US Participation in Workshop: Enumeration and Bounds in Real Algebraic Geometry

主要研究人员：Sottile，Frank J.Proposal Numbers：DMS-0800253机构：德克萨斯A&M研究基金会标题：实代数几何中的计数和界限研讨会关于多项式方程组的实解的先验信息对于数学应用非常重要，数学应用通常只需要实解。关于这个问题的当前知识状况，特别是计算或限制解决方案的数量，将是2008年4月21日至4月25日在瑞士洛桑EPFL(瑞士联邦理工学院)的Bernoulli中心举行的研讨会的焦点。本课程包含在伯努利中心长达一学期的实数和热带代数几何课程中。这个项目将资助来自美国的科学家参加这个工作坊。众所周知，如果不首先通过找到所有复杂的解来完全解决方程，就很难提取关于方程的真实解的任何信息。最近的突破导致了对一些结构多项式方程组的实解的数量的上界和下界的新的见解。也有只计算实数解的新算法。与此同时，人们对一般情况下的行为有了更好的理解，其他科学家也开始应用这些结果。本次研讨会规模较小(约20人参加)，非常注重了解多项式方程的真实解这一重要问题的当前发展和未来前景。

ABSTRACTPrincipal Investigator: Sottile, Frank J.Proposal Number: DMS - 0800253 Institution: Texas A&M Research Foundation Title: Workshop on Enumeration and Bounds in Real Algebraic GeometryA priori information about the real solutions to a system of polynomial equations is important for the applications of mathematics, which typically only need the real solutions. The current state of knowledge about this problem, particularly counting or bounding the numbers of solutions, will be the focus of a workshop to be held at the Bernoulli Center in the EPFL (Swiss Federal Technological Institute) in Lausanne, Switzerland the week of April 21 to April 25 in 2008. This is embedded within a semester long program at the Bernoulli Center on real and tropical algebraic geometry. This project will fund the participation of scientists from the United States at this workshop.It is notoriously difficult to extract any information about real solutions to equations without first solving them completely by finding all complex solutions. Recent breakthroughs have led to new insights about bounds, both upper and lower for the number of real solutions to some structured systems of polynomial equations. There are also new algorithms for only computing the real solutions. At the same time, a better understanding has arisen about average-case behavior, and other scientists have begun to apply these results. This workshop will be small (about 20 participants) and very focused on current developments and future prospects for this important problem of understanding the real solutions to polynomial equations.