The 8th Lunch Seminar
Date and time: Tuesday, March 31st, from 12:20.
Place: 理学部会議室 (Science Building 3, Room 207-209)
Speaker: Xavier Dahan
Some aspects of Polynomial systems solving by elimination
After systems of linear equations, polynomial systems are the simplest functions met in mathematics and have been extensively studied with respect to many aspects. As such they are easy to implement in a computer, but additionally are also able to approximate arbitrarily closely any “regular” functions. They play important roles in various applications, ranging from motion planning of robot “arms”, cryptography, modelling (Differential-Algebraic equations) and celestial mechanic.
The main drawback is the representation of these data which is growing exponentially with the number of parameters involved (variables) or with the degree of precision (degree). Moreover, processing a system of polynomials into a “useful form” (e.g. like a solving process) is more complicated than solving a system of linear equations because of the more complicated geometry. The “useful form” in question, whether it is a Groebner basis or another related kind of system, is very often much bigger than the initial polynomial system. The degree indeed can increase significantly, and along with the exponential growth aforementioned make the computations very heavy. This difficulty in processing efficiently polynomial systems has called for progresses, realized through techniques ranging from mathematics, computational complexity to data structures and modern high-performance computing. Some aspects (hopefully “recreational”) of to this realm of research will be presented during this lunch seminar.