Final Report Summary - A3 (Algebraic Algorithms and Applications)
Traditional techniques in algorithms usually treat/compute linear objects and quantities. In recent years, there have been efforts to extend the range of our techniques using tools from (real) algebraic geometry, which pertain to basic questions about the real roots of equations, to handle non-linear objects. A3 addressed the challenge to provide solid mathematical and algorithmic foundations, and efficient implementations for computations with curved objects.
The overall objectives of the project span the following main axes.
• Algebraic algorithms
Improved algorithms for solving univariate polynomials.
• Non-linear computational geometry
Computations with non-linear geometric objects using algebraic tools.
• Game theory
Applications of algebraic algorithms and techniques to problems in game theory.
• Effective implementations
Software libraries for computations with real algebraic numbers and solving of polynomials. There has been progress in all main axes, which is detailed in the next section (Work progress and achievements during the period)
Our work on the foundations of algebraic algorithms resulted optimal, up to poly-logarithmic factors, algorithms for approximating the real and complex roots of univariate polynomials, improving the previously known bounds by a factor. In addition, we presented new zero-bounds for the roots of polynomial systems that lead to novel effective bounds for polynomial optimization problems.
From the application point of view, we considered problems in non-linear computational geometry and game theory. We extended the limits of the state-of-the-art in non-linear computational geometry by proposing the first complete and exact algorithm for computing the Voronoi diagram of ellipses in the plane. In the game-theoretic field we presented exact bounds for the optimal strategies of matrix games based on zero-bounds.
Finally, we developed an open source, complete, robust and efficient software package for isolating and refining the real roots of univariate polynomials able to compute the roots of univariate polynomials having degree 1 000 and coefficients of 10 000 bits in about 30 seconds.
Currently, the fellow of the project has obtained a permanent researcher position at Inria, working in the research team POLSYS.
The overall objectives of the project span the following main axes.
• Algebraic algorithms
Improved algorithms for solving univariate polynomials.
• Non-linear computational geometry
Computations with non-linear geometric objects using algebraic tools.
• Game theory
Applications of algebraic algorithms and techniques to problems in game theory.
• Effective implementations
Software libraries for computations with real algebraic numbers and solving of polynomials. There has been progress in all main axes, which is detailed in the next section (Work progress and achievements during the period)
Our work on the foundations of algebraic algorithms resulted optimal, up to poly-logarithmic factors, algorithms for approximating the real and complex roots of univariate polynomials, improving the previously known bounds by a factor. In addition, we presented new zero-bounds for the roots of polynomial systems that lead to novel effective bounds for polynomial optimization problems.
From the application point of view, we considered problems in non-linear computational geometry and game theory. We extended the limits of the state-of-the-art in non-linear computational geometry by proposing the first complete and exact algorithm for computing the Voronoi diagram of ellipses in the plane. In the game-theoretic field we presented exact bounds for the optimal strategies of matrix games based on zero-bounds.
Finally, we developed an open source, complete, robust and efficient software package for isolating and refining the real roots of univariate polynomials able to compute the roots of univariate polynomials having degree 1 000 and coefficients of 10 000 bits in about 30 seconds.
Currently, the fellow of the project has obtained a permanent researcher position at Inria, working in the research team POLSYS.