Pythagore's Dilemma, Symbolic-Numeric Computation, and the Border Basis Method

In this tutorial paper, we first discuss the motivation of doing symbolic-numeric computation, with the aim of developing efficient and certified polynomial solvers. We give a quick overview of fundamental algebraic properties, used to recover the roots of a polynomial system, when we know the multiplicative structure of its quotient algebra. Then, we describe the border basis method, justifying and illustrating the approach on several simple examples. In particular, we show its usefulness in the context of solving polynomial systems, with approximate coefficients. The main results are recalled and we prove a new result on the syzygies, naturally associated with commutation properties. Finally, we describe an algorithm and its implementation for computing such border bases