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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.

##### MSC:
 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.)
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##### References:
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