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I’m trying to understand the simplest cases where a Diophantine equation has solutions in the reals and modulo all prime powers, but not in the integers. I know classes of examples of degree 6 in 1 variable, and of degree 4 in 3 variables, and I know there are no examples of degree 2 in any number of variables.

  1. what’s the best (smallest) degree achievable in 1, 2, and 3 variables?
  2. in 2 variables, does the answer change if the polynomial is required to be separable (expressible as f(x)-g(y)=0 not just f(x,y) = 0)?
Joe Shipman
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  • Auxiliary question: in 1 variable, how does one algorithmically test whether a polynomial has zeroes modulo all prime powers? – Joe Shipman Dec 03 '23 at 00:11
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    https://math.stackexchange.com/questions/55119/on-the-equation-3x3-4y3-5z3-0 – Will Jagy Dec 03 '23 at 00:14
  • For one variable, see, e.g., Berend and Bilu, Polynomials with roots modulo every integer, Proc Amer Math Soc 124 (1996) 1663-1671, https://www.ams.org/journals/proc/1996-124-06/S0002-9939-96-03210-8/S0002-9939-96-03210-8.pdf Such polynomials are sometimes called intersective. – Gerry Myerson Dec 03 '23 at 00:47
  • Thanks, that’s exactly what I was looking for in the 1-variable case! – Joe Shipman Dec 03 '23 at 02:41
  • Will that link is not what I asked for, I was well aware of Selmer’s equation but I was very specifically intending “no solutions in Z” not “no nontrivial solutions in Z”. – Joe Shipman Dec 03 '23 at 04:39
  • It might have been a good idea to mention in the question that you knew the Selmer example, since that's the first thing anyone familiar with these problems is going to think of. – Gerry Myerson Dec 03 '23 at 06:29
  • But I said “solutions” not “nontrivial solutions”. – Joe Shipman Dec 03 '23 at 10:44

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