All Dissertations

8-2014

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Legacy Department

Mathematical Science

Dr. Shuhong Gao

Dr. Michael Burr

Committee Member

Dr. Gretchen Matthews

Dr. Hui Xue

Abstract

In this thesis, we study two problems related to GrÃ¶bner basis theory: degree bounds for general ideals and GrÃ¶bner bases structure for generic ideals. We start by giving an introduction to GrÃ¶bner bases and their basic properties and presenting a recent algorithm by Gao, Volny and Wang. Next, we survey degree bounds for the ideal membership problem, the effective Nullstellensatz, and polynomials in minimal GrÃ¶bner bases. We present general upper bounds, and bounds for several classes of special ideals. We provide classical examples showing some of these bounds cannot be improved in general. We present a comprehensive study of a result by Lazard, that gives a bound on the degree of GrÃ¶bner bases after a generic change of variables. The maximum degree of minimal generators of the initial ideal obtained this way is related to the regularity of the ideal, an important concept in algebraic geometry. We give a complete proof of Lazard's bound, filling in the details omitted in his paper. Finally, we study GrÃ¶bner bases structure for generic ideals. It was conjectured by Moreno-SocÃ­as that the initial ideal of generic ideals is almost reverse lexicographic, which implies a conjecture by FrÃ¶berg on Hilbert series of generic algebras. In the literature, these conjectures were attacked using indirect methods. We use a direct incremental approach, based on a method by Gao, Guan and Volny. We show how a GrÃ¶bner basis for the ideal $\langle I, g\rangle$ can be obtained from that of $I$ when adding a generic polynomial $g$, using properties of the standard basis of $I$. For a generic ideal $I = \langle f_1, \dots, f_n \rangle$ in $K[x_1, \dots, x_n]$, with $\deg f_i = d_i$, we are able to give a complete description of the ideal of leading terms of $I$ in the case where $d_i \geq \left(\sum_{j=1}^{i-1}d_j\right) - i -2$. As a result, we obtain a partial answer to Moreno-SocÃ­as Conjecture: the initial ideal of $I$ is almost reverse lexicographic if the degrees of generators satisfy the condition above. This result slightly improves a result by Cho and Park. We hope this approach can be strengthened to prove the conjecture in full.

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