Titlebook: Ideals, Varieties, and Algorithms; An Introduction to C David Cox,John Little,Donal O’Shea Textbook 19972nd edition Springer Science+Busine

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Groebner Bases,shion. The method of Groebner bases is also used in several powerful computer algebra systems to study specific polynomial ideas that arise in applications. In Chapter 1, we posed many problems concerning the algebra of polynomial ideals and the geometry of affine varieties. In this chapter and the next, we will focus on four of these problems.

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Elimination Theory,theory of resultants. The geometric interpretation of elimination will also be explored when we discuss the Closure Theorem. Of the many applications of elimination theory, we will treat two in detail: the implicitization problem and the envelope of a family of curves.

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Projective Algebraic Geometry, projective point of view. By working in projective space, we will get a much better understanding of the Extension Theorem from Chapter 3. The chapter will end with a discussion of the geometry of quadric hypersurfaces and an introduction to Bezout’s Theorem.

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The Algebra-Geometry Dictionary, arising out of the Hilbert Basis Theorem: notably the possibility of decomposing a variety into a union of simpler varieties and the corresponding algebraic notion of writing an ideal as an intersection of simpler ideals.

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Geometry, Algebra, and Algorithms,her dimensional objects) defined by polynomial equations. To understand affine varieties, we will need some algebra, and in particular, we will need to study . in the polynomial ring .[.., ..., ..]. Finally, we will discuss polynomials in one variable to illustrate the role played by ..

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Groebner Bases,apter, we will study the method of Groebner bases, which will allow us to solve problems about polynomial ideals in an aIgorithmic or computational fashion. The method of Groebner bases is also used in several powerful computer algebra systems to study specific polynomial ideas that arise in applica

Induction