Diophantine equations over global function fields I: The Thue equation

We solve completely Thue equations in function fields over arbitrary finite fields. In the function field case such equations were formerly only solved over algebraically closed fields (of characteristic zero and positive characteristic). Our method can be applied to similar types of Diophantine equations, as well.     

A parametric family of cubic Thue equations

In this paper, we solve a family of Diophantine equations associated with families of number fields of degree 3. In fact, we find all solutions to the Thue equation

     

On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields

Let m?−1 be an integer. We give a correspondence between integer solutions to the parametric family of cubic Thue equations

X³−mX²Y−(m+3)XY²−Y³=λ     

On a Family of Thue Equations Over Function Fields

Many families of parametrized Thue equations over number fields have been solved recently. In this paper we consider for the first time a family of Thue equations over a polynomial ring. In particular, we calculate all solutions of over for all . The first author was supported by the Austrian Science Foundation, grants S8307-MAT and J2407-N12. The second author was supported by the Austrian Science Foundation, grant S8307-MAT.     

Tetranomial Thue equations

Let F(x,y)$F(x,y)$ be an irreducible binary form of degree n?6$n?6$ with exactly four nonzero terms. Assuming certain conditions relating the coefficients and the degrees of the different terms are satisfied, we prove upper bounds on the number of equivalent pairs of nontrivial solutions of the Thue equation |F(x,y)|=1$|F(x,y)|=1$. Improved bounds are provided for a variety of cases, where more information about the form is known.     

A Parametric Family of Quintic Thue Equations II

 In this paper we completely solve the family of Thue equations

On the number of solutions of a diophantine equation with symmetric entries

For a class of strictly increasing real valued functions f(n) we obtain an upper bound for the number of solutions of the equation

     

On algebraic approximations of certain algebraic numbers

Using hypergeometric functions and the Thue-Siegel method we give an effective improvement of Liouville's approximation theorem. As an application, we derive effective upper bounds for the solutions (X,Y) of the two-parametric family of quartic Thue inequalities

|BX4−AX3Y−6BX2Y2+AXY3+BY4|?N     

Thue equations and torsion groups of elliptic curves

A new characterization of rational torsion subgroups of elliptic curves is found, for points of order greater than 4, through the existence of solution for systems of Thue equations.     

The algebraic functional equation of Selmer groups for CM fields

Because the analytic functional equation holds for Katz p-adic L-function for CM fields, the algebraic functional equation of the Selmer groups for CM fields is expected to hold. In this note we prove it following the specialization principle developed by T. Ochiai (2005) in [Och05].     

On Explicit Bounds for the Solutions of a Class of Parametrized Thue Equations of Arbitrary Degree

 In a recent paper [7] the author considered the family of parametrized Thue equations

On the number of solutions of the generalized Ramanujan-Nagell equation D1X2 + DM2 = 2N+2

Let D₁, D₂ be coprime odd integers with min (D₁, D₂) > 1, and let N (D₁, D₂) denote the number of positive integer solutions (x, m, n) of the equation D₁x²+D^m₂ = 2ⁿ⁺². In this paper, we prove that N (D₁, D₂) ≤ 2 except for N (3, 5) = N (5, 3) = 4 and N (13, 3) = N (31, 97) = 3.     

On small solutions to quadratic congruences

We estimate the deviation of the number of solutions of the congruence

     

Isolated semidefinite solutions of the continuous-time algebraic riccati equation

The set of all negative-semidefinite solutions of the CAREA ^* X+XA+XBB ^* X–C ^* C=0 is homeomorphic to a well defined set ofA-invariant subspaces provided that the purely imaginary eigenvalues ofA are controllable. Based on that homeomorphism isolated n.s.d. solutions of the CARE are characterized by properties of their kernels.Research supported by Deutsche Forschungsgemeinschaft (Wi 1219/1-1).