What percentage of polynomials with integer coefficients have roots over integers?

What is the probability that the homogeneous polynomial of degree $d$ in $n$ variables over integer numbers has non-zero integer solutions? Bjorn Poonen and José Felipe Voloch explain in their laconic paper that if $d$ is bigger than $n$, then the probability is equal to zero. Otherwise, it is not necessarily zero. Using the conjecture of Jean-Louis Colliot-Thélène and the Brauer–Manin obstruction we try to study the explanation of Bjorn Poonen and José Felipe Voloch.