Suppose , if is zero on the following set

(a) A real hyperplane in

(b) The real two dimensional plane

(c) the arc

Show that must be zero on .

*Proof:* (a) Suppose the hyperplane has normal vector and passing through origin, then will be in this hyperplane for any .

Fix , is a holomorphic function in . Because zero point of holomorphic function is isolated unless a constant, will imply .

(b) Let , , then

is also a holomorphic function in . As we did in part (a), fix , imply for any . Since is arbitrary, then .

(c) Use the transformation of part (b), is equivalent

Consider , then has a sequence of roots converging to 0, then for any . From the Taylor expansion, there exists such that

Since is zero on , then is zero on when . By continuity, is zero on .

Then we can repeat the above procedure, such that

Then we must have , otherwise we can continue this process infinitely.

**Remark**: The idea of part (b) is due to Lun Zhang.

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