In this note we construct several infinite families of diagonal quartic surfaces ax 4 + by 4 + cz... more In this note we construct several infinite families of diagonal quartic surfaces ax 4 + by 4 + cz 4 + dw 4 = 0, where a, b, c, d ∈ Z \ {0} with infinitely many rational points and satisfying the condition abcd =. In particular, we present an infinite family of diagonal quartic surfaces defined over Q with Picard number equal to one and possessing infinitely many rational points. Further, we present some sextic surfaces of type ax 6 + by 6 + cz 6 + dw i = 0, i = 2, 3, or 6, with infinitely many rational points.
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service... more This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ON CERTAIN DIOPHANTINE EQUATIONS OF THE FORM z 2 = f (x) 2 ± g(y) 2 SZ. TENGELY AND M. ULAS
In this paper we investaigate Diophantine equations of the form T 2 = G(X), X = (X 1 ,. .. , Xm),... more In this paper we investaigate Diophantine equations of the form T 2 = G(X), X = (X 1 ,. .. , Xm), where mainly m = 3 or m = 4 and G specific homogenous quintic form. First, we prove that if F (x, y, z) = x 2 + y 2 + az 2 + bxy + cyz + dxz ∈ Z[x, y, z] and (b − 2, 4a − d 2 , d) = (0, 0, 0), then the Diophantine equation t 2 = nxyzF (x, y, z) has solution in polynomials x, y, z, t with integer coefficients, without polynomial common factor of positive degree. In case a = d = 0, b = 2 we prove that there are infinitely many primitive integer solutions of the Diophantine equation under consideration. As an application of our result we prove that for each n ∈ Q \ {0} the Diophantine equation 2010 Mathematics Subject Classification. 11D41.
Let p be a prime number and consider a p-automatic sequence u = (un) n∈N and its generating funct... more Let p be a prime number and consider a p-automatic sequence u = (un) n∈N and its generating function U (X) = ∞ n=0 unX n ∈ Fp[[X]]. Moreover, let us suppose that u 0 = 0 and u 1 = 0 and consider the formal power series V ∈ Fp[[X]] which is a compositional inverse of U (X), i.e., U (V (X)) = V (U (X)) = X. In this note we initiate the study of arithmetic properties of the sequence of coefficients of the power series V (X). We are mainly interested in the case when un = tn, where tn = s 2 (n) (mod 2) and t = (tn) n∈N is the Prouhet-Thue-Morse sequence defined on the two letter alphabet {0, 1}. More precisely, we study the sequence c = (cn) n∈N which is the sequence of coefficients of the compositional inverse of the generating function of the sequence t. This sequence is clearly 2-automatic. We describe the sequence a characterizing solutions of the equation cn = 1. In particular, we prove that the sequence a is 2-regular. We also prove that an increasing sequence characterizing solutions of the equation cn = 0 is not k-regular for any k. Moreover, we present a result concerning some density properties of a sequence related to a.
In this paper we deal with a problem of Pethő related to existence of quartic algebraic integer α... more In this paper we deal with a problem of Pethő related to existence of quartic algebraic integer α for which 2010 Mathematics Subject Classification. Primary 11D61; Secondary 11Y50.
We prove several theorems concerning arithmetic properties of Stern polynomials defined in the fo... more We prove several theorems concerning arithmetic properties of Stern polynomials defined in the following way: B 0 (t) = 0, B 1 (t) = 1, B 2n (t) = tBn(t), and B 2n+1 (t) = Bn(t) + B n+1 (t). We study also the sequence e(n) = deg t Bn(t) and give various properties of it. ∞ i=0 ǫ i 2 i , where ǫ i ∈ {0, 1, 2}. These are called hiperbinary representations. The connections of the Stern sequence with continued fractions and the Euclidean algorithm are considered in [6] and [7]. An interesting application of the Stern sequence to the problem of construction a bijection between N + and Q + is given in [1]. In this paper it is shown that the sequence s(n)/s(n + 1), for n ≥ 1, encounters every positive rational number exactly once. A comprehensive survey of properties of the Stern sequence can be found in [12]. An interesting survey of known results and applications of the Stern sequence can also be found in [8]. Recently two distinct polynomial analogues of the Stern sequence appeared. The sequence of polynomials a(n; x) for n ≥ 0, defined by a(0; x) = 0, a(1; x) = 1, and for n ≥ 2: a(2n; x) = a(n; x 2), a(2n + 1; x) = xa(n; x 2) + a(n + 1; x 2), was considered in [2]. It is easy to see that a(n; 1) = s(n). Remarkably, as was proved in the cited paper, xa(2n − 1; x) ≡ A n+1 (x) (mod 2), where A n (x) = n j=0 S(n, j)x j and S(n, j) are the Stirling numbers of the second kind. Further 2010 Mathematics Subject Classification. 11B83. Key words and phrases. Stern diatomic sequence, Stern polynomials. The first named author is holder of START scholarship funded by the Foundation for Polish Science (FNP).
The material of this paper was presented in part at the 11th International Workshop for Young Mat... more The material of this paper was presented in part at the 11th International Workshop for Young Mathematicians - NUMBER THEORY, Kraków, 14th-20th september 2008.
Let K be a non-algebraically closed field and f ∈ K[x] be an irreducible polynomial of degree d ≥... more Let K be a non-algebraically closed field and f ∈ K[x] be an irreducible polynomial of degree d ≥ 3. In this paper we study the existence of a polynomial h ∈ K[x] of degree ≤ d−1 such that the polynomial f (h(x)) is reducible in K[x]. We prove the existence of such polynomial in case of d ≤ 4. A similar (and quite general) result is true in the case when f represents an even function, i.e., f (x) = f (−x). In particular, if char(K) = 2, then we prove that the existence of a required polynomial h.
Let σ i (x 1 ,. .. , xn) = 1≤k 1 <k 2 <...<k i ≤n x k 1. .. x k i be the i-th elementary symmetri... more Let σ i (x 1 ,. .. , xn) = 1≤k 1 <k 2 <...<k i ≤n x k 1. .. x k i be the i-th elementary symmetric polynomial. In this note we generalize and extend the results obtained in a recent work of Zhang and Cai [8, 9]. More precisely, we prove that for each n ≥ 4 and rational numbers a, b with ab = 0, the system of diophantine equations σ 1 (x 1 ,. .. , xn) = a, σn(x 1 ,. .. , xn) = b, has infinitely many solutions depending on n − 3 free parameters. A similar result is proved for the system σ i (x 1 ,. .. , xn) = a, σn(x 1 ,. .. , xn) = b, with n ≥ 4 and 2 ≤ i < n. Here, a, b are rational numbers with b = 0. We also give some results concerning the general system of the form σ i (x 1 ,. .. , xn) = a, σ j (x 1 ,. .. , xn) = b, with suitably chosen rational values of a, b and i < j < n. Finally, we present some remarks on the systems involving three different symmetric polynomials.
G. Campbell described a technique for producing infinite families of quartic elliptic curves cont... more G. Campbell described a technique for producing infinite families of quartic elliptic curves containing a length-9 arithmetic progression. He also gave an example of a quartic elliptic curve containing a length-12 arithmetic progression. In this note we give a construction of an infinite family of quartics on which there is an arithmetic progression of length 10. Then we show that there exists an infinite family of quartics containing a sequence of length 12.
Let m be a positive integer and bm(n) be the number of partitions of n with parts being powers of... more Let m be a positive integer and bm(n) be the number of partitions of n with parts being powers of 2, where each part can take m colors. We show that if m = 2 k − 1, then there exists the natural density of integers n such that bm(n) can not be represented as a sum of three squares and it is equal to 1/12 for k = 1, 2 and 1/6 for k ≥ 3. In particular, for m = 1 the equation b 1 (n) = x 2 + y 2 + z 2 has a solution in integers if and only if n is not of the form 2 2k+2 (8s + 2ts + 3) + i for i = 0, 1 and k, s are non-negative integers, and where tn is the nth term in the Prouhet-Thue-Morse sequence. A similar characterization is obtained for the solutions in n of the equation b 2 k −1 (n) = x 2 + y 2 + z 2 .
In this note we consider Diophantine equations of the form a(x p − y q) = b(z r − w s), where 1 p... more In this note we consider Diophantine equations of the form a(x p − y q) = b(z r − w s), where 1 p + 1 q + 1 r + 1 s = 1, with even positive integers p, q, r, s. We show that in each case the set of rational points on the underlying surface is dense in the Zariski topology. For the surface with (p, q, r, s) = (2, 6, 6, 6) we prove density of rational points in the Euclidean topology. Moreover, in this case we construct infinitely many parametric solutions in coprime polynomials. The same result is true for (p, q, r, s) ∈ {(2, 4, 8, 8), (2, 8, 4, 8)}. In the case (p, q, r, s) = (4, 4, 4, 4), we present some new parametric solutions of the equation x 4 − y 4 = 4(z 4 − w 4).
Let t n = (−1) s 2 (n) , where s 2 (n) is the sum of binary digits function. The sequence (t n) n... more Let t n = (−1) s 2 (n) , where s 2 (n) is the sum of binary digits function. The sequence (t n) n∈N is the well-known Prouhet-Thue-Morse sequence. In this note we initiate the study of the sequence (h n) n∈N , where h 0 = 0, h 1 = 1 and for n ≥ 2 we define h n recursively as follows: h n = t n h n−1 + h n−2. We prove several results concerning arithmetic properties of the sequence (h n) n∈N. In particular, we prove non-vanishing of h n for n ≥ 5, automaticity of the sequence (h n (mod m)) n∈N for each m, and other results.
Given an elliptic quartic of type Y 2 = f (X) representing an elliptic curve of positive rank ove... more Given an elliptic quartic of type Y 2 = f (X) representing an elliptic curve of positive rank over Q, we investigate the question of when the Y-coordinate can be represented by a quadratic form of type ap 2 + bq 2. In particular, we give examples of equations of surfaces of type c 0 + c 1 x + c 2 x 2 + c 3 x 3 + c 4 x 4 = (ap 2 + bq 2) 2 , a, b, c ∈ Q where we can deduce the existence of infinitely many rational points. We also investigate surfaces of type Y 2 = f (ap 2 + bq 2) where the polynomial f is of degree 3.
Let A ⊂ N + and by P A (n) denotes the number of partitions of an integer n into parts from the s... more Let A ⊂ N + and by P A (n) denotes the number of partitions of an integer n into parts from the set A. The aim of this paper is to prove several result concerning the existence of integer solutions of Diophantine equations of the form P A (x) = P B (y), where A, B are certain finite sets.
Let f ∈ Q[x] be a square-free polynomial of degree ≥ 3 and m ≥ 3 be an odd positive integer. Base... more Let f ∈ Q[x] be a square-free polynomial of degree ≥ 3 and m ≥ 3 be an odd positive integer. Based on our earlier investigations we prove that there exists a function D 1 ∈ Q(u, v, w) such that the Jacobians of the curves C 1 : D 1 y 2 = f (x), C 2 : y 2 = D 1 x m + b, C 3 : y 2 = D 1 x m + c, have all positive ranks over Q(u, v, w). Similarly, we prove that there exists a function D 2 ∈ Q(u, v, w) such that the Jacobians of the curves C 1 : D 2 y 2 = h(x), C 2 : y 2 = D 2 x m + b, C 3 : y 2 = x m + cD 2 , have all positive ranks over Q(u, v, w). Moreover, if f (x) = x m + a for some a ∈ Z \ {0}, we prove the existence of a function D 3 ∈ Q(u, v, w) such that the Jacobians of the curves C 1 : y 2 = D 3 x m + a, C 2 : y 2 = D 3 x m + b, C 3 : y 2 = x m + cD 3 , have all positive ranks over Q(u, v, w). We present also some applications of these results. Finally, we present some results concerning the torsion parts of the Jacobians of the superelliptic curves y p = x m (x + a) and y p = x m (a − x) k for a prime p and 0 < m < p − 2 and k < p and apply our result in order to prove the existence of a function D ∈ Q(u, v, w, t) such that the Jacobians of the curves C 1 : Dy p = x m (x + a), Dy p = x m (x + b) have both positive rank over Q(u, v, w, t).
Let Xn = (x 1 ,. .. , xn) and σ i (Xn) = x k 1. .. x k i be i-th elementary symmetric polynomial.... more Let Xn = (x 1 ,. .. , xn) and σ i (Xn) = x k 1. .. x k i be i-th elementary symmetric polynomial. In this note we prove that there are infinitely many triples of integers a, b, c such that for each 1 ≤ i ≤ n the system of Diophantine equations σ i (X 2n) = a, σ 2n−i (X 2n) = b, σ 2n (X 2n) = c has infinitely many rational solutions. This result extend the recent results of Zhang and Cai, and the author. Moreover, we also consider some Diophantine systems involving sums of powers. In particular, we prove that for each k there are at least k n-tuples of integers with the same sum of i-th powers for i = 1, 2, 3. Similar result is proved for i = 1, 2, 4 and i = −1, 1, 2. 2000 Mathematics Subject Classification. 11G05. Key words and phrases. symmetric polynomials, elliptic curves.
In this note we construct several infinite families of diagonal quartic surfaces ax 4 + by 4 + cz... more In this note we construct several infinite families of diagonal quartic surfaces ax 4 + by 4 + cz 4 + dw 4 = 0, where a, b, c, d ∈ Z \ {0} with infinitely many rational points and satisfying the condition abcd =. In particular, we present an infinite family of diagonal quartic surfaces defined over Q with Picard number equal to one and possessing infinitely many rational points. Further, we present some sextic surfaces of type ax 6 + by 6 + cz 6 + dw i = 0, i = 2, 3, or 6, with infinitely many rational points.
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service... more This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ON CERTAIN DIOPHANTINE EQUATIONS OF THE FORM z 2 = f (x) 2 ± g(y) 2 SZ. TENGELY AND M. ULAS
In this paper we investaigate Diophantine equations of the form T 2 = G(X), X = (X 1 ,. .. , Xm),... more In this paper we investaigate Diophantine equations of the form T 2 = G(X), X = (X 1 ,. .. , Xm), where mainly m = 3 or m = 4 and G specific homogenous quintic form. First, we prove that if F (x, y, z) = x 2 + y 2 + az 2 + bxy + cyz + dxz ∈ Z[x, y, z] and (b − 2, 4a − d 2 , d) = (0, 0, 0), then the Diophantine equation t 2 = nxyzF (x, y, z) has solution in polynomials x, y, z, t with integer coefficients, without polynomial common factor of positive degree. In case a = d = 0, b = 2 we prove that there are infinitely many primitive integer solutions of the Diophantine equation under consideration. As an application of our result we prove that for each n ∈ Q \ {0} the Diophantine equation 2010 Mathematics Subject Classification. 11D41.
Let p be a prime number and consider a p-automatic sequence u = (un) n∈N and its generating funct... more Let p be a prime number and consider a p-automatic sequence u = (un) n∈N and its generating function U (X) = ∞ n=0 unX n ∈ Fp[[X]]. Moreover, let us suppose that u 0 = 0 and u 1 = 0 and consider the formal power series V ∈ Fp[[X]] which is a compositional inverse of U (X), i.e., U (V (X)) = V (U (X)) = X. In this note we initiate the study of arithmetic properties of the sequence of coefficients of the power series V (X). We are mainly interested in the case when un = tn, where tn = s 2 (n) (mod 2) and t = (tn) n∈N is the Prouhet-Thue-Morse sequence defined on the two letter alphabet {0, 1}. More precisely, we study the sequence c = (cn) n∈N which is the sequence of coefficients of the compositional inverse of the generating function of the sequence t. This sequence is clearly 2-automatic. We describe the sequence a characterizing solutions of the equation cn = 1. In particular, we prove that the sequence a is 2-regular. We also prove that an increasing sequence characterizing solutions of the equation cn = 0 is not k-regular for any k. Moreover, we present a result concerning some density properties of a sequence related to a.
In this paper we deal with a problem of Pethő related to existence of quartic algebraic integer α... more In this paper we deal with a problem of Pethő related to existence of quartic algebraic integer α for which 2010 Mathematics Subject Classification. Primary 11D61; Secondary 11Y50.
We prove several theorems concerning arithmetic properties of Stern polynomials defined in the fo... more We prove several theorems concerning arithmetic properties of Stern polynomials defined in the following way: B 0 (t) = 0, B 1 (t) = 1, B 2n (t) = tBn(t), and B 2n+1 (t) = Bn(t) + B n+1 (t). We study also the sequence e(n) = deg t Bn(t) and give various properties of it. ∞ i=0 ǫ i 2 i , where ǫ i ∈ {0, 1, 2}. These are called hiperbinary representations. The connections of the Stern sequence with continued fractions and the Euclidean algorithm are considered in [6] and [7]. An interesting application of the Stern sequence to the problem of construction a bijection between N + and Q + is given in [1]. In this paper it is shown that the sequence s(n)/s(n + 1), for n ≥ 1, encounters every positive rational number exactly once. A comprehensive survey of properties of the Stern sequence can be found in [12]. An interesting survey of known results and applications of the Stern sequence can also be found in [8]. Recently two distinct polynomial analogues of the Stern sequence appeared. The sequence of polynomials a(n; x) for n ≥ 0, defined by a(0; x) = 0, a(1; x) = 1, and for n ≥ 2: a(2n; x) = a(n; x 2), a(2n + 1; x) = xa(n; x 2) + a(n + 1; x 2), was considered in [2]. It is easy to see that a(n; 1) = s(n). Remarkably, as was proved in the cited paper, xa(2n − 1; x) ≡ A n+1 (x) (mod 2), where A n (x) = n j=0 S(n, j)x j and S(n, j) are the Stirling numbers of the second kind. Further 2010 Mathematics Subject Classification. 11B83. Key words and phrases. Stern diatomic sequence, Stern polynomials. The first named author is holder of START scholarship funded by the Foundation for Polish Science (FNP).
The material of this paper was presented in part at the 11th International Workshop for Young Mat... more The material of this paper was presented in part at the 11th International Workshop for Young Mathematicians - NUMBER THEORY, Kraków, 14th-20th september 2008.
Let K be a non-algebraically closed field and f ∈ K[x] be an irreducible polynomial of degree d ≥... more Let K be a non-algebraically closed field and f ∈ K[x] be an irreducible polynomial of degree d ≥ 3. In this paper we study the existence of a polynomial h ∈ K[x] of degree ≤ d−1 such that the polynomial f (h(x)) is reducible in K[x]. We prove the existence of such polynomial in case of d ≤ 4. A similar (and quite general) result is true in the case when f represents an even function, i.e., f (x) = f (−x). In particular, if char(K) = 2, then we prove that the existence of a required polynomial h.
Let σ i (x 1 ,. .. , xn) = 1≤k 1 <k 2 <...<k i ≤n x k 1. .. x k i be the i-th elementary symmetri... more Let σ i (x 1 ,. .. , xn) = 1≤k 1 <k 2 <...<k i ≤n x k 1. .. x k i be the i-th elementary symmetric polynomial. In this note we generalize and extend the results obtained in a recent work of Zhang and Cai [8, 9]. More precisely, we prove that for each n ≥ 4 and rational numbers a, b with ab = 0, the system of diophantine equations σ 1 (x 1 ,. .. , xn) = a, σn(x 1 ,. .. , xn) = b, has infinitely many solutions depending on n − 3 free parameters. A similar result is proved for the system σ i (x 1 ,. .. , xn) = a, σn(x 1 ,. .. , xn) = b, with n ≥ 4 and 2 ≤ i < n. Here, a, b are rational numbers with b = 0. We also give some results concerning the general system of the form σ i (x 1 ,. .. , xn) = a, σ j (x 1 ,. .. , xn) = b, with suitably chosen rational values of a, b and i < j < n. Finally, we present some remarks on the systems involving three different symmetric polynomials.
G. Campbell described a technique for producing infinite families of quartic elliptic curves cont... more G. Campbell described a technique for producing infinite families of quartic elliptic curves containing a length-9 arithmetic progression. He also gave an example of a quartic elliptic curve containing a length-12 arithmetic progression. In this note we give a construction of an infinite family of quartics on which there is an arithmetic progression of length 10. Then we show that there exists an infinite family of quartics containing a sequence of length 12.
Let m be a positive integer and bm(n) be the number of partitions of n with parts being powers of... more Let m be a positive integer and bm(n) be the number of partitions of n with parts being powers of 2, where each part can take m colors. We show that if m = 2 k − 1, then there exists the natural density of integers n such that bm(n) can not be represented as a sum of three squares and it is equal to 1/12 for k = 1, 2 and 1/6 for k ≥ 3. In particular, for m = 1 the equation b 1 (n) = x 2 + y 2 + z 2 has a solution in integers if and only if n is not of the form 2 2k+2 (8s + 2ts + 3) + i for i = 0, 1 and k, s are non-negative integers, and where tn is the nth term in the Prouhet-Thue-Morse sequence. A similar characterization is obtained for the solutions in n of the equation b 2 k −1 (n) = x 2 + y 2 + z 2 .
In this note we consider Diophantine equations of the form a(x p − y q) = b(z r − w s), where 1 p... more In this note we consider Diophantine equations of the form a(x p − y q) = b(z r − w s), where 1 p + 1 q + 1 r + 1 s = 1, with even positive integers p, q, r, s. We show that in each case the set of rational points on the underlying surface is dense in the Zariski topology. For the surface with (p, q, r, s) = (2, 6, 6, 6) we prove density of rational points in the Euclidean topology. Moreover, in this case we construct infinitely many parametric solutions in coprime polynomials. The same result is true for (p, q, r, s) ∈ {(2, 4, 8, 8), (2, 8, 4, 8)}. In the case (p, q, r, s) = (4, 4, 4, 4), we present some new parametric solutions of the equation x 4 − y 4 = 4(z 4 − w 4).
Let t n = (−1) s 2 (n) , where s 2 (n) is the sum of binary digits function. The sequence (t n) n... more Let t n = (−1) s 2 (n) , where s 2 (n) is the sum of binary digits function. The sequence (t n) n∈N is the well-known Prouhet-Thue-Morse sequence. In this note we initiate the study of the sequence (h n) n∈N , where h 0 = 0, h 1 = 1 and for n ≥ 2 we define h n recursively as follows: h n = t n h n−1 + h n−2. We prove several results concerning arithmetic properties of the sequence (h n) n∈N. In particular, we prove non-vanishing of h n for n ≥ 5, automaticity of the sequence (h n (mod m)) n∈N for each m, and other results.
Given an elliptic quartic of type Y 2 = f (X) representing an elliptic curve of positive rank ove... more Given an elliptic quartic of type Y 2 = f (X) representing an elliptic curve of positive rank over Q, we investigate the question of when the Y-coordinate can be represented by a quadratic form of type ap 2 + bq 2. In particular, we give examples of equations of surfaces of type c 0 + c 1 x + c 2 x 2 + c 3 x 3 + c 4 x 4 = (ap 2 + bq 2) 2 , a, b, c ∈ Q where we can deduce the existence of infinitely many rational points. We also investigate surfaces of type Y 2 = f (ap 2 + bq 2) where the polynomial f is of degree 3.
Let A ⊂ N + and by P A (n) denotes the number of partitions of an integer n into parts from the s... more Let A ⊂ N + and by P A (n) denotes the number of partitions of an integer n into parts from the set A. The aim of this paper is to prove several result concerning the existence of integer solutions of Diophantine equations of the form P A (x) = P B (y), where A, B are certain finite sets.
Let f ∈ Q[x] be a square-free polynomial of degree ≥ 3 and m ≥ 3 be an odd positive integer. Base... more Let f ∈ Q[x] be a square-free polynomial of degree ≥ 3 and m ≥ 3 be an odd positive integer. Based on our earlier investigations we prove that there exists a function D 1 ∈ Q(u, v, w) such that the Jacobians of the curves C 1 : D 1 y 2 = f (x), C 2 : y 2 = D 1 x m + b, C 3 : y 2 = D 1 x m + c, have all positive ranks over Q(u, v, w). Similarly, we prove that there exists a function D 2 ∈ Q(u, v, w) such that the Jacobians of the curves C 1 : D 2 y 2 = h(x), C 2 : y 2 = D 2 x m + b, C 3 : y 2 = x m + cD 2 , have all positive ranks over Q(u, v, w). Moreover, if f (x) = x m + a for some a ∈ Z \ {0}, we prove the existence of a function D 3 ∈ Q(u, v, w) such that the Jacobians of the curves C 1 : y 2 = D 3 x m + a, C 2 : y 2 = D 3 x m + b, C 3 : y 2 = x m + cD 3 , have all positive ranks over Q(u, v, w). We present also some applications of these results. Finally, we present some results concerning the torsion parts of the Jacobians of the superelliptic curves y p = x m (x + a) and y p = x m (a − x) k for a prime p and 0 < m < p − 2 and k < p and apply our result in order to prove the existence of a function D ∈ Q(u, v, w, t) such that the Jacobians of the curves C 1 : Dy p = x m (x + a), Dy p = x m (x + b) have both positive rank over Q(u, v, w, t).
Let Xn = (x 1 ,. .. , xn) and σ i (Xn) = x k 1. .. x k i be i-th elementary symmetric polynomial.... more Let Xn = (x 1 ,. .. , xn) and σ i (Xn) = x k 1. .. x k i be i-th elementary symmetric polynomial. In this note we prove that there are infinitely many triples of integers a, b, c such that for each 1 ≤ i ≤ n the system of Diophantine equations σ i (X 2n) = a, σ 2n−i (X 2n) = b, σ 2n (X 2n) = c has infinitely many rational solutions. This result extend the recent results of Zhang and Cai, and the author. Moreover, we also consider some Diophantine systems involving sums of powers. In particular, we prove that for each k there are at least k n-tuples of integers with the same sum of i-th powers for i = 1, 2, 3. Similar result is proved for i = 1, 2, 4 and i = −1, 1, 2. 2000 Mathematics Subject Classification. 11G05. Key words and phrases. symmetric polynomials, elliptic curves.
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