On the monodromy conjecture for non-degenerate hypersurfaces
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DOI10.4171/JEMS/1241zbMATH Open1506.14106arXiv1309.0630WikidataQ113691990 ScholiaQ113691990MaRDI QIDQ2087382
Author name not available (Why is that?)
Publication date: 27 October 2022
Published in: (Search for Journal in Brave)
Abstract: The monodromy conjecture is an umbrella term for several conjectured relationships between poles of zeta functions, monodromy eigenvalues and roots of Bernstein-Sato polynomials in arithmetic geometry and singularity theory. Even the weakest of these relations -- the Denef--Loeser conjecture on topological zeta functions -- is open for surface singularities. We prove it for a wide class of multidimensional singularities that are non-degenerate with respect to their Newton polyhedra, including all such singularities of functions of four variables. A crucial difference from the known case of three variables is the existence of degenerate singularities arbitrarily close to a non-degenerate one. Thus, even aiming at the study of non-degenerate singularities, we have to go beyond this setting. We develop new tools to deal with such multidimensional phenomena, and conjecture how the proof for non-degenerate singularities of arbitrarily many variables might look like.
Full work available at URL: https://arxiv.org/abs/1309.0630
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