(From the Introduction)
In number theory, discriminants often appear at key points in proving theorems which assert the finiteness in number (or even non-existence) of certain kinds of arithmetic objects. A typical argument is to show two estimates for the discriminant of opposite directions which eventually contradict; one from above which is often done algebraically, and the other which is done by some other methods, say, analytically. As this seems rather prevalent, I try in this paper to explain some of these phenomena, including classical theorems such as Minkowski's as well as my recent results obtained jointly with H.Moon.
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