Subsections

• Norm Equations
• Thue Equations

NormEquation(O, a) : RngOrd, RngOrdElt -> BoolElt, RngOrdElt

NormEquation(K, m) : FldNum, RngIntElt -> BoolElt, RngOrdElt

NormEquation(K, a) : FldNum, RngOrdElt -> BoolElt, RngOrdElt

    All: BoolElt                        Default: false

    Solutions: RngIntElt                Default: --

    Exact: BoolElt                      Default: false

Given an order O and an element of the ground ring of O (a positive integer m or an element a of a suborder), this function returns a Boolean indicating whether an element alpha in O exists such that N_(K/Q)(alpha), the norm of alpha, equals m or a, and if so, a sequence of solutions alpha too. If a number field K is given rather than the order O, solutions in the equation order of K are found.

The optional argument Exact can be used to indicate whether an exact solution is required (with Exact := true) or a solution up to a unit suffices. The maximal number of required solutions can be indicated with the Solutions variable, but setting All := true will override this and the search will find all solutions.

Given a monic irreducible polynomial of degree at least 3 over the integers, this function returns the `Thue object' corresponding to f; such object is used in the functions for solving Thue equations, and print as the homogeneous version of f.

Thue(O) : RngOrd -> Thue

Given an order O with Z as its coefficient ring, this function returns the Thue object corresponding to the defining polynomial of O.

Given a Thue object t and integers a, b, return the evaluation of the homogeneous polynomial f involved in t at (a, b), that is f(a, b).

Given a Thue object t and an integer a this function return a sequence consisting of sequences of two integers [x, y] together providing all solutions to the (homogeneous form of) the Thue equation associated with t. Currently a must be positive.

Given a Thue object t and an integer a this function return a sequence consisting of sequences of two integers [x, y] together providing all solutions to the equation |f(X, Y)|=a, where f is the (homogeneous form of) the polynomial associated with t.