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Failure of the Weierstrass Preparation Theorem in quasi-analytic Denjoy–Carleman rings

Francesca Acquistapace^a, Fabrizio Broglia^a, Michail Bronshtein^b, Andreea C. Nicoara^{c d}, Nahum Zobin^e✉

^aDipartimento di Matematica, Università degli Studi di Pisa, Largo Bruno Pontecorvo, 5, 56127 Pisa, Italy
^bDepartment of Mathematics, Kazan National Research Technological University, 68 Karl Marx St., Kazan 420015, Republic of Tatarstan, Russian Federation
^cDepartment of Mathematics, University of Pennsylvania, 209 South 33rd St., Philadelphia, PA 19104, United States
^dDepartment of Mathematics University of Pennsylvania, 209 South 33rd St., Philadelphia, PA 19104, United States
^eDepartment of Mathematics, College of William and Mary, P.O. Box 8795, Williamsburg, VA 23187-8795, United States
✉Corresponding author

Advances in Mathematics Elsevier BV2014-03-28 12 citations14 references

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Abstract

Abstract It is shown that Denjoy–Carleman quasi-analytic rings of germs of functions in two or more variables fail to satisfy the Weierstrass Preparation Theorem. The result is proven via a non-extension theorem.

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Failure of the Weierstrass Preparation Theorem in quasi-analytic Denjoy–Carleman rings

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