Spring 1999
Computer Algebra (3) MAS-5731
Instructor: Professor Teo Mora

Professor Mora is visiting FSU from the University of Genoa (Italy) during  the Spring Semester 1999.  He is one of the world leaders of the theory of Groebner bases - a central topic in computer algebra.  
Also those students who have taken Computer Algebra previously can take this class.

Topics to be covered:

  • Recall on Euclidean Algorithm, Chinese Remainder Theorem  and Lagrange Formula
  • The introduction of the imaginary number
  • Multiplicity of roots and Field Theory
  • Kronecker Theory: algebraic extension, splitting field, equation  solving
  • Finite field and applications
  • Duval representation of algebraic numbers and equation solving
  • Representation of real numbers
  • Recall on Groebner bases and Buchberger Algorithm
  • Application of Groebner bases to commutative algebra (syzygies and Hilbert function)
  • Ideal decomposition: Lasker-Noether Theorem and the  Gianni-Trager-Zacharias decomposition techniques
  • Root multiplicity of 0-dimension ideal: Groebner duality
  • Solving 0-dimensional polynomial equation systems: Trinks and  Auzinger-Stetter Algorithms
  • Kronecker-Duval representation of the roots of a 0-dimensional  Polynomial Equation Systems: Shape Lemma and the ``natural'' representation

    • What does it mean to solve a polynomial equation system?  For a univariate polynomial   f(X)   in  k[X]  apparently the answer is obvious:  e.g. it is clear that the solutions of   f(X) = X2 - X  are  0 and  1.
    Analogously, we could then say that ``the solutions of  f(X) = X2 - 2  are   21/2 and   -21/2''.

    Yes, yes, ...  unless somebody asks you for a definition of  21/2  ... Well, whatever approach you use, your only possible answer is: ``21/2 and   -21/2 are the solutions of  X2- 2''.  Apparently, we have a funny tautology:  the solutions of   X2- 2  are  the solutions of   X2 - 2!

    If you are not really convinced by this, let me try a stronger example: you will agree that the solutions of the polynomial   X2 + 1   are   i  and -i and that the  imaginary number  i can be defined only as that number whose square is  -1,  i.e. to be a solution of the polynomial   X2 + 1. So we truly have a  tautology:   The solutions of the polynomial equation   X2 + 1=0  are the two solutions of the polynomial equation    X2 + 1=0.

    To base a Solving Polynomial Equation Systems theory on a tautology is not a  clever idea. So it is essential to understand better what does it means  ``solving''.

    In the classical interpretation ``solving'' polynomial equations meant to  be able to compute the roots by applying the operations to the  coefficients of the equations; it is even a temptation to translate this notion  of ``solving'' in a more modern language, saying that solving an equation means writing a program whose input is the coefficients  of the equation and the output is the roots, the allowed operations being the  arithmetical operations, testing equalities and branching.

    When Abel-Ruffini Theorem proved the impossibility of solving univariate  polynomials with the classical notion of ``solving'', the way of getting out  of this impasse was  Kronecker's solution to interpret  ``solving'' via the theory of algebraic extensions as   producing programs  which compute  with  the roots of a polynomial equation.  The recent break-through of Computer Algebra in solving polynomial equation  systems is based on following Kronecker's approach as much as possible.

    For the most advanced Computer Algebra research, the most effective way for  solving polynomial equation  systems is just to interpret such system as a tool  for solving itself, by building programs which use this tool to manipulate its  own roots. In other words this means that instead of working hard to build programs which   compute  the solutions, one should work hard to build  programs which use the given equations in order to  manipulate  the  solutions, without even computing them.

    The course will presents the most recent Computer Algebra advances to  solving polynomial equation systems and algebraic number representation,  using Kronecker's approach as a constant guideline.