Superelliptic Equation Research Articles

Explicit bounds for the solutions of superelliptic equations over number fields

Abstract Let f be a polynomial with coefficients in the ring O S {O_{S}} of S-integers of a number field K, b a non-zero S-integer, and m an integer ≥ 2 {\geq 2} . We consider the following equation ( ⋆ ) {(\star)} : f ⁢ ( x ) = b ⁢ y m {f(x)=by^{m}} in x , y ∈ O S {x,y\in O_{S}} . Under the well-known LeVeque condition, we give fully explicit upper bounds in terms of K , S , f , m {K,S,f,m} and the S-norm of b for the heights of the solutions x of equation ( ⋆ ) {(\star)} . Further, we give an explicit bound C in terms of K , S , f {K,S,f} and the S-norm of b such that if m > C {m>C} equation ( ⋆ ) {(\star)} has only solutions with y = 0 {y=0} or a root of unity. Our results are more detailed versions of work of Trelina, Brindza, Shorey and Tijdeman, Voutier and Bugeaud, and extend earlier results of Bérczes, Evertse, and Győry to polynomials with multiple roots. In contrast with the previous results, our bounds depend on the S-norm of b instead of its height.

Most odd-degree binary forms fail to primitively represent a square

Let $F$ be a separable integral binary form of odd degree $N \geq 5$ . A result of Darmon and Granville known as ‘Faltings plus epsilon’ implies that the degree- $N$ superelliptic equation $y^2 = F(x,z)$ has finitely many primitive integer solutions. In this paper, we consider the family $\mathscr {F}_N(f_0)$ of degree- $N$ superelliptic equations with fixed leading coefficient $f_0 \in \mathbb {Z} \smallsetminus \pm \mathbb {Z}^2$ , ordered by height. For every sufficiently large $N$ , we prove that among equations in the family $\mathscr {F}_N(f_0)$ , more than $74.9\,\%$ are insoluble, and more than $71.8\,\%$ are everywhere locally soluble but fail the Hasse principle due to the Brauer–Manin obstruction. We further show that these proportions rise to at least $99.9\,\%$ and $96.7\,\%$ , respectively, when $f_0$ has sufficiently many prime divisors of odd multiplicity. Our result can be viewed as a strong asymptotic form of ‘Faltings plus epsilon’ for superelliptic equations and constitutes an analogue of Bhargava's result that most hyperelliptic curves over $\mathbb {Q}$ have no rational points.

A variant of the Nagell–Ljunggren superelliptic equation

A variant of the Nagell–Ljunggren superelliptic equation

On a special superelliptic equation

On a special superelliptic equation

On k-partitions of multisets with equal sums

We study the number of ordered k-partitions of a multiset with equal sums, having elements alpha _1,ldots ,alpha _n and multiplicities m_1,ldots ,m_n. Denoting this number by S_k(alpha _1, ldots , alpha _n; m_1, ldots , m_n), we find the generating function, derive an integral formula, and illustrate the results by numerical examples. The special case involving the set {1,dots ,n} presents particular interest and leads to the new integer sequences S_k(n), Q_k(n), and R_k(n), for which we provide explicit formulae and combinatorial interpretations. Conjectures in connection to some superelliptic Diophantine equations and an asymptotic formula are also discussed. The results extend previous work concerning 2- and 3-partitions of multisets.

Irreducible binary cubics and the generalised superelliptic equation over number fields

For a large class (heuristically most) of irreducible binary cubic forms $F(x,y) \in \mathbb Z[x,y]$, Bennett and Dahmen proved that the generalized superelliptic equation $F(x,y)=z^l$ has at most finitely many solutions in $x,y \in \mathbb Z$ coprime, $z \in \mathbb Z$ and exponent $l \in \mathbb Z_{\geq 4} $. Their proof uses, among other ingredients, modularity of certain mod $l$ Galois representations and Ribet's level lowering theorem. The aim of this paper is to treat the same problem for binary cubics with coefficients in $\mathcal O_K$, the ring of integers of an arbitrary number field $K$, using by now well-documented modularity conjectures.

On special extrema of polynomials with applications to Diophantine problems

We prove that apart from explicitly given cases, described in terms of Dickson polynomials, a polynomial fin mathbb {Q}[x] can have at most one shift f(x)-lambda (lambda in mathbb {C}) of the form u(g(x))^q(h(x))^k with uin mathbb {C}, g,hin mathbb {C}[x] and either deg (g)=2, k is even, q=k/2 or deg (g)le 1, kge 2, qge 1. This is shown by handling the case of two possible shifts, which was an open issue. As an application, we give a precise statement yielding a description of polynomials f having infinitely many shifted power (S-integral) values, and a complete description of superelliptic equations having infinitely many S-integral solutions when the polynomial involved is composite. In the case where there are finitely many solutions, our results yield effective bounds for them. Finally, as further applications, we give effective results for polynomial values in the solutions of Pell equations and in non-degenerate binary recurrence sequences.

Superelliptic equations arising from sums of consecutive powers

Using only elementary arguments, Cassels solved the Diophantine equation (x−1)3+x3+(x+1)3=z2 in integers x, z. The generalization (x−1)k+xk+(x+1)k=zn (with x, z, n integers and n≥2) was considered by Zhongfeng Zhang who solved it for k E {2, 3, 4} using Frey-Hellegouarch curves and their Galois representations. In this paper, by employing some sophisticated refinements of this approach, we show that the only solution for k=5 is x=z=0, and that there are no solutions for k=6. The chief innovation we employ is a computational one, which enables us to avoid the full computation of data about cuspidal newforms of high level.

Effective results for hyper- and superelliptic equations over number fields

We consider hyper- and superelliptic equations $f(x)=by^m$ with unknowns x,y from the ring of S-integers of a given number field K. Here, f is a polynomial with S-integral coefficients of degree n with non-zero discriminant and b is a non-zero S-integer. Assuming that n>2 if m=2 or n>1 if m>2, we give completely explicit upper bounds for the heights of the solutions x,y in terms of the heights of f and b, the discriminant of K, and the norms of the prime ideals in S. Further, we give a completely explicit bound C such that $f(x)=by^m$ has no solutions in S-integers x,y if m>C, except if y is 0 or a root of unity. We will apply these results in another paper where we consider hyper- and superelliptic equations with unknowns taken from an arbitrary finitely generated integral domain of characteristic 0.

Klein forms and the generalized superelliptic equation

If F (x;y)2 Z[x;y] is an irreducible binary form of degree k 3, then a theorem of Darmon and Granville implies that the generalized superelliptic equation F (x;y) = z l has, given an integer l maxf2; 7 kg, at most nitely many solutions in coprime integers x;y and z. In this paper, for large classes of forms of degree k = 3; 4; 6 and 12 (including, heuristically, \most cubic forms), we extend this to prove a like result, where the parameter l is now taken to be variable. In the case of irreducible cubic forms, this provides the rst examples where such a conclusion has been proven. The method of proof combines classical invariant theory, modular Galois representations, and properties of elliptic curves with isomorphic mod-n Galois representations.

A superelliptic equation involving alternating sums of powers

In this short note, we solve completely the Diophantine equation 1 − 3 + 5 − · · · + (4x− 3) − (4x− 1) = −y, for 3 ≤ k ≤ 6. This may be viewed as a “character-twisted” analogue of a classic equation of Schaffer (in which context, it was previously considered by Dilcher). In our proof, we appeal primarily to techniques based upon the modularity of Galois representations and, in particular, to a combination of these ideas with suitable local information.

Solving superelliptic Diophantine equations by Baker's method

We describe a method for complete solution of the superelliptic Diophantine equation ay=f(x). The method is based on Baker‘s theory of linear forms in the logarithms. The characteristic feature of our approach (as compared with the classical method) is that we reduce the equation directly to the linear forms in logarithms, without intermediate use of Thue and linear unit equations. We show that the reduction method of Baker and Davenport [3] is applicable for superelliptic equations, and develop a very efficient method for enumerating the solutions below the reduced bound. The method requires computing the algebraic data in number fields of degree pn(n-1)/2 at most; in many cases this number can be reduced. Two examples with p=3 and n=4 are given.

Bounds for the solutions of superelliptic equations

In this work, we study the diophantine equation \renewcommand{\theequation}{1.1} \begin{equation} f(X) = Y^m, \end{equation} where -integral solutions. After that, LeVeque [13] gave a necessary and sufficient condition for the algebraic curve defined by (1.1) to have positive genus. However, all these results are based on Thue's method, and hence are ineffective.

An Upper Bound for the Size of Integral Solutions to Ym=ƒ(X)

In this paper, we establish improved upper bounds on the size of integral solutions to hyper- and super-elliptic equations under the conditions of LeVeque (Acta. Arith.IX (1964), 209- 219). The proof follows the classical argument of Siegel (J. London Math. Soc.1 (1926), 66-68), using upper bounds on the size of solutions of unit equations derived from lower bounds for linear forms in logarithms.

On the number ofS-integral solutions toY m =f(X)

In this paper, we establish new upper bounds on the number ofS-integral solutions to hyper- and super-elliptic equations under the conditions ofLeVeque [3]. Our method is based upon that ofEvertse andSilverman [1].

On superelliptic equations

On superelliptic equations

Bounds for the solutions of some diophantine equations in terms of discriminants

AbstractSeveral effective upper bounds are known for the solutions of Thue equations, Thue-Mahler equations and superelliptic equations. One of the basic parameters occurring in these bounds is the height of the polynomial involved in the equation. In the present paper it is shown that better (and, in certain important particular cases, best possible) upper bounds can be obtained in terms of the height, if one takes into consideration also the discriminant of the polynomial.