## Master's Theses

#### Title

An F4-Style Involutive Basis Algorithm

Summer 2010

Masters Thesis

#### Degree Name

Master of Science (MS)

Mathematics

Haiyan Tian

Mathematics

#### Abstract

How to solve a linear equation system? The echelon form of this system will be obtained by Gaussian elimination then give us the solution. Similarly, Grebner Basis is the "nice form" of nonlinear equation systems that can span all the polynomials in the given ideal [4]. So we can use Grebner Basis to analyze the solution of a nonlinear equation system. But how to compute a Grebner Basis? There exist several ways to do it. Buchberger's algorithm is the original method [2]. Gebauer-Moller algorithm [6] is a refined Buchberger's algorithm. The F4 algorithm [5] uses matrix reduction to compute efficiently. Involutive Basis algorithm [8, 1, 12] is an effective method avoiding much ambiguity in the other algorithms. In Chapters 1 and 2 we describe two well-known methods of computing Grebner Basis called Buchberger's and F4 algorithm. In Chapter 3 after presenting the definition of involutive division we give a detailed formulation of basic and improved Involutive Basis algorithm. We will see that there exists ambiguity both in Buchberger's and F4 algorithm. But in the method of Involutive Basis Algorithm, the ambiguity for the choice of prolonagation has been avoided. So in Chapter 4 we combine the F4 algorithm and Involutive Basis algorithm in order to obtain a new approach that can reduce polynomials faster as well as avoid ambiguity. The combined algorithm called F4-involutive is a partial result due to its efficiency. More work such as implementing Buchberger's criteria would be done in the future.

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