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

Erman, Daniel; Sam, Steven; Snowden, Andrew

doi:10.1090/bull/1652

Cubics in 10 variables vs. cubics in 1000 variables: Uniformity phenomena for bounded degree polynomials
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by Daniel Erman, Steven V Sam and Andrew Snowden PDF

Bull. Amer. Math. Soc. 56 (2019), 87-114 Request permission

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.

References

Tigran Ananyan and Melvin Hochster, Ideals generated by quadratic polynomials, Math. Res. Lett. 19 (2012), no. 1, 233–244. MR 2923188, DOI 10.4310/MRL.2012.v19.n1.a18
Tigran Ananyan and Melvin Hochster, Small subalgebras of polynomial rings and Stillman’s Conjecture, arXiv:1610.09268v1 (2016).
Tigran Ananyan and Melvin Hochster, Strength conditions, small subalgebras, and Stillman bounds in degree $\leq 4$, arXiv:1810.00413 (2018).
Edoardo Ballico and Luca Chiantini, On smooth subcanonical varieties of codimension $2$ in $\textbf {P}^{n},$ $n\geq 4$, Ann. Mat. Pura Appl. (4) 135 (1983), 99–117 (1984). MR 750529, DOI 10.1007/BF01781064
Wolf Barth, Submanifolds of low codimension in projective space, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 409–413. MR 0422294
W. Barth and A. Van de Ven, A decomposability criterion for algebraic $2$-bundles on projective spaces, Invent. Math. 25 (1974), 91–106. MR 379515, DOI 10.1007/BF01389999
W. Barth and A. Van de Ven, On the geometry in codimension $2$ of Grassmann manifolds, Classification of algebraic varieties and compact complex manifolds, Lecture Notes in Math., Vol. 412, Springer, Berlin, 1974, pp. 1–35. MR 0354661
David Bayer and Michael Stillman, A criterion for detecting $m$-regularity, Invent. Math. 87 (1987), no. 1, 1–11. MR 862710, DOI 10.1007/BF01389151
Jesse Beder, Jason McCullough, Luis Núñez-Betancourt, Alexandra Seceleanu, Bart Snapp, and Branden Stone, Ideals with larger projective dimension and regularity, J. Symbolic Comput. 46 (2011), no. 10, 1105–1113. MR 2831475, DOI 10.1016/j.jsc.2011.05.011
Aaron Bertram, Lawrence Ein, and Robert Lazarsfeld, Vanishing theorems, a theorem of Severi, and the equations defining projective varieties, J. Amer. Math. Soc. 4 (1991), no. 3, 587–602. MR 1092845, DOI 10.1090/S0894-0347-1991-1092845-5
Arthur Bik, Jan Draisma, and Rob Eggermont, Polynomials and tensors of bounded strength, arXiv:1805.01816v2 (2018).
Winfried Bruns, “Jede” endliche freie Auflösung ist freie Auflösung eines von drei Elementen erzeugten Ideals, J. Algebra 39 (1976), no. 2, 429–439. MR 399074, DOI 10.1016/0021-8693(76)90047-8
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
Lindsay Burch, A note on the homology of ideals generated by three elements in local rings, Proc. Cambridge Philos. Soc. 64 (1968), 949–952. MR 230718, DOI 10.1017/s0305004100043632
Giulio Caviglia, Marc Chardin, Jason McCullough, Irena Peeva, and Mateo Varbaro, Regularity of prime ideals. https://orion.math.iastate.edu/jmccullo/papers/regularityofprimes.pdf., DOI 10.1007/s00209-018-2089-y
Thomas Church, Jordan S. Ellenberg, and Benson Farb, FI-modules and stability for representations of symmetric groups, Duke Math. J. 164 (2015), no. 9, 1833–1910. MR 3357185, DOI 10.1215/00127094-3120274
Harm Derksen, Rob H. Eggermont, and Andrew Snowden, Topological noetherianity for cubic polynomials, Algebra Number Theory 11 (2017), no. 9, 2197–2212. MR 3735467, DOI 10.2140/ant.2017.11.2197
Jan Draisma, Noetherianity up to symmetry, Combinatorial algebraic geometry, Lecture Notes in Math., vol. 2108, Springer, Cham, 2014, pp. 33–61. MR 3329086, DOI 10.1007/978-3-319-04870-3_{2}
Jan Draisma, Topological noetherianity for polynomial functors, arXiv:1705.01419v1 (2017).
Jan Draisma, MichałLasoń, and Anton Leykin, Stillman’s Conjecture via generic initial ideals, arXiv:1802.10139v1 (2018).
Lawrence Ein and Robert Lazarsfeld, The gonality conjecture on syzygies of algebraic curves of large degree, Publ. Math. Inst. Hautes Études Sci. 122 (2015), 301–313. MR 3415069, DOI 10.1007/s10240-015-0072-2
Rob H. Eggermont, Finiteness properties of congruence classes of infinite-by-infinite matrices, Linear Algebra Appl. 484 (2015), 290–303. MR 3385063, DOI 10.1016/j.laa.2015.06.035
Bahman Engheta, A bound on the projective dimension of three cubics, J. Symbolic Comput. 45 (2010), no. 1, 60–73. MR 2568899, DOI 10.1016/j.jsc.2009.06.005
Bahman Engheta, On the projective dimension and the unmixed part of three cubics, J. Algebra 316 (2007), no. 2, 715–734. MR 2358611, DOI 10.1016/j.jalgebra.2006.11.018
Daniel Erman, Steven V. Sam, and Andrew Snowden, Stillman type bounds via twisted commutative algebra, arXiv:1804.09807v1 (2018).
Daniel Erman, Steven V. Sam, and Andrew Snowden, Big Polynomial Rings and Stillman’s Conjecture, arXiv:1801.09852v4 (2018).
Daniel Erman, Steven V. Sam, and Andrew Snowden, Big polynomial rings with imperfect coefficient fields, arXiv:1806.04208v1 (2018).
Daniel Erman, Steven V. Sam, and Andrew Snowden, Strength and Hartshorne’s Conjecture in high degree, arXiv:1804.09730v1 (2018).
Gerd Faltings, Ein Kriterium für vollständige Durchschnitte, Invent. Math. 62 (1981), no. 3, 393–401 (German). MR 604835, DOI 10.1007/BF01394251
Mark L. Green, Koszul cohomology and the geometry of projective varieties, J. Differential Geom. 19 (1984), no. 1, 125–171. MR 739785
Mark L. Green, Koszul cohomology and the geometry of projective varieties. II, J. Differential Geom. 20 (1984), no. 1, 279–289. MR 772134
Mark Green and Robert Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Invent. Math. 83 (1986), no. 1, 73–90. MR 813583, DOI 10.1007/BF01388754
Robin Hartshorne, Varieties of small codimension in projective space, Bull. Amer. Math. Soc. 80 (1974), 1017–1032. MR 384816, DOI 10.1090/S0002-9904-1974-13612-8
David Hilbert, Ueber die Theorie der algebraischen Formen, Math. Ann. 36 (1890), no. 4, 473–534 (German). MR 1510634, DOI 10.1007/BF01208503
David Hilbert, Ueber die vollen Invariantensysteme, Math. Ann. 42 (1893), no. 3, 313–373 (German). MR 1510781, DOI 10.1007/BF01444162
Craig Huneke, Paolo Mantero, Jason McCullough, and Alexandra Seceleanu, A tight bound on the projective dimension of four quadrics, J. Pure Appl. Algebra 222 (2018), no. 9, 2524–2551. MR 3783004, DOI 10.1016/j.jpaa.2017.10.005
Craig Huneke, Paolo Mantero, Jason McCullough, and Alexandra Seceleanu, Multiple structures with arbitrarily large projective dimension supported on linear subspaces, J. Algebra 447 (2016), 183–205. MR 3427633, DOI 10.1016/j.jalgebra.2015.09.019
David Kazhdan and Tomer M. Schlank, On bias and rank, arXiv:1808.05801 (2018).
David Kazhdan and Tamar Ziegler, On ranks of polynomials, arXiv:1802.04984v2 (2018).
David Kazhdan and Tamar Ziegler, Extending weakly polynomial functions from high rank varieties, arXiv:1808.09439 (2018).
Peter Kohn, Ideals generated by three elements, Proc. Amer. Math. Soc. 35 (1972), 55–58. MR 296064, DOI 10.1090/S0002-9939-1972-0296064-1
Daniel R. Grayson and Michael E. Stillman, Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/.
Paolo Mantero and Jason McCullough, The projective dimension of three cubics is at most 5, arXiv:1801.08195v1 (2018).
Jason McCullough, A family of ideals with few generators in low degree and large projective dimension, Proc. Amer. Math. Soc. 139 (2011), no. 6, 2017–2023. MR 2775379, DOI 10.1090/S0002-9939-2010-10792-X
Jason McCullough, Stillman’s question for exterior algebras and Herzog’s conjecture on Betti numbers of syzygy modules, J. Pure Appl. Algebra 223 (2019), no. 2, 634–640. MR 3850562, DOI 10.1016/j.jpaa.2018.04.012
Jason McCullough and Alexandra Seceleanu, Bounding projective dimension, Commutative algebra, Springer, New York, 2013, pp. 551–576. MR 3051385, DOI 10.1007/978-1-4614-5292-8_{1}7
Rohit Nagpal, Steven V. Sam, and Andrew Snowden, Noetherianity of some degree two twisted commutative algebras, Selecta Math. (N.S.) 22 (2016), no. 2, 913–937. MR 3477338, DOI 10.1007/s00029-015-0205-y
Irena Peeva and Mike Stillman, Open problems on syzygies and Hilbert functions, J. Commut. Algebra 1 (2009), no. 1, 159–195.
Z. Ran, On projective varieties of codimension $2$, Invent. Math. 73 (1983), no. 2, 333–336., DOI 10.1007/BF01394029
Hans Schoutens, The use of ultraproducts in commutative algebra, Lecture Notes in Mathematics, vol. 1999, Springer-Verlag, Berlin, 2010., DOI 10.1007/978-3-642-13368-8
Andrew Snowden, Syzygies of Segre embeddings and $\Delta$-modules, Duke Math. J. 162 (2013), no. 2, 225–277.
Richard P. Stanley, The upper bound conjecture and Cohen-Macaulay rings, Studies in Appl. Math. 54 (1975), no. 2, 135–142., DOI 10.1002/sapm1975542135
L. van den Dries and K. Schmidt, Bounds in the theory of polynomial rings over fields. A nonstandard approach, Invent. Math. 76 (1984), no. 1, 77–91., DOI 10.1007/BF01388493
Claire Voisin, Green’s canonical syzygy conjecture for generic curves of odd genus, Compos. Math. 141 (2005), no. 5, 1163–1190., DOI 10.1112/S0010437X05001387
Claire Voisin, Green’s generic syzygy conjecture for curves of even genus lying on a $K3$ surface, J. Eur. Math. Soc. (JEMS) 4 (2002), no. 4, 363–404., DOI 10.1007/s100970200042
Claire Voisin, Some results on Green’s higher Abel-Jacobi map, Ann. of Math. (2) 149 (1999), no. 2, 451–473., DOI 10.2307/120970