Something that may not be clear when reading this is the distinction between complex numbers and the ring of integers adjoined with <i>i</i>.<p>"Complex numbers" are of the form a+bi where a and b can be any real number -- 1, 5, pi, the square root of 2, -2.14, etc.<p>The ring of integers adjoined with <i>i</i> are numbers of the form a+bi where a and b are both integers (-1, 5, 34, etc).<p>You can also, in addition to using <i>i</i>, adjoin any real number to the integers and get a new field with numbers of the form a+bx where a and b are integers and x is any additional number you want to add-- frequently square roots like the square root of two.<p>This result shows undecidability of diophantine equations in all those fields of integers, but not complex numbers, for which it's easy to prove that there are _always_ solutions.
Math is such an interesting field. People can work for decades and not make progress, then discover something in a moment of clarity from some seemingly unrelated problem. As a programmer, I don't have that type of patience.
Could Matiyasevich's result be viewed in the same vein as Turing’s Halting Problem, but in an even more compact form regarding the limits of mathematics and logic? Diophantine equations are expressed in a simpler form than Turing machines, which makes the size and expressiveness of the problem particularly interesting for study.
> Where is the cutoff<p>Is there a reason to believe there is a "cutoff"? As in, do we know that if the undecidability property holds for some ring A, then it holds for every subring of A?