2017/02/27 by Ajai Choudhry, Choudhry, Ajai
Computer Science · Mathematics · #11D25 #11D41 #14G05 #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1702.08136
openalex publication_date 2017/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we present a new method of solving certain quartic and higher degree homogeneous polynomial diophantine equations in four variables. The method can also be extended to solve simultaneous homogeneous polynomial diophantine equations, in five or more variables, with one of the equations being of degree ≥ 4. We show that, under certain conditions, the method yields an arbitrarily large number of integer solutions of such diophantine equations and diophantine systems, some examples being a sextic equation in four variables, a tenth degree equation in six variables, and two simultaneous equations of degrees four and six in six variables. The method of solving these homogeneous equations also simultaneously yields arbitrarily many rational solutions of certain related nonhomogeneous equations of high degree. In contrast to existing methods, we obtain the arbitrarily large number of solutions without finding a parametric solution of the equations under consideration and without relating the solutions to rational points on an elliptic curve of positive rank. It appears from the examples given in the paper that there may exist projective varieties on which there are an arbitrarily large number of integer points and on which a curve of genus 0 or 1 does not exist.