I know of an example, found in this paper here, where a polynomial is taken and, by using the Hasse-Minkowski theorem and investigating the behaviour over some primes, to conclude that it has a non-trivial root over $\Bbb Q$.
I am looking for more examples of this type to try and consolidate my understanding. Can we just take the polynomial and replace the coefficients with others and end up coming to a similar conclusion? For example, can we take say $f(x,y,z)=3x^2+11y^2-17z^2$ or are there specific conditions on the coefficients that will allow the conclusion of a non-trivial root over $\Bbb Q$?
Thanks in advance.