Calcul exact ou certifié avec des systèmes algébriques – EXACTA

Algebraic systems are fundamental mathematical objects used in the formulation, modeling, and investigation of scientific, engineering, and industrial problems, of complex information, biological, and social systems, and of natural, financial, and economic phenomena. Computational studies of algebraic systems are foundational research in information science that requires sophisticated mathematical theories and methods and that has numerous applications in other domains. The main objective of this project is to study and compute the solutions of nonlinear algebraic systems and their structures and properties with selected target applications using exact or certified computation. The project consists of one main task of basic research on the design and implementation of fundamental algorithms and four tasks of applied research on computational geometry, algebraic cryptanalysis, global optimization, and algebraic biology. The foundational research will be based on the classical methods of Gröbner bases, characteristic sets, and cylindrical algebraic decomposition introduced by three distinguished scientists, B. Buchberger, W.-T. Wu, and G. E. Collins, as well as many subsequent and significant developments and extensions made by prominent researchers, including some of the participants of this project, in the last two decades. The four domains of applied research have been chosen in view of their scientific and technological importance and the expertise and ongoing work of the project participants therein. The focus of investigation is placed on the design and analysis of fundamental algorithms for basic operations on polynomials and advanced computations with algebraic systems, efficient implementations of fundamental algorithms and software tools, and applications of the algorithms and tools to challenging problems from the four selected domains. The methodology of using exact or certified computation will take and combine the advantages of symbolic computation in producing rigorous results and numeric computation with high speed. The project is proposed for three years with 300 person-months of workforce. Its consortium is composed of increasingly strong research teams from France and China in the area of solving algebraic systems with applications. It is expected that this project will strengthen the already close cooperation between French and Chinese specialists for productive research and development on exact and certified computation, result in substantial advances on the design and implementation of fundamental algorithms and software tools for studying algebraic systems, expand the applications of such algorithms and tools in the four application domains, and produce a number of joint publications and software packages.