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Study of Invariants of Diophantine Forms and Equations

Najafi Amin, Amin | 2013

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 44899 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Jafari, Amir
  7. Abstract:
  8. The aim of this thesis is to study invariants of Diophantine forms and equations. These invariants are the transformations that do not change the original equations and forms. This thesis has a different view to the concept of invariants from Hlibertian view. Based on this new definition, by having one of the solutions of the equation, new solutions can be found with invariant method. The number of these solutions can be finite or infinite. In case of infinite solutions this method is more important. The focus of this thesis is on forms and linear invariants. The most important result in this text is about two main theorems that can classify the invariants of all completely decomposable forms of any number of variables, into the linear factors in ℂ. More results are obtained for the ring of integers. In general, there is no systematic way to show if a given equation has finite or infinite number of solutions. Invariants help to find this answer for some sets of equations. These types of solutions along with parametric solutions are considered as two main categories for Diophantine equations which have infinite number of solutions. So, one chapter in this thesis is devoted to study the relation between Hilbert invariants and parametric solutions of some famous solved Diophantine equations. Most of the content of this thesis are new conclusion and are not available in other books or articles
  9. Keywords:
  10. Invariant Form ; Diophantine Forms ; Diophantine Equations ; Parametric Solution

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