Course Code：S070104ZJ006
Course Category：Professional Basic Course
Period/Credits：40/2
Prerequisites：Higher algebra, abstract algebra
Aims & Requirements：
This course is a basic course of mathematics, robotics, coding and other calculation and application of algebraic geometry. Then contents include the basic knowledge of algebraic geometry, such as the decomposition of the ideal theory of algebraic varieties, dimension theory; Groebner base in algebraic geometry, including the problem of elimination ideals,Hilbert polynomial, Buchberger algorithms; elimination methods in algebraic geometry, such as knot theory, quantifier elimination in algebraically closed field.
Primary Coverage：
Chapter 1, affine varieties, Zariski topology, resultant and U-resultant, Extension theorem, Hilbert zero theorem
Chapter 2, The concept of Groebner basis, Buchberger’s algorithm, determination of root ideal of polynomial ideal, zero dimensional ideal, saturation ideal.
Chapter 3, algebra-geometry dictionary, calculations of algebraic varieties, Shape lemma, decomposition of algebraic varieties.
Chapter 4, Dimensional theory, transcend degree, dimension of algebraic varieties and ideals, Hilbertplynomial, Noether Lemma, dimensional theorem of affine varieties.
References：
1.Cox, Little, O＇Shea: Ideals, Varieties and Algorithms, Second Edition, Springer, 1996.
2.Cox, Little, O＇Shea：Using Algebraic Geometry, Second Edition, Springer, 2004.
Author：Ziming Li（Academy of Mathematics and Systems Science ）
Date：June, 2009
