( z − a1 )α i ; Q = ∏ ( z − bt ) βt ; R = ∏ ( z − c j ) s ⇒ P g '/ g =− =− Q f '/ f βt r γj ∑ z −b −∑ z −c t =1 m t αi j =1 r i j γj ∑ z −a −∑ z −c i =1 γj j =1 j The last quotient has the same denominator and numerator and the denominator is: N0= ∏ ( z − ).∏ ( z − bt ).∏ ( z − c j ) It is a polynomial of degree n0(PQR) Then, N0 f '/ f N0 g '/ g are polynomials of degrees which are not greater n0(PQR)-1 On the other hand, we have: P N g '/ g = Q N f '/ f Since (P, Q) = then we have deg P ≤ r + s + m − degQ ≤ r + s + m − Similarly, we get: deg R ≤ r + s + m − Therefore, max {deg P, degQ, degR} ≤ r + m + s − = n0 ( PQR) − (QED) SOME THEOREMS OF POLYNOMIALS SIMILAR TO THEOREMS AND HYPOTHESISES OF ARITHMETIC In this section we present some similarities among numbers and polynomials through some theorems, hypotheisies of arithmetic, and theorems of polynomials 3.1 Fermat’s theorem For all integers n ≥ , the equation: xn + yn = zn does not have solutions which are integers (except trivial solutions) Scientific Journal − No27/2018 31 3.2 Consequences of Mason’s theorem (Similar to Fermat's theorem) For all integers n ≥ , there not exist non-constant polynomials P, Q, R with complex relatively-prime coefficients satisfying the equation: pn + Qn = Rn Proof Assume polynomials P, Q, R satisfy the above equation It is clear that the number of distinct solutionS of the polynomial PnQnRn is not greater then deg P + deg Q + deg R Applying Mason’s theorem, we have: { } m ax deg P n , deg Q n , deg R n ≤ n0 ( P n Q n R n ) − Then n.max {deg P, deg Q,deg R} ≤ n0 ( PQR) − = deg( PQR ) − = deg P + deg Q + deg R − ⇒ n.deg P ≤ deg P + deg Q + deg R − n.deg Q ≤ deg P + deg Q + deg R − n.deg R ≤ deg P + deg Q + deg R − Add these above inequalities by sides, we get: n (deg P + deg Q + deg R ) ≤ 3(deg P + deg Q + deg R ) − (This is a contradiction for n ≥ 3) , QED 3.2.1 Arithmetic Theorem (Hall hypothesis) Let x, y be positive integers such that: x3 ≠ y2 Then for any ε > , there always exists a constant number C (ε ) > such that: y − x > C x1/ −ε 3.2.2 Davenport theorem (Similar to Hall hypothesis) If P and Q are relatively prime polynomials and P ≠ Q3 , then we have: deg( P − Q3 ) ≥ 1/ 2deg Q + Proof Put R = P − Q3 ⇔ R + Q3 = P Applying Mason’s theorem, we have: Ha Noi Metroplolitan University 32 { } max deg R, deg Q3 deg P ≤ n0 ( P RQ3 ) − 2 ⇒ deg P ≤ n0 ( P Q R) − deg Q ≤ n0 ( P 2Q R ) − ⇒ deg P ≤ deg P + deg Q + deg R − deg Q ≤ deg P + deg Q + deg R − ⇒ deg P + 3deg Q ≤ 2(deg P + deg Q + deg R − 1) ⇒ deg R ≥ / deg Q + (QED) 3.3 Arithmetic hypothesis There not exist integers x, y , z ≠ pairwise relatively prime such that x 2018 + y 2019 = z 2020 Theorem (Similar to the hypothesis 3.3.1) There not exist polynomials P, Q, R, pairwise relatively prime such that P 2018 + Q 2019 = R 2020 Proof Suppose that there exist polynomials P,Q,R, pairwise relatively prime such that P 2018 + Q 2019 = R 2020 Applying Mason’s theorem, we have: { } m ax deg P 2018 , deg Q 2019 , deg R 2020 ≤ n0 ( P 2018 Q 2019 R 2020 ) − ⇔ max {2018deg P, 2019 deg Q, 2020 deg R} ≤ deg P + deg Q + deg R − Then 2018 deg P ≤ deg P + deg Q + deg R − 2019 deg Q ≤ deg P + deg Q + deg R − 2020 deg R ≤ deg P + deg Q + deg R − 2018 deg P + 2019 deg Q + 2020 deg R ≤ 3(deg P + deg Q + deg R − 1) 2015 deg P + 2016 deg Q + 2017 deg R ≤ −3 (This is a contradiction) QED 3.4 Fermat’s expanding theorem If m, n, k are positive integers satisfying: numbers a, b, c such that: a m + b n = c k 1 + + < , then there not exist m n k Scientific Journal − No27/2018 33 Theorem (Similar to Fermat’s expanding theorem) 1 + + < , then there not exist m n k non-zero degree polynomials P, Q, R such that P m + Q n = R k If there are positive integers m, n, k satisfying Proof Suppose that there exist non-zero degree polynomials P, Q, R such that: P + Q n = R k Applying Mason’s theorem, we have: m { } m ax deg P m , deg Q n , deg R k ≤ n0 ( P m Q n R k ) − ⇔ max {m deg P, n deg Q, k deg R} ≤ deg P + deg Q + deg R − Then m deg P ≤ deg P + deg Q + deg R − (*) n deg Q ≤ deg P + deg Q + deg R − k deg R ≤ deg P + deg Q + deg R − It follows from (*) that m deg P < deg P + deg Q + deg R Then deg P > m deg P + degQ + degR (1) Similarly, we have: degQ > n deg P + degQ+ degR (2) degR > k deg P + degQ+ degR (3) Add (1), (2), (3) by sides we get 1 + + > (This is a contradiction) QED m n k 3.5 Arithmetic hypothesis Let x and y be integers such that x7 ≠ y Then for every ε > , there exists a constant number C depending on ε such that x − y > C (ε ) 23/ +ε Theorem (Similar to the hypothesis 3.5) Let P and Q be polynomials with coefficients which are integers and P ≠ Q5 Prove 23 that: deg( P − Q ) ≥ deg Q + Proof Put: P − Q5 = R ⇔ P + R = Q5 Applying Mason’s theorem, we get: { } m ax deg P , deg Q , deg R ≤ n0 ( P Q R ) − Ha Noi Metroplolitan University 34 ⇔ max {7 deg P,5deg Q,deg R} ≤ deg P + deg Q + deg R − ⇒ deg P ≤ deg P + deg Q + deg R − (1) ⇒ deg Q ≤ deg P + deg Q + deg R − ⇒ 30 deg Q ≤ deg P + deg Q + deg R − (2) Add (1) and (2) by sides, we get: deg P + 30 deg Q ≤ deg P + deg Q + deg R − ⇔ deg R ≥ 23deg Q + ⇔ deg R ≥ 23 deg Q + (QED) CONCLUSION In the above study of some similarities among the integers and polynomials, we used Mason’s theorem to prove the consequences of Mason’s theorem, the Davenport theorem and propose three theorems (Theorem 1, Theorem 2, Theorem 3) of polynomials which are similar to theorems and hypotheses of arithmetic, then we can develop other theorems of polynomials similar to other theorems and hypothesises of arithmetic We hope that there will be more results in arithmetic study REFERENCES Pham Xuan Hinh (2000), “About the similarities among numeric field and field functional field”, Master Thesis of Mathematics, Institute of Mathematics Ha Huy Khoai, Pham Huy Dien (2003), Arithmetic Algorithm Theoretical Basis and Practical Computation, - Hanoi National University Publishing House Nitaj (1996), “La conjecture abc”, - Einsengment Matthématique, N042, pp.3-24 Langevin (1993), “Cas d’égalité pour le theorem de Mason et applications de la conjecture (abc)”, - C R Acad Sei Paris, t 317, Série I, pp.441-444 Stewart.C.L and Kunrui YU (1991), “On the abc - conjecture”, - Math Ann 291, pp.225-230 ĐỊNH LÝ MASON VÀ SỰ TƯƠNG TỰ GIỮA TRƯỜNG SỐ VÀ VÀNH ĐA THỨC Tóm tắ tắt: Trong báo nghiên cứu số tương tự trường số vành đa thức, Chúng đề xuất số định lý đa thức tương tự với định lý giả thuyết số học Đồng thời sử dụng định lý Mason để chúng minh định lý nêu Từ khóa: Tương tự trường số vành đa thức  ... Mason’s theorem to prove the consequences of Mason’s theorem, the Davenport theorem and propose three theorems (Theorem 1, Theorem 2, Theorem 3) of polynomials which are similar to theorems and. .. (QED) SOME THEOREMS OF POLYNOMIALS SIMILAR TO THEOREMS AND HYPOTHESISES OF ARITHMETIC In this section we present some similarities among numbers and polynomials through some theorems, hypotheisies... similar to theorems and hypotheses of arithmetic, then we can develop other theorems of polynomials similar to other theorems and hypothesises of arithmetic We hope that there will be more results