Invited Talks 
Linz, Austria
Symbolic Computation and Finite Element Methods:
During the past few decades there have been many examples where computer algebra methods have been applied successfully in the analysis and construction of numerical schemes, including the computation of approximate solutions to partial differential equations. The methods range from Groebner basis computations and Cylindrical Algebraic Decomposition to algorithms for symbolic summation and integration. Among the numerical methods to approximately solve partial differential equations on complicated domains, finite element methods are often the preferred tool. In this talk some recent applications of computer algebra methods to problems arising in finite element methods are presented.
Paris, France
Bounds for the Condition Number of Polynomial Systems with Integer Coefficients:
Polynomial systems of equations are a central object of study in computer algebra. Among the many existing algorithms for solving polynomial systems, perhaps the most successful numerical ones are the homotopy algorithms. The number of operations that these algorithms perform depends on the condition number of the roots of the polynomial system. Roughly speaking the condition number expresses the sensitivity of the roots with respect to small perturbation of the input coefficients.
A natural question to ask is how can we bound, in the worst case, the condition number when the input polynomials have integer coefficients? We address this problem and we provide effective bounds that depend on the number of variables, the degree and the maximum coefficient bitsize of the input polynomials. Such bounds allows us to estimate the bit complexity of the algorithms that depend on the condition number, like the homotopy algorithms, for solving polynomial systems.
