(TENTATIVE) OUTLINE OF CBMS LECTURES BY BERND STURMFELS

  • Lecture 1: Polynomials in One Variable
    Basic properties of univariate polynomials, companion matrices, Fundamental Theorem of Algebra, solvability in terms of radicals, Sturm sequences, Descartes' rule of signs, numerical root finding.
  • Lecture 2: Hypergeometric Functions
    We study the roots of the general polynomial of degree n as an algebraic function of all n+1 coefficients, and we determine the system of linear differential equations which characterizes this algebraic function. This leads to 2^{n-1} distinct Puiseux series formulas for the roots.
  • Lecture 3: Groebner Bases and Numerical Linear Algebra
    We show how Groebner bases for any term order can be used to solve zero-dimensional polynomial systems, by setting up an appropriate eigenvalue problem. The trace form can be used to count the number of real roots.
  • Lecture 4: Rational Univariate Representation
    Faugere and Roullier have the currently best Groebner based method for finding all complex (and all real) roots of a polynomial system. One key idea is that of a rational univariate representation of the solution set. We explain how this works, and we discuss the state of the art on Groebner bases methods for solving zero-dimensional systems.
  • Lecture 5: Resultants
    Resultants are very useful for studying square systems, where the number of equations equals the number of variables. We give summary of known exact determinantal formulas for resultants, specifically for dense systems in few variables. This includes references to Chow forms and sparse resultants.
  • Lecture 6: Homotopy Methods
    We give a survey of numerical homotopy methods for solving polynomial equations, including sparse and determinantal systems, including recent work of Li, Sommese, Verschelde, and their collaborators.
  • Lecture 7: The Real Nullstellensatz
    The real Nullstellensatz is a common generalization of the usual Nullstellensatz (over an algebraically closed field) and Linear Programming Duality (for linear inequalities). It says that a polynomial system over the reals either has a solution or there exists an obvious certificate for its non-solvability. A special case concerns the representability of positive polynomials as sums of squares.
  • Lecture 8: Polynomial Optimization and Semidefinite Programming
    We discuss the problem of minimizing real polynomial functions, possibly subject to semi-algebraic constraints. We show how the highly efficient technique of semi-definite programming can be used to solve such problems.
  • Lecture 9: Primary Decomposition
    Solving a polynomial system means to decompose the solution set into irreducible components, or more generally into its primary components. We show how this can be done, first for the simpler cases of monomial and binomial ideals, and then for general polynomial ideals.
  • Lecture 10: The Palamodov-Ehrenpreis Theorem
    A system of polynomial equations can be regarded as a linear system of differential equations with constant coefficients. In this lecture we explain what can be gained by this point of view. We argue that the Palamodov-Ehrenpreis Theorem, a classical result in analysis, can be seen as the ultimate representation of the solutions to a polynomial system. In particular, we discuss recent work of Oberst which makes the Palamodov-Ehrenpreis Theorem effective.