caa a22 4500 475740645 CHVBK 20180406123458.0 cr unu---uuuuu 170329e20001101xx s 000 0 eng 10.1023/A:1026612817194 doi (NATIONALLICENCE)springer-10.1023/A:1026612817194 Petrogradskii V. Ul'yanovsk State University, Russia aut On the Complexity Functions for T -Ideals of Associative Algebras [Elektronische Daten] [V. Petrogradskii] Let $$c_n \left( V \right)$$ be the sequence of codimension growth for a variety V of associative algebras. We study the complexity function $$\mathcal{C}\left( {V,z} \right) = \sum\nolimits_{n = 0}^\infty {c_n } {{\left( V \right)z^n } \mathord{\left/ {\vphantom {{\left( V \right)z^n } {n!}}} \right. \kern-\nulldelimiterspace} {n!}}$$ , which is the exponential generating function for the sequence of codimensions. Earlier, the complexity functions were used to study varieties of Lie algebras. The objective of the note is to start the systematic investigation of complexity functions in the associative case. These functions turn out to be a useful tool to study the growth of varieties over a field of arbitrary characteristic. In the present note, the Schreier formula for the complexity functions of one-sided ideals of a free associative algebra is found. This formula is applied to the study of products of T-ideals. An exact formula is obtained for the complexity function of the variety U c of associative algebras generated by the algebra of upper triangular matrices, and it is proved that the function $$c_n \left( {U_c } \right)$$ is a quasi-polynomial. The complexity functions for proper identities are investigated. The results for the complexity functions are applied to study the asymptotics of codimension growth. Analogies between the complexity functions of varieties and the Hilbert--Poincaré series of finitely generated algebras are traced. Plenum Publishing Corporation, 2000 variety of associative algebras nationallicence complexity function nationallicence growth of a variety nationallicence Schreier formula nationallicence Hilbert--Poincaré series of a finitely generated algebra nationallicence T -ideal nationallicence Mathematical Notes Kluwer Academic Publishers-Plenum Publishers 68/5-6(2000-11-01), 751-759 0001-4346 68:5-6<751 2000 68 11006 https://doi.org/10.1023/A:1026612817194 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1023/A:1026612817194 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Petrogradskii V. Ul'yanovsk State University, Russia aut NATIONALLICENCE 773 0- Mathematical Notes Kluwer Academic Publishers-Plenum Publishers 68/5-6(2000-11-01), 751-759 0001-4346 68:5-6<751 2000 68 11006 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer