Rays of small integer solutions of homogeneous ternary quadratic equations.
Item
-
Title
-
Rays of small integer solutions of homogeneous ternary quadratic equations.
-
Identifier
-
AAI9130353
-
identifier
-
9130353
-
Creator
-
Mishra, Sudhakara.
-
Contributor
-
Adviser: Harvey Cohn
-
Date
-
1991
-
Language
-
English
-
Publisher
-
City University of New York.
-
Subject
-
Mathematics | Physics, Astronomy and Astrophysics | Computer Science
-
Abstract
-
We have dealt with the general ternary quadratic equation: {dollar}ax\sp{lcub}2{rcub} + by\sp{lcub}2{rcub} + cz\sp{lcub}2{rcub} + dxy + exz + fyz{dollar} = 0 with integer coefficients. After giving a matrix-reduction formula for a quadratic equation in any number of variables, of which the reduction of the above ternary equation is an easy consequence, we have devoted our attention to the reduced equation: {dollar}ax\sp{lcub}2{rcub} + by\sp{lcub}2{rcub} + cz\sp{lcub}2{rcub}{dollar} = 0.;We have devised an algorithm for reducing Dirichlet's possibly larger solutions to this prescribed range of Holzer's.;Then we have generalized Holzer's theorem to the case of the ternary equation: {dollar}ax\sp{lcub}2{rcub} + by\sp{lcub}2{rcub} + cz\sp{lcub}2{rcub} + dxy + exz + fyz{dollar} = 0, giving in this context a new range called the CM-range, of which the Holzer's range is a particular case when d = e = f = 0. We have described an algorithm for getting a solution of the general ternary within this CM-range.;After that we have devised an algorithm for getting all the solutions of the Legendre's equation {dollar}ax\sp{lcub}2{rcub} + by\sp{lcub}2{rcub} + cz\sp{lcub}2{rcub}{dollar} = 0 within the Holzer's range--and have shown that if we regard this Legendre's equation as a double cone, these solutions within the Holzer's range lie along some definite rays, here called the CM-rays, which are completely determined by the prime factors of the coefficients a, b and c.;After giving an algorithm for detecting these CM-rays of the reduced equation: {dollar}ax\sp2{dollar} + {dollar}by\sp2{dollar} + {dollar}cz\sp2{dollar} = 0, we have shown how one can produce some similar rays of solutions of the above general ternary quadratic equation: {dollar}ax\sp{lcub}2{rcub} + by\sp{lcub}2{rcub} + cz\sp{lcub}2{rcub} + dxy + exz + fyz{dollar} = 0.;Note that apart from the method of exhausting all the possibilities, so far there has been no precisely stated algorithm to find the minimum solutions of the above ternary equations.;Towards the end, observing in the context of our main result an inequality involving two functions, namely C and PCM from {dollar}\doubz\sbsp{lcub}*{rcub}{lcub}3{rcub}{dollar} to {dollar}\doubz\sb+{dollar}, and simultaneously presenting some tables of these positive CM-rays or PCM-rays lying in the positive octant, we have concluded this work with a number of hints for some possible future investigations. (Abstract shortened with permission of author.).
-
Type
-
dissertation
-
Source
-
PQT Legacy CUNY.xlsx
-
degree
-
Ph.D.
