Intersections of polynomial rings and modules with applications. We are concerned with two problems, namely with computing the intersection of (some) graded structures of commutative algebra and computing the stratification of linear actions of compact Lie groups. The connection between these seemingly unrelated problems is invariant theory, more precisely, the construction of fundamental invariants and, additionally, of fundamental equivariants of compact Lie groups. The principle of symmetry breaking allows to decompose in a unique way the representation and orbit space in finitely many disjoint basic open semi-algebraic sets, called strata. A set of fundamental invariants allows to construct this stratification of the representation- and the orbit space. We provide algorithms for computing the intersection of finitely generated graded subalgebras, which can be specialized to compute invariants of algebraic groups and invariant rings of compact Lie groups, and for computing the intersection of graded submodules, which can be specialized to compute equivariants of algebraic groups. Moreover we propose a new approach for stratifying actions of compact Lie groups and we present algorithms for computing s stratification of the representation space and of all or selected strata (and their closures) of the orbit space. In addition, we show that the dimension of a stratum of the orbit space is an upper and lower bound for the number of inequalities needed for a description. Finally we describe the computer algebra package STRATIFY for computing stratifications of compact Lie groups.
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