# day6 - Hilberts Tenth Diophantine Equations are style Click...

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Click to edit Master subtitle style 10/4/11 Hilbert’s Tenth Diophantine Equations are Semi-decidable One Variable Diophantine Equations are Solvable

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10/4/11 Dec 2, 2007/COT5310 © UCF (Charles E. Hughes) 22 Hilbert’s 10th is Semi- Decidable Consider over one variable: P(x) = 0 Can semi-decide by plugging in 0, 1, -1, 2, -2, 3, -3, … This terminates and says “yes” if P(x) evaluates to 0, eventually. Unfortunately, it never terminates if there is no x such that P(x) =0. Can easily extend to P(x1,x2,..,xk) = 0.
10/4/11 Dec 2, 2007/COT5310 © UCF (Charles E. Hughes) 33 P(x) = 0 is Decidable cn xn + cn-1 xn-1 +… + c1 x + c0 = 0 xn = -(cn-1 xn-1 + … + c1 x + c0)/cn |xn| ° cmax(|xn-1| + … + |x| + 1|)/|cn| |xn| ° cmax(n |xn-1|)/|cn|, since | x|°1

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10/4/11 Dec 2, 2007/COT5310 © UCF (Charles E. Hughes) 44 P(x) = 0 is Decidable Can bound the search to values of x in range [± n * ( cmax / cn )], where n = highest order exponent in polynomial cmax = largest absolute value coefficient cn = coefficient of highest order term Once we have a search bound and we are
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