## Area under holomorphic maps

Let be a map from to . The area form gives the Jacobian , where .

Now consider the complex setting, where . Let be a map from to . Then . So this time the Jacobian becomes .

Suppose is a holomorphic map on the unit disk . Then

, the area of is .

Using polar coordinate, we have , ,

and if , and if .

So .

Let be an irreducible polynomial with integer coefficients, be an irrational number such that . Then for some .

Proof. Let . For each rational number , we have . Eliminating the denominator, we have . Each item being integral, we see that and hence . On the other hand, . Combining them, we have , where .

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