Bounding Projective Dimension

McCullough, Jason; Seceleanu, Alexandra

doi:10.1007/978-1-4614-5292-8_17

Cited by 18 publications

(16 citation statements)

References 33 publications

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“…As a consequence, he proves that if I is an ideal generated by three cubics, then the projective dimension of S/I is at most 36 [7]. This answers a specific case to Stillman's question: Furthermore, the bound found by Engheta is not optimal see [8] and [19], as the expected projective dimension of three cubics is 5. In [12], Huneke et.al.…”

Section: Introductionmentioning

confidence: 88%

On the projective dimension of $5$ quadric almost complete intersections with low multiplicities

Khoury¹

2019

Rocky Mountain J. Math.

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Section: Introductionmentioning

confidence: 88%

On the projective dimension of $5$ quadric almost complete intersections with low multiplicities

Khoury¹

2019

Rocky Mountain J. Math.

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“…Finally, there is work on producing tight bounds in special cases like three cubics or four quadrics; see [23,24,36,43]. There are a great many open questions in this area and the expository article [46] provides a nice introduction (though it was written before many of these recent advances).…”

Section: Effective Boundsmentioning

confidence: 99%

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

Erman

Sam

Snowden

2018

Bull. Amer. Math. Soc.

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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. remains bounded, at least according to a wide variety of measures. We refer to this phenomenon as Stillman uniformity, as Stillman's conjecture is the model case. The purpose of this paper is to give an exposition of Stillman uniformity and some of the work around it.Our account is focused on four closely related threads of work, which we now introduce and briefly summarize.I. Stillman's conjecture. The first indication, as far as we are aware, of the general phenomenon of Stillman uniformity can be found in a conjecture posed by Michael Stillman around the year 2000. 1 Recall that the projective dimension of a module is the minimal length of a projective resolution (see §7 for a review). This is a fundamental, albeit rather technical, invariant. The Hilbert Syzygy Theorem is exactly the statement that every module over an n-variable polynomial ring has projective dimension at most n. Stillman's conjecture refines this theorem: it asserts that the projective dimension of an ideal in an n-variable polynomial ring generated by r homogeneous polynomials 2 of degrees ≤ d can be bounded in terms of r and d, but is independent of the number n of variables. In other words, in this particular regime, the Hilbert Syzygy Theorem holds uniformly in n. Stillman's conjecture was proved by Ananyan and Hochster [2] in 2016, and has subsequently been reproven by us [26] and Draisma, Lasoń, and Leykin [20].

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“…The study of Stillman's question is further motivated by the following parallel question: Caviglia proved that an affirmative answer to Stillman's Question 4.1 implies an affirmative answer to Question 4.2 and vice versa; see [MS,Theorem 2.4] and [Pe,Theorem 29.5]. We focus our attention on Question 4.1 for the rest of this section as all non-trivial known positive answers to Question 4.2 follow from Caviglia's result.…”

Section: Stillman's Questionmentioning

confidence: 99%

“…If the number of terms appearing in the minimal generators of I is bounded (for example, ideals generated by monomials or toric ideals), then Question 4.1 has a positive answer because in this situation there is an immediate bound on the number of variables appearing in the minimal generators of I; thus the minimal resolution of I may be computed over a smaller polynomial ring with a bounded number of variables, and so Hilbert's Syzygy Theorem 2.1 yields a bound. On the other hand, even in the case g = 3 and d 1 = d 2 = d 3 = 2 in which we consider the projective dimension of ideals generated by three quadric forms, it requires a non-trivial argument to show pd S (S/I) ≤ 4; see [MS,Theorem 3.1]. One of the first positive results was proved by Engheta [En1,En2], who showed that the projective dimension of an ideal generated by three cubics is at most 36, while the expected upper bound is 5.…”

Section: Stillman's Questionmentioning

confidence: 99%

Three themes of syzygies

Fløystad

McCullough

Peeva

2016

Bull. Amer. Math. Soc.

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Abstract. We present three exciting themes of syzygies, where major progress has been made recently: Boij-Söderberg theory, Stillman's question, and syzygies over complete intersections.Free resolutions are both central objects and fruitful tools in commutative algebra. They have many applications in algebraic geometry, computational algebra, invariant theory, hyperplane arrangements, mathematical physics, number theory, and other fields. We introduce and motivate free resolutions and their invariants in Sections 1 and 3. The other sections focus on three hot topics, where major progress has been made recently:• Syzygies over complete intersections (see Section 2), • Stillman's question (see Section 4), • Boij-Söderberg theory (see Sections 5 and 6).Of course, there are a number of other interesting aspects of syzygies. The expository papers [MP, PS] contain many open problems and conjectures.

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Bounding Projective Dimension

Cited by 18 publications

References 33 publications

On the projective dimension of $5$ quadric almost complete intersections with low multiplicities

On the projective dimension of $5$ quadric almost complete intersections with low multiplicities

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

Three themes of syzygies

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