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Solving quadratic equations using reduced unimodular quadratic forms. (English) Zbl 1078.11072
The author considers indefinite quadratic forms with integer coefficients and gives a polynomial time algorithm (based on LLL) to reduce it to a diagonal form. Further, an algorithm is given for the minimization of a ternary quadratic form: if a quadratic equation \(q(x,y,z)=0\) is solvable over the rationals, a solution can be deduced from another quadratic form equation of determinant \(\pm 1\). Combining these methods we obtain a polynomial time algorithm to solve any ternary quadratic equation over the rationals. The paper is illustrated with interesting examples.

11Y50 Computer solution of Diophantine equations
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
11H55 Quadratic forms (reduction theory, extreme forms, etc.)
Full Text: DOI
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