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Are there any known non-slow methods for solving diophantine systems?

I can't find books of mathematics that appear methods explaining how to solve diophantine systems in a manner "not slow", e.g. not force brute enumeration.

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Which diophantine systems? Linear systems are very well understood (see, eg, the vast literature on "integer programming"), quadratic equations well understood for a couple of centuries (search for "Pell's equation"), general nonlinear systems undecidable. So, take your pick. –  Igor Rivin Dec 14 '10 at 1:58
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In addition to Igor's comment there is a Matiyasevich's theorem that the general biquadratic equation cannot be solved algorithmically. But there is an open problem is any cubic equation is solvable algorithmically or not ;) –  zroslav Dec 14 '10 at 2:03
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Voted to close. The OP did not even google "Diophantine equations". –  Mark Sapir Dec 14 '10 at 2:32
    
To user h10, Hilbert's 10th problem is for one equation, it's not enough for a system of equations. –  Ecologicyborg Dec 14 '10 at 2:33
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@Ecologicyborg: If one equation is bad, a system is worse. –  Felipe Voloch Dec 14 '10 at 2:42
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closed as too localized by Andres Caicedo, Mark Sapir, Will Jagy, Felipe Voloch, José Figueroa-O'Farrill Dec 14 '10 at 2:59

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2 Answers

You must not have looked very hard. This will get you started:

Nigel P. Smart. The Algorithmic Resolution of Diophantine Equations. London Mathematical Society Student Texts 41. Cambridge University Press, 1998.

However, as Igor and zroslav have mentioned, some problems are unsolvable, others believed to be very hard, so don't be surprised that there are no "easy" methods.

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