Speciality:
05.13.11 (Mathematical and programme software for computers, computer systems, and networks)
Birth date:
11.04.1946
E-mail: , ,
Website: http://www.ccas.ru/zavar/abrsa/ Keywords: computer algebra; symbolic computation; linear ordinary differential equations; linear (q-)difference equations; identtities proving.
Subject:
The main results are related to computer algebra (symbolic computation), linear ordinary differential and (q-)difference equations. Algorithms to solve the following problems were proposed: the decomposition of indefinite sums of rational functions (an analogue of integral algorithms by Hermite and Ostrogradskii); the same problem for indefinite sums of hypergeometric terms (with M. Petkovsek); constructing rational solutions of linear differential and (q-)difference equations with polynomial coefficients; the same problem for systems of equations (with M. Barkatou and M. Bronstein); constructing q-hypergeometric solutions of linear q-difference equations with polynomial coefficients (with M. Petkovsek and P. Paule). In the context of the theory of non-commutative Ore polynomials a number of algorithms were designed (these algorithms are adjustable for differential, difference, q-difference and some other cases): an algorithm for "accurate integration" of solutions of equations (with M. van Hoeij); an algorithm for the peripheral factorization of Ore polynomials (with S. P. Tsarev); an algorithm to search for d'Alembertian solutions (with M. Petkovsek for the homogeneous case, with E. V. Zima for equations with d'Alembertian right-hand sides). Some algorithms, related to power series solutions were proposed: given a linear ordinary differential equation with polynomial coefficients, to find the points where the equation has power series solution with hypergeometric coefficients (with M. Petkovsek and A. A. Ryabenko); to find the points where the equation has power series solution with sparse sequence of coefficients, etc. A correct algorithmical solution of the orbit problem for algebraic numbers (with M. Bronstein). The well-known Zeilberger's algorithm, which is a useful tool to prove combinatorial identities, was improved. First, the problem of recognizing if Zeilberger's algorithm is applicable to a given hypergeometric term (i.e., the algorithm terminates in finite time) was solved. Second, a method to reduce the complexity of the item-by-item examination inherent in Zeilberger's algorithm was proposed (with H. Q. Le). The Wilf–Zeilberger conjecture that a hypergeometric term is proper iff it is holonomic was proven (with M. Petkovsek). Outside of symbolic computation, e.g., the "multiple cards algorithm" to control the questions of a learning system (with G. G. Gnezdilova) and the optimal-in-average algorithm to search for the maximal and the minimal elements of finite set of numbers were proposed.
Main publications:
Abramov S., Petkovsek M., Ryabenko A. Special formal series solutions of linear operator equations // Discrete Math., 2000, 210, 3–25.
Abramov S., van Hoeij M. Integration of solutions of linear functional equations // Integral transforms and special functions, 1999, V. 8, no. 1–2, 3–12.
Abramov S. m-Sparse solutions of linear ordinary differential equations with polynomial coefficients // Discrete Math., 2000, 217, 3–15.
Abramov S., Bronstein M. On solutions of linear functional systems // Proceedings of ISSAC'01, 2001, London, ACM Press, 1–6.
ISTINA
