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Big polynomial rings and Stillman’s conjecture

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Notes

  1. In this paper, all graded rings are supported in non-negative degrees.

  2. To see this, let \(F=\sum _i x_i^d\). If F decomposes as \(F=f_0g_0+ \cdots +f_sg_s\) then the singular locus of V(F) would contain \(V(f_0,\dots ,f_s,g_0,\dots ,g_s)\) and hence would have codimension at most \(2s+2\). However, if d is invertible, then the singular locus of V(F) has infinite codimension.

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Acknowledgements

We thank Craig Huneke and Gregory G. Smith for useful conversations. We also thank Alexandru Chirvasitu for informing us about his work on the non-commutative analogue of Theorem 1.1 and the reference in [14].

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  1. Steven V. Sam

    Present address: Department of Mathematics, University of California, San Diego, La Jolla, CA, USA

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  1. Department of Mathematics, University of Wisconsin, Madison, WI, USA

    Daniel Erman & Steven V. Sam

  2. Department of Mathematics, University of Michigan, Ann Arbor, MI, USA

    Andrew Snowden

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Correspondence to Steven V. Sam.

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DE was partially supported by NSF DMS-1302057 and NSF DMS-1601619. SS was partially supported by NSF DMS-1500069 and DMS-1651327 and a Sloan Fellowship. AS was supported by NSF DMS-1303082 and DMS-1453893 and a Sloan Fellowship.

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Erman, D., Sam, S.V. & Snowden, A. Big polynomial rings and Stillman’s conjecture. Invent. math. 218, 413–439 (2019). https://doi.org/10.1007/s00222-019-00889-y

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