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Hilbert's Tenth Problem (HTP) asks for an effective algorithm to test whether an arbitrary polynomial equation P(x_1, ... ,x_n)=0 (with integer coefficients) has solutions over the ring Z of the integers.
This was finally solved by Matiyasevich in 1970 negatively. In this talk we introduce the speaker's further results on HTP.
In particular, we present a sketch of the proof of the speaker's main result that there is no effective algorithm to determine whether an arbitrary polynomial equation P(x_1, ... ,x_{11})=0 (with integer coefficients) in 11 unknowns has integral solutions or not.
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