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Finding Invariants and Parametric Solutions for Some Systems of Diophantine Equations with Arbitrary Coefficients and Variables Over Q

Najafi Amin, Amin | 2022

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 55507 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Jafari, Amir
  7. Abstract:
  8. The main topic of this dissertation is to find methods for obtaining parametric solutions and linear/nonlinear invariants of Diophantine equations and consists of 4 chapters. The first chapter consists of some introductory discussions. The second chapter begins with a review of linear invariant, U-invariants, covariants and the concept of semi-invariants. The generators of maximum degree 3 for producing all linear invariants are introduced in Chapter 2 as well. Moreover, the relations between 3rd and 4th degree Hilbert invariants in terms of Procesi bases are included in this chapter.In chapter 3, a general conjecture is given to check whether there are finitely many solutions to a Diophantine equation. This conjecture also mentions some conditions under which there is or there is not a linear/nonlinear invariant. Next, an algorithm for solving a wide range of Diophantine equations as well as a set of Diophantine equations on Q and Z is outlined. The presented algorithm needs some computer calculation to find one or more initial solutions. Starting from the computed initial answer (by a computer), the suggested algorithm obtains the parametric solutions as well as nonlinear invariants of the given equations. We note that there was not a unified method for solving such equations before. Theoretically, this algorithm can be used for arbitrary Diophantine equation with any degree and coefficients if there are enough variables in the equation. However, for equations of degree 5 or 6, we encounter computational challenges. Therefore, in this chapter, we focus on relations in equations with degree at most 5.In chapter 4, the 3-variable Frobenius coin problem is discussed in details, a new algorithm is proposed to calculate the exact solution and a new upper bound is given to quickly estimate the solution. In addition, a brief discussion on generalization to the case of system of equations is presented. According to Hardy-Littlewood’s conjecture, we know that the number of solutions to a linear equation is associated with prime numbers. This motivate us to present a generalization of this statement and some unproven conjectures at the end of Chapter 4.
  9. Keywords:
  10. Diophantine Equations ; Parametric Solution ; Linear Invariants ; Nonlinear Invariants ; Frobenius Coin Problem

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