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Bernstein polynomials and Milnor algebras.

F Kochman1

  • 1Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138.

Proceedings of the National Academy of Sciences of the United States of America
|August 1, 1976
PubMed
Summary

Researchers proved that analytic functions with isolated critical zeros can be related to differential operators. This establishes a functional equation connecting function values at shifted arguments.

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Area of Science:

  • Complex analysis
  • Algebraic geometry
  • Differential geometry

Background:

  • Analytic germs on C(n+1) are fundamental objects in complex analysis.
  • Understanding the behavior of functions near critical points is a key challenge.
  • The Jacobian ideal provides insights into the singularity structure of analytic functions.

Purpose of the Study:

  • To prove the existence of a specific analytic linear partial differential operator.
  • To establish a functional equation relating an analytic germ f to its shifted values.
  • To provide a geometric interpretation for functions with isolated critical zeros.

Main Methods:

  • Construction of an analytic linear partial differential operator P.
  • Demonstration of polynomial dependence on the variable s.
  • Derivation of the functional equation Pf(s+1) = b(s)f(s).

Main Results:

  • Existence of an operator P and polynomial b(s) satisfying the functional equation.
  • The proof is simplified for analytic germs with isolated critical zeros.
  • Geometric interpretation is provided for this specific case.

Conclusions:

  • The study establishes a novel connection between analytic germs and differential operators.
  • The findings offer a new perspective on the structure of functions near critical points.
  • The functional equation provides a powerful tool for analyzing analytic functions.

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