Thue equations and CM-fields
نویسندگان
چکیده
We obtain a polynomial type upper bound for the size of the integral solutions of Thue equations F (X,Y ) = b defined over a totally real number field K, assuming that F (X, 1) has a root α such that K(α) is a CM-field. Furthermore, we give an algorithm for the computation of the integral solutions of such an equation.
منابع مشابه
On Correspondence between Solutions of a Parametric Family of Cubic Thue Equations and Isomorphic Simplest Cubic Fields
We give a correspondence between non-trivial solutions to a parametric family of cubic Thue equations X − mXY − (m + 3)XY 2 − Y 3 = k where k | m+3m+9 and isomorphic simplest cubic fields. By applying R. Okazaki’s result for isomorphic simplest cubic fields, we obtain all solutions to the family of cubic Thue equations for k | m + 3m + 9.
متن کاملOn Correspondence between Solutions of a Parametric Family of Cubic Thue Equations and Non-isomorphic Simplest Cubic Fields
We give a correspondence between non-trivial solutions to a parametric family of cubic Thue equations X − mXY − (m + 3)XY 2 − Y 3 = k where k | m + 3m+ 9 and non-isomorphic simplest cubic fields. By applying R. Okazaki’s result for non-isomorphic simplest cubic fields, we obtain all solutions to the family of cubic Thue equations for k | m + 3m+ 9.
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is called a Thue equation, due to Thue [22] who proved, in the case R = Z, that such an equation has finitely many solutions. In the last decade, starting with the result of Thomas in [21], several families (at the moment up to degree 8; see [9] and the references mentioned therein) of Thue equations have been considered, where the coefficients of the form Fc(X,Y ) depend on an integral paramet...
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To each non totally real cubic extension K of Q and to each generator α of the cubic field K, we attach a family of cubic Thue equations, indexed by the units of K, and we prove that this family of cubic Thue equations has only a finite number of integer solutions, by giving an effective upper bound for these solutions.
متن کاملCalculating Power Integral Bases by Solving Relative Thue Equations
In our recent paper I. Gaál: Calculating “small” solutions of relative Thue equations, J. Experiment. Math. (to appear) we gave an efficient algorithm to calculate “small” solutions of relative Thue equations (where “small” means an upper bound of type 10500 for the sizes of solutions). Here we apply this algorithm to calculating power integral bases in sextic fields with an imaginary quadratic...
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