  We show that under mild conditions, a Gaussian analytic function $\boldsymbol
F$ that a.s. does not belong to a given weighted Bergman space or Bargmann-Fock
space has the property that a.s. no non-zero function in that space vanishes
where $\boldsymbol F$ does. This establishes a conjecture of Shapiro (1979) on
Bergman spaces and allows us to resolve a question of Zhu (1993) on
Bargmann-Fock spaces. We also give a similar result on the union of two (or
more) such zero sets, thereby establishing another conjecture of Shapiro (1979)
on Bergman spaces and allowing us to strengthen a result of Zhu (1993) on
Bargmann-Fock spaces.
