Shawki Al-Rashed

• Polynomial systems arise in many applications: robotics, kinematics, chemical kinetics, computer vision, truss design, geometric modeling, and many others. Many polynomial systems have solutions sets, called algebraic varieties, having several irreducible components. A fundamental problem of the numerical algebraic geometry is to decompose such an algebraic variety into its irreducible components. The witness point sets are the natural numerical data structure to encode irreducible algebraic varieties. Sommese, Verschelde and Wampler represented the irreducible algebraic decomposition of an affine algebraic variety \(X\) as a union of finite disjoint sets \(\cup_{i=0}^{d}W_i=\cup_{i=0}^{d}\left(\cup_{j=1}^{d_i}W_{ij}\right)\) called numerical irreducible decomposition. The \(W_i\) correspond to the pure i-dimensional components, and the \(W_{ij}\) represent the i-dimensional irreducible components. The numerical irreducible decomposition is implemented in BERTINI. We modify this concept using partially Gröbner bases, triangular sets, local dimension, and the so-called zero sum relation. We present in the second chapter the corresponding algorithms and their implementations in SINGULAR. We give some examples and timings, which show that the modified algorithms are more efficient if the number of variables is not too large. For a large number of variables BERTINI is more efficient. Leykin presented an algorithm to compute the embedded components of an algebraic variety based on the concept of the deflation of an algebraic variety. Depending on the modified algorithm mentioned above, we will present in the third chapter an algorithm and its implementation in SINGULAR to compute the embedded components. The irreducible decomposition of algebraic varieties allows us to formulate in the fourth chapter some numerical algebraic algorithms. In the last chapter we present two SINGULAR libraries. The first library is used to compute the numerical irreducible decomposition and the embedded components of an algebraic variety. The second library contains the procedures of the algorithms in the last Chapter to test inclusion, equality of two algebraic varieties, to compute the degree of a pure i-dimensional component, and the local dimension.