61 research outputs found

    A general framework for Noetherian well ordered polynomial reductions

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    Polynomial reduction is one of the main tools in computational algebra with innumerable applications in many areas, both pure and applied. Since many years both the theory and an efficient design of the related algorithm have been solidly established. This paper presents a general definition of polynomial reduction structure, studies its features and highlights the aspects needed in order to grant and to efficiently test the main properties (noetherianity, confluence, ideal membership). The most significant aspect of this analysis is a negative reappraisal of the role of the notion of term order which is usually considered a central and crucial tool in the theory. In fact, as it was already established in the computer science context in relation with termination of algorithms, most of the properties can be obtained simply considering a well-founded ordering, while the classical requirement that it be preserved by multiplication is irrelevant. The last part of the paper shows how the polynomial basis concepts present in literature are interpreted in our language and their properties are consequences of the general results established in the first part of the paper.Comment: 36 pages. New title and substantial improvements to the presentation according to the comments of the reviewer

    Using elimination to describe Maxwell curves

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    Cartesian ovals are curves in the plane that have been studied for hundreds of years. A Cartesian oval is the set of points whose distances from two fixed points called foci satisfy the property that a linear combination of these distances is a fixed constant. These ovals are a special case of what we call Maxwell curves. A Maxwell curve is the set of points with the property that a specific weighted sum of the distances to n foci is constant. We shall describe these curves geometrically. We will then examine Maxwell curves with two foci and a special case with three foci by deriving a system of equations that describe each of them. Since their solution spaces have too many dimensions, we will eliminate all but two variables from these systems in order to study the curves in xy-space. We will show how to do this first by hand. Then, after some background from algebraic geometry, we will discuss two other methods of eliminating variables, Groebner bases and resultants. Finally we will find the same elimination polynomials with these two methods and study them

    Non-acyclicity of coset lattices and generation of finite groups

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