Uniform Upper Bounds in Computational Commutative Algebra

Yihui Liang, Purdue University

Abstract

Let S be a polynomial ring K[x1, . . . , xn] over a field K and let Fbe a non-negatively graded free module over S generated by m basis elements. In this thesis, we study four kinds of upper bounds: degree bounds for Gr¨obner bases of submodules of F, bounds for arithmetic degrees of S-ideals, regularity bounds for radicals of S-ideals, and Stillman bounds. Let M be a submodule of F generated by elements with degrees bounded above by D and dim(F/M) = r. We prove that if M is graded, the degree of the reduced Gr¨obner basis of M for any term order is bounded above by 2 [1/2((Dm) n−rm + D)]2 r−1 . If M is not graded, the bound is 2 h 1/2((Dm) (n−r) 2m + D) i2 r. This is a generalization of bounds for ideals in a polynomial ring due to Dub´e (1990) and Mayr-Ritscher (2013). Our next results are concerned with a homogeneous ideal I in S generated by forms of degree at most d with dim(S/I) = r. In Chapter 4, we show how to derive from a result of Hoa (2008) an upper bound for the regularity of √ I, which denotes the radical of I. More specifically we show that reg(√ I) ≤ d (n−1)2r−1 . In Chapter 5, we show that the i-th arithmetic degree of I is bounded above by 2 · d 2 n−i−1. This is done by proving upper bounds for arithmetic degrees of strongly stable ideals and ideals of Borel type. In the last chapter, we explain our progress in attempting to make Stillman bounds explicit. Ananyan and Hochster (2020) were the first to show the existence of Stillman bounds. Together with G. Caviglia, we observe that a possible way of making their results explicit is to find an effective bound for an invariant called D(k, d) and supplement it into their proof. Although we are able to obtain this bound D(k, d) and realize Stillman bounds via an algorithm, it turns out that the computational complexity of Ananyan and Hochster’s inductive proof would make the bounds too large to be meaningful. We explain the bad behavior of these Stillman bounds by giving estimates up to degree 3.

Degree

Ph.D.

Advisors

Caviglia, Purdue University.

Subject Area

Mathematics

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