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Week
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Topics
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Study Metarials
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1
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The relation between geometry and algebra, polynomials and affine space; the definition and basic properties of an affine variety, and its examples
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2
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Parametrizations of affine varieties, the ideal of an affine variety and the relations between a variety and its ideal
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3
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The problem of orderings on the monomials in the ring of n-variable polynomials over a field k, and a generalization of a division algorithm in one variable polynomials to n-variable polynomials
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4
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Monomial ideals and Dickson`s lemma, the Hilbert basis theorem
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5
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Groebner bases of ideals and its properties
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6
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The problem of finding the Groebner basis of an ideal, Buchberger`s algorithm, first applications of Groebner bases
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7
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The elimination and extension theorems, the geometry of elimination, and the closure theorem
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8
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Implicitization
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9
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Singular points of a curve and envelopes of a family of curves
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10
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Irreducible polynomials, unique factorization and resultans
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11
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Proof of the extension theorem by using resultans
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12
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Hilbert`s Nullstellensatz, radical ideals and ideal-variety correspondence
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13
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Sums, products, and intersections of ideals, the Zariski closure of a set in an affine space and quotients of ideals
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14
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Irreducible varieties and prime ideals, decomposition of a variety into irreducibles
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