A numeric-symbolic interface was designed for the REDUCE computer algebra system, which allows to call FORTRAN and C codes from LISP functions in interpretation as well as in compilation regimes, and inversely, that is, calling both compiled and interpreted LISP functions from numerical codes written in C or FORTRAN. An extra debugging tool was also developed, such that dynamically loaded codes in C, FORTRAN and LISP can be debugged in running process .
A new general algorithmic technique for the involutive analysis of polynomial ideals and polynomial equation systems was designed. It is based on a new concept of involutive monomial division. Based of the latter, a new efficient algorithm for transformation of algebraic nonlinear equation systems into involutive form, turning up into a certain Grebner based form, was developed. It provides the user with universal tools for solving nonlinear algebraic systems .
A new computer algebra algorithm was designed and implemented in C for manipulation of finitely presented Lie algebra and superalgebra. The package is useful for solving problems of theoretical and mathematical physics dealing with string theories, quantum groups and integral analysis of nonlinear differential equations .
In the REDUCE system, has been developed a program for Birkhoff-Gustavson normalization of polynomial Hamiltonians . A new algorithmic method for computation of multi-loop Feynman diagrams with masses and external momenta was developed. The method was successfully applied to the following computational problems of great importance in theoretical high energy physics: three- loop QCD corrections to the electroweak r parameter; two-loop corrections to the same parameter with the massive Higgs boson and massive top-quark; momenta of the structure functions in two-loop approximation with massive particles .