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Bulletin of the American Mathematical Society

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Cubics in 10 variables vs. cubics in 1000 variables: Uniformity phenomena for bounded degree polynomials


Authors: Daniel Erman, Steven V Sam and Andrew Snowden
Journal: Bull. Amer. Math. Soc. 56 (2019), 87-114
MSC (2010): Primary 13A02, 13D02
DOI: https://doi.org/10.1090/bull/1652
Published electronically: October 5, 2018
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Abstract: Hilbert famously showed that polynomials in $ n$ variables are not too complicated, in various senses. For example, the Hilbert Syzygy Theorem shows that the process of resolving a module by free modules terminates in finitely many (in fact, at most $ n$) steps, while the Hilbert Basis Theorem shows that the process of finding generators for an ideal also terminates in finitely many steps. These results laid the foundations for the modern algebraic study of polynomials.

Hilbert's results are not uniform in $ n$: unsurprisingly, polynomials in $ n$ variables will exhibit greater complexity as $ n$ increases. However, an array of recent work has shown that in a certain regime--namely, that where the number of polynomials and their degrees are fixed--the complexity of polynomials (in various senses) remains bounded even as the number of variables goes to infinity. We refer to this as Stillman uniformity, since Stillman's conjecture provided the motivating example. The purpose of this paper is to give an exposition of Stillman uniformity, including the circle of ideas initiated by Ananyan and Hochster in their proof of Stillman's conjecture, the followup results that clarified and expanded on those ideas, and the implications for understanding polynomials in many variables.


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Additional Information

Daniel Erman
Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin
Email: \href{mailto:derman@math.wisc.edu}{derman@math.wisc.edu}

Steven V Sam
Affiliation: Department of Mathematics, University of California, San Diego, California
Email: \href{mailto:ssam@ucsd.edu}{ssam@ucsd.edu}

Andrew Snowden
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan
Email: \href{mailto:asnowden@umich.edu}{asnowden@umich.edu}

DOI: https://doi.org/10.1090/bull/1652
Received by editor(s): September 18, 2018
Published electronically: October 5, 2018
Additional Notes: The first author was partially supported by NSF DMS-1601619.
The second author was partially supported by NSF DMS-1500069, NSF DMS-1812462, and a Sloan Fellowship.
The third author was partially supported by NSF DMS-1453893.
Article copyright: © Copyright 2018 American Mathematical Society

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