Grobner Bases: Degree Bounds and Generic Ideals

In this thesis, we study two problems related to Gröbner basis theory: degree bounds for general ideals and Gröbner bases structure for generic ideals. We start by giving an introduction to Gröbner 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 Gröbner 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 Gröbner 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\u27s bound, filling in the details omitted in his paper. Finally, we study Gröbner bases structure for generic ideals. It was conjectured by Moreno-Socías that the initial ideal of generic ideals is almost reverse lexicographic, which implies a conjecture by Fröberg 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 Gröbner 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-Socías 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