Rays of Small Integer Solutions of Homogeneous Ternary Quadratic Equations

Details Rays of Small Integer Solutions of Homogeneous Ternary Quadratic Equations Rays of Small Integer Solutions of Homogeneous Ternary Quadratic Equations

Feb 1991

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1991phdt........80m&link_type=abstract

Thesis (PH.D.)--CITY UNIVERSITY OF NEW YORK, 1991.Source: Dissertation Abstracts International, Volume: 52-07, Section: B, page:

Computer Science

Quadratic Equations

Scientific paper

We have dealt with the general ternary quadratic equation: ax2 + by^ {2} + cz2 + dxy + exz + fyz = 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: ax^ {2} + by2 + cz^{2 } = 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: ax^{2 } + by2 + cz2 + dxy + exz + fyz = 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 ax 2 + by2 + cz^ {2} = 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: ax^2 + by^2 + cz^2 = 0, we have shown how one can produce some similar rays of solutions of the above general ternary quadratic equation: ax2 + by2 + cz2 + dxy + exz + fyz = 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 doubz_sp{*} {3} to doubz_+, 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.).

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