Strongly stable ideals are important in algebraic geometry, commutative algebra, and combinatorics. Prompted, for example, by combinatorial approaches for studying Hilbert schemes and the existence of maximal total Betti numbers among saturated ideals with a given Hilbert polynomial, three algorithms are presented. Each of these algorithms produces all strongly stable ideals with some prescribed property: the saturated strongly stable ideals with a given Hilbert polynomial, the almost lexsegment ideals with a given Hilbert polynomial, and the saturated strongly stable ideals with a given Hilbert function. Bounds for the complexity of our algorithms are included. Also included are some applications for these algorithms and some estimates for counting strongly stable ideals with a fixed Hilbert polynomial.

Tropical geometry is an area of mathematics that interfaces algebraic geometry and combinatorics. The main object of study in tropical geometry is the tropical variety, which is the combinatorial counterpart of a classical variety. A classical variety is converted into a tropical variety by a process called tropicalization, thus reducing the problems of algebraic geometry to problems of combinatorics. This new tropical variety encodes several useful information about the original variety, for example an algebraic variety and its tropical counterpart have the same dimension. In this thesis, we look at the some of the computational aspects of tropical algebraic geometry. We study a generalization of Grobner basis theory of modules which unlike the standard Grobner basis also takes the valuation of coefficients into account. This was rst introduced in (Maclagan & Sturmfels, 2009) in the settings of polynomial rings and its computational aspects were first studied in (Chan & Maclagan, 2013) for the polynomial ring case. The motivation for this comes from tropical geometry as it can be used to compute tropicalization of varieties. We further generalize this to the case of modules. But apart from that it has many other computational advantages. For example, in the standard case the size of the initial submodule generally grows with the increase in degree of the generators. But in this case, we give an example of a family of submodules where the size of the initial submodule remains constant. We also develop an algorithm for computation of Grobner basis of submodules of modules over Z=p`Z[x1; : : : ; xn] that works for any weight vector. We also look at some of the important applications of this new theory. We show how this can be useful in efficiently solving the submodule membership problem. We also study the computation of Hilbert polynomials, syzygies and free resolutions.

One of the fundamental problems in commutative algebra and algebraic geometry is to understand the nature of the solution space of a system of multivariate polynomial equations over a field k, such as real or complex numbers. An important algorithmic tool in this study is the notion of Groebner bases (Buchberger (1965)). Given a system of polynomial equations, f1= 0,..., fm = 0, Groebner basis is a “canonical" generating set of the ideal generated by f1,...., fm, that can answer, constructively, many questions in computational ideal theory. It generalizes several concepts of univariate polynomials like resultants to the multivariate case, and answers decisively the ideal membership problem. The dimension of the solution set of an ideal I called the affine variety, an important concept in algebraic geometry, is equal to the Krull dimension of the corresponding coordinate ring, k[x1,...,xn]/I. Groebner bases were first introduced to compute k-vector space bases of k[x1,....,xn]/I and use that to characterize zero-dimensional solution sets. Since then, Groebner basis techniques have provided a generic algorithmic framework for computations in control theory, cryptography, formal verification, robotics, etc, that involve multivariate polynomials over fields. The main aim of this thesis is to study problems related to computational ideal theory over Noetherian commutative rings (e.g: the ring of integers, Z, the polynomial ring over a field, k[y1,…., ym], etc) using the theory of Groebner bases. These problems surface in many domains including lattice based cryptography, control systems, system-on-chip design, etc. Although, formal and standard techniques are available for polynomial rings over fields, the presence of zero divisors and non units make developing similar techniques for polynomial rings over rings challenging. Given a polynomial ring over a Noetherian commutative ring, A and an ideal I in A[x1,..., xn], the first fundamental problem that we study is whether the residue class polynomial ring, A[x1,..., xn]/I is a free A-module or not. Note that when A=k, the answer is always ‘yes’ and the k-vector space basis of k[x1,..., xn]/I plays an important role in computational ideal theory over fields. In our work, we give a Groebner basis characterization for A[x1,...,xn]/I to have a free A-module representation w.r.t. a monomial ordering. For such A-algebras, we give an algorithm to compute its A-module basis. This extends the Macaulay-Buchberger basis theorem to polynomial rings over Noetherian commutative rings. These results help us develop a theory of border bases in A[x1,...,xn] when the residue class polynomial ring is finitely generated. The theory of border bases is handled as two separate cases: (i) A[x1,...,xn]/I is free and (ii) A[x1,...,xn]/I has torsion submodules. For the special case of A = Z, we show how short reduced Groebner bases and the characterization for a free A-module representation help identify the cases when Z[x1,...,xn]/I is isomorphic to ZN for some positive integer N. Ideals in such Z-algebras are called ideal lattices. These structures are interesting since this means we can use the algebraic structure, Z[x1,...,xn]/I as a representation for point lattices and extend all the computationally hard problems in point lattice theory to Z[x1,...,xn]/I . Univariate ideal lattices are widely used in lattice based cryptography for they are a more compact representation for lattices than matrices. In this thesis, we give a characterization for multivariate ideal lattices and construct collision resistant hash functions based on them using Groebner basis techniques. For the construction of hash functions, we define a worst case problem, shortest substitution problem w.r.t. an ideal in Z[x1,...,xn], and establish hardness results for this problem. Finally, we develop an approach to compute the Krull dimension of A[x1,...,xn]/I using Groebner bases, when A is a Noetherian integral domain. When A is a field, the Krull dimension of A[x1,...,xn]/I has several equivalent algorithmic definitions by which it can be computed. But this is not true in the case of arbitrary Noetherian rings. We introduce the notion of combinatorial dimension of A[x1,...,xn]/I and give a Groebner basis method to compute it for residue class polynomial rings that have a free A-module representation w.r.t. a lexicographic ordering. For such A-algebras, we derive a relation between Krull dimension and combinatorial dimension of A[x1,...,xn]/I. For A-algebras that have a free A-module representation w.r.t. degree compatible monomial orderings, we introduce the concepts of Hilbert function, Hilbert series and Hilbert polynomials and show that Groebner basis methods can be used to compute these quantities. We then proceed to show that the combinatorial dimension of such A-algebras is equal to the degree of the Hilbert polynomial. This enables us to extend the relation between Krull dimension and combinatorial dimension to A-algebras with a free A-module representation w.r.t. a degree compatible ordering as well.