For n>24d and d≥17, the proportion of coefficient vectors a on the thin set P(a)=0 for which f_a(x)=0 satisfies the Hasse principle tends to 1 as A grows.
Yeon, The Hasse principle for homogeneous polynomials with random coefficients over thin sets, Proc
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The Hasse principle for random homogeneous polynomials in thin sets
For n>24d and d≥17, the proportion of coefficient vectors a on the thin set P(a)=0 for which f_a(x)=0 satisfies the Hasse principle tends to 1 as A grows.