Fachbereich Mathematik und Statistik

Many problems in applied mathematics can be formulated as a system of nonlinear equa-tions or inequalities, and a broad subset are those problems consisting either partially or completely of multivariate polynomials. Broadly speaking, an ‘enclosure ’ method attempts to solve such a problem over a specified region of interest, commonly a box subset of Rn, where n is the number of unknowns. Subdivision (or branch-and-bound) is a commonly-applied scheme, wherein a starting box is successively subdivided into sub-boxes; sub-boxes for which a proof of non-existence of a solution can be completed are discarded. As a component of such, a boundary method attempts to exploit the properties of component functions over the boundary of a sub-box, without considering their behaviour within it. Two main types of non-existence proof are considered for polynomial systems over boxes or sub-boxes. Firstly, the topological degree of a system of equations (considered as a mapping from Rn to Rn) can be computed over a sub-box, which possesses a root-counting property. Together with an enclosure for the determinant of the Jacobian, the existence (or otherwise) of roots in the sub-box can be ascertained. Secondly, and alternatively,