The Zero Set of a Real Analytic Function

TL;DR

The authors respond to requests for a clear reference or explicit statement of the zero-set result by offering a brief proof. Using local Implicit Function Theorem arguments and finite coverings, they establish zero measure for the zero set of a nontrivial real-analytic function.

• Problem

  The authors note requests for a good reference or explicit statement of the zero-set result.

  Source: p. 1

• Method

  The proof uses local Implicit Function Theorem arguments and finite coverings of compact sets to show measure zero.

  Source: p. 2

• Results

  The zero set of a nontrivial real-analytic function on a connected open domain in R^d has measure zero.

  Source: p. 1

• Takeaways & Limitations

  The paper states that some claims also remain valid for real functions in certain quasi-analytic classes.

  Source: p. 3

  The zero-set conclusion assumes the function is not identically zero; the stated extension covers only some quasi-analytic classes.

  Source: pp. 2–3