Khovanskiĭ, A. G. Multidimensional results on the nonrepresentability of functions by quadratures. (English. Russian original) Zbl 1078.32006 Funct. Anal. Appl. 37, No. 4, 302-310 (2003); translation from Funks. Anal. Prilozh. 37, No. 4, 74-85 (2003). Summary: The paper completes the construction of a multidimensional topological version of differential Galois theory. We construct a rich class of germs of functions of several variables which is closed under superpositions and other natural operations. The main theorem describes the behavior of the monodromy groups of such germs under the natural operations. As a result, we obtain topological obstructions to the representability of functions by quadratures, which give the strongest known statements about unsolvability of equations in closed form. MSC: 32B99 Local analytic geometry 41A55 Approximate quadratures 65D30 Numerical integration 58K10 Monodromy on manifolds Keywords:multivalued function; monodromy group; differential Galois theory; representability by quadratures PDFBibTeX XMLCite \textit{A. G. Khovanskiĭ}, Funct. Anal. Appl. 37, No. 4, 302--310 (2003; Zbl 1078.32006); translation from Funks. Anal. Prilozh. 37, No. 4, 74--85 (2003) Full Text: DOI