Giorgio Ottaviani
Monomial orders. Groebner basis and the Buchberger algorithm. The Elimination Theorem. Ideals and algebraic varieties. The Zariski topology. Hilbertnullstellensatz. Cartesian and parametric equations for algebraic varieties. Hilbert polynomial and the dimension of an algebraic variety. Solving polynomial equations via Eigenvalues. Counting the real solutions.Primary decomposition for zero dimensional ideals. Extension to many variables.
Notes bt the teacher available on Moodle.
D.Cox, J.Little, D.O'Shea, Ideals, Varieties and Algorithms, Springer 1992, capp. 1,2,3,7,9
D.Cox, J.Little, D.O'Shea, Using Algebraic Geometry, Springer 1998, cap. 2
Learning Objectives
Introduction to algebriac varieties, from a constructive point of view. Learning constructive techniques to manage polynomials. Understanding the differences between the linear and the nonlinear case.
Prerequisites
Algebra I, Geometria I
Teaching Methods
Lectures and computer-aided exercises.
Total hours of the course (including the time spent in attending lectures, seminars, private study, examinations, etc...): 48
Hours reserved to private study and other indivual formative activities: 102
Further information
Interactive informations on moodle web site
Type of Assessment
Oral examination and preparation of two scripts executable with Macaulay2, chosen by the students from a list of 4/5 suggested at the end of the course. The oral exam concerns in particular the topics of the scripts chosen.
Course program
Noetheria rings and basissatz. Monomial orders and their properties. Division algorithm. Monomial ideals. Groebner basis. They generate. Normal form with respect to an ideal. Ideal membership criterion. S-pairs. Technical lemma which shows that any cancellation is generated by S-pairs. Buchberger criterion. Buchberger algorithm to construct a Groebner basis. Existence and uniqueness of a reduced Groebner basis. Elimination Theorem. Parametric representation of a variety. Tangential surfaces. Intersection of ideals and their computation. Lcm and GCD. Product among ideals. Radical ideal. Definition of algebraic variety. Ideals of a subset of K^n. Prime ideals and radical. Zariski topology. Hilbertnullstellenstatz. The resultant and the elimination ideal. The extension theorem. Consistency algorithm for solutions of a polynomial system. The closure theorem. Fermat curves. Polynomial and rational parametrizations.
The quotient of polynomial ideal with respect to an ideal, the case V(I) finite.
Hamilton-Cayley theorem. Minimal polynomial and diagonalization. Symultaneous diagonalization. Generalized eigenspaces. Companion matrix and its properties. Trace form and chinese remainder theorem in polynomial setting. Sylvester theorem on counting the real roots of a polynomial. Primary decomposition of zerodimensional ideals. Eigenvalues of companion matrices in many variables. Stickelberger Theorem. Trace form for zerodimensional polynomial systems in many variables. Sylvester theorem on counting the real roots of zerodimensional polynomial systems in many variables.