Invited Talks
We are proud to present the invited speakers of CASC conference:
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Prof. Carsten Schneider, Director of Research Institute for Symbolic Computation, Johannes Kepler University, Linz, Austria
This talk provides an overview of the fundamental principles used to simplify multi-sums into indefinite nested sums over hypergeometric products in the setting of difference rings. We place special emphasis on the algorithmic translation between hypergeometric sums and the formal difference ring setting. Furthermore, we detail the core summation paradigms of telescoping, creative telescoping, and recurrence solving within difference rings, illustrating these techniques and their underlying algorithms with concrete examples.
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Dr. Joseph Tooby-Smith, University of Bath, UK
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Thi Xuan Vu - Université de Lille, France
In the context of solving systems of polynomial equations, homotopy refers to the process of deforming a simple system G(X), where X is a sequence of variables, whose solutions are known, into a target system F(X) that we aim to solve. There are two main approaches to homotopy methods: numerical and symbolic. While the core mathematical idea, by using the continuous transformation H(t, X) = (1-t) G(X) + t F(X), is the same, the way we handle that transformation differs between the symbolic and numerical worlds.
Numerical homotopy methods approximate solutions by tracking solution paths as the parameter t varies from 0 to 1, using predictor-corrector techniques. In contrast, symbolic homotopy methods treat t as an algebraic parameter and study the structure of the ideal generated by the homotopy system in order to obtain a parametric description of the solutions for all values of t simultaneously. Then specilizing this formula at t = 1 gives a parametrization for the solution set of the target system F(X).
In this talk, we focus on symbolic homotopy techniques for solving multivariate polynomial systems, with a particular emphasis on determinantal systems arising from minors of polynomial matrices, illustrating how exploiting algebraic structure can lead to efficient solution methods.