Some geometric results are presented which can be derived from algebraic formulae for topological degree and Euler characteristic. In particular, it is shown that Euler characteristics of configuration spaces and work spaces of mechanical linkages can be computed in an algorithmic way. We also find the expected gradient degree of rotation invariant Gaussian random polynomial on an even-dimensional space.
Mapping degree, Euler characteristic, mechanical linkage, configuration space, work space, Gaussian random polynomial.
MSC 2000: 14P25, 58R45