Gr(o|¨)bner Basis Over A Noetherian Valuation Ring

Although the development of Gr(o|¨)bner basis theory is just more than forty years,it has a wide range of applications in various fields, such as, solving algebraicequations, computational algebraic geometry and commutative algebra, algebraicmanifold decomposition, cryptography and coding, image processing and so on. Sincethe Gr(o|¨)bner bases appeared, the scholars in mathematics, computer science, systemscience and other field, especially those, in (computational) algebra, algebraicgeometry, have been paying much attention to it and been developing both the theoryand applications of the Gr(o|¨)bner basis rapidly. Currently, there are a large number ofarticles and monographs in this area, among them, T. Becker and V.Weispfenning [8],David Cox, John Little and Donal O’Shea [11], both contain a detailed description ofGr&o&bner basis theory and applications. In addition, many scholars study the Gr(o|¨)bnerbasis on other algebraic structures (see Byrne [10], Kacem [15], Kosir [16], Norton[21,22], Pauer [23], Weispfenning [28,29], Yengui [30], etc) or try to find moreeffective methods to calculate Gr(o|¨)bner basis over other specific rings(see Faugere[12], Mnif [18], Montes [19]), and also many scholars study its further applications,Shi [27] employs it recently to investigate the graph representation of projectiveresolutions, Byrne [10] and Norton [20] explore its other applications.In this paper, we mainly study the properties and algorithm of Gr(o|¨)bner basis overa noetherian valuation ring. The paper is divided into four chapters.The first chapter reviews briefly the origin and development history as well assome applications of the Gr(o|¨)bner basis theory.The second chapter introduces some definitions and fundamental facts in abstractalgebra that will be used in the next chapter.The third chapter, the main topic of this paper, focues on studying the propertiesof the Gr(o|¨)bner base of the polynomial ring over a noetherian valuation ring, andparticularly the properties---the existence and uniqueness and algorithms---of theminimal and respectively the reduced Gr(o|¨)bner bases, generalizing the existing resultsof polynomial rings over a field.The last chapter is a brief introduction to the applications, especially those to theideal membership problem for polynomial rings over a noetherian valuation ring, andon cyclic codes over an artinian chain ring.