# The solution of triangularly connected decomposable form equations

@article{Smart1995TheSO, title={The solution of triangularly connected decomposable form equations}, author={Nigel P. Smart}, journal={Mathematics of Computation}, year={1995}, volume={64}, pages={819-840} }

An algorithm is given to solve the equations of the title. It generalizes an earlier algorithm to solve discriminant form equations. An application is given to finding curves of genus 2 with good reduction outside a given finite set of primes and Weierstrass points in given number fields

## 32 Citations

Solving Norm Form Equations Via Lattice

- Mathematics
- 2007

The author uses irrationality and linear independence measures for certain algebraic numbers to derive explicit upper bounds for the solutions of related norm form equations. The Lenstra-Lenstra-Lovv…

Thue and Thue–Mahler Equations over Rings of Integers

- Mathematics
- 1997

A method is given to solve any Thue–Mahler equation when the coefficients and variables come from the ring of integers of a number field. The method involves using an algorithm for solving an S‐unit…

On the resolution of relative Thue equations

- MathematicsMath. Comput.
- 2002

An efficient algorithm is given for the resolution of relative Thue equations by the application of an appropriate version of Wildanger's enumeration procedure based on the ellipsoid method of Fincke and Pohst.

The Resolution of Norm Form Equations

- MathematicsDiophantine Equations and Power Integral Bases
- 2019

Although there is an extensive literature of the explicit resolution of Thue equations (see for example Chap. 3), the problem of algorithmic resolution of norm form equations was not investigated…

Solving Thue Equations of High Degree

- Mathematics
- 1996

Abstract We propose a general method for numerical solution of Thue equations, which allows one to solve in reasonable time Thue equations of high degree (provided necessary algebraic number theory…

Applications of S-unit equations to the arithmetic of elliptic curves

- Mathematics, Computer Science
- 2016

This thesis develops a new algorithmic method of computing EK;S by solving S{unit equations] by solving Diophantine equations, implemented in the mathematical software Sage.

S-unit equations and curves of genus 2 with good reduction away from 3

- Mathematics
- 2016

The Shafarevich conjecture (now a theorem of Faltings) guarantees that for any genus g ≥ 1, there are only finitely many isomorphism classes of curves over Q with good reduction outside any given…

Solving Thue equations without the full unit group

- Computer Science, MathematicsMath. Comput.
- 2000

It is shown that the knowledge of a subgroup of finite index is in fact sufficient and two examples linked with the primitive divisor problem for Lucas and Lehmer sequences are given.

Exceptional units in a family of quartic number fields

- MathematicsMath. Comput.
- 1998

All exceptional units among the elements of certain groups of units in quartic number fields arise from a one-parameter family of polynomials with two real roots.

Determining the small solutions to S-unit equations

- MathematicsMath. Comput.
- 1999

The method uses a minor generalization of the LLL based techniques used to reduce the bounds derived from transcendence theory, followed by an enumeration strategy based on the Fincke-Pohst algorithm to generalize the method of Wildanger for finding small solutions to unit equations.

## References

SHOWING 1-10 OF 29 REFERENCES

Solving a quartic discriminant form equation

- Mathematics
- 1993

All algebraic integers of discriminant and norm a power of 2 and 3 only in the quartic field x4 + 12x2 + 18 are calculated. This involves solving a discriminant form equation of Mahler type.

Curves of genus 2 with good reduction away from 2 with a rational Weierstrass point

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 1993

Abstract All curves of the title are calculated up to an equivalence relation which is coarser than the relation of isogeny between the associated Jacobian varieties.

A hyperelliptic diophantine equation related to imaginary quadratic number fields with class number 2.

- Mathematics
- 1991

In my paper [dW], recently published in this Journal, there appears to be an error in the arguments leading to the upper bound (36) for J5, namely in the lower bound for linear forms in logarithms of…

Products of prime powers in binary recurrence sequences, Part I: the hyperbolic case, with an application to the generalized Ramanujan-Nagell equation

- Mathematics, Computer Science
- 1986

It is shown how the Gelfond-Baker theory and diophantine approximation techniques can be applied to solve explicitly the Diophantine equation [formula], which is a generalization of the Ramanujan-Nagell equation.

Solving a specific Thue-Mahler equation

- Mathematics
- 1991

The diophantine equation x3 _ 3xy2 _y3 = ?3no 17 n l9n2 is completely solved as follows. First, a large upper bound for the variables is obtained from the theory of linear forms in p-adic and real…

Factoring polynomials with rational coefficients

- Mathematics
- 1982

In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into…

Algebraic Number Theory

- Mathematics
- 1971

This is a corrected printing of the second edition of Lang's well-known textbook. It covers all of the basic material of classical algebraic number theory, giving the student the background necessary…

Matrix analysis

- MathematicsStatistical Inference for Engineers and Data Scientists
- 2018

This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications.