Ilash N. Asymptotic behavior of the Poincaré series of algebras of –invariants.

This thesis studies the Poincaré series of the algebras of –invariants as well as the first two coefficients of the Laurent expansion of the Poincaré series at The first coefficient of this expansion is called the degree of the algebra.The main part of the thesis consists of introduction, four chapters, conclusion and references.The introduction provides the well-known results, highlights the object, subject and research methods, gives the aim of the research, defines the scientific novelty and relations with scientific programs. In addition, it contains the information about the results obtained in the dissertation, author’s relevant publications, and the structure of the thesis.Chapter one gives the results previously obtained by the other scientists, bibliography review on the thesis topic, introduces basic definitions.The second chapter analyses the Poincaré series of the algebra of covariants for binary form. It was calculated the first two terms of the Laurent series expansion of at the point of this algebra. Also, it was calculated both an integral representation and the asymptotic behavior of the terms. In addition, in this section it is proved that the degree of the algebra of covariants of binary form is a positive number. In chapter three we computed both the degree of the algebra of joint invariants of two binary forms and the degree of the algebra of joint covariants of two binary forms. Moreover, several new combinatorial identities were found. Chapter 4 deals with the study of the Poincaré series of the algebra of joint invariants of n linear forms, the algebra of joint covariants of n linear forms, the algebra of joint invariants of n quadratic forms, the algebra of joint covariants of n quadratic forms. In particular, we expressed the Poincaré serries of these algebras in terms of Narayana polynomials, and got recurrence relations for these series. This chapter gives the first two terms of the Laurent series expansion at the point of the Poincaré series of the algebras of joint invariants and covariants of n linear and n quadratic forms. In addition, it has been calculated both an integral representation and the asymptotic behavior of the terms. In this chapter it has been expressed the Hilbert polynomials of those algebras in terms of quasi-polynomial. Moreover, we have got several binomial identities.