SI2-SSE: Collaborative Proposal: Symbolic-Numeric Approaches to Polynomials

多项式系统在人类奋进的许多领域中自然出现。这些包括肿瘤生长的建模;机器人设备的设计;从燃烧到血液凝固等领域产生的化学系统;物理学中的各种问题;加上许多数学领域。多项式系统的解决方案回答了这些努力的关键问题。本研究将致力于开发下一代的Bertini，一个开放源码的软件包，它已被许多研究人员成功地使用在许多问题，其中包括所有这些上面提到的Bertini将重写C++的脚本和模块化，这将允许它与符号软件接口透明。新的Bertini将包括允许用户根据Bertini的输出构建和操作同伦的工具。研究的一个主要重点将是微分方程系统的离散化所产生的多项式系统。这些非常大的多项式系统的巨大挑战是由新的高效和鲁棒的解决方案的巨大潜在影响所平衡的。

Polynomial systems arise naturally in many areas of human endeavor. These include the modeling of tumor growth; the design of robotic devices; chemical systems arising in areas ranging from combustion to blood clotting; assorted problems in physics; plus many areas with mathematics. The solution of the polynomial systems answers questions critical to these endeavors. This research will be devoted to developing the next generation of Bertini, an open source software package, which has been used successfully by many researchers on many problems, which include all those mentioned above.Bertini will be rewritten in C++ to be scriptable and modular, which will allow it to be interfaced transparently with symbolic software. The new Bertini will include tools allowing the user to construct and manipulate homotopies based on the output of Bertini. A major focus of the research will be given to systems of polynomials arising from the discretization of systems of differential equations. The great challenge of these very large systems of polynomials is balanced by the great potential impact new efficient and robust methods of solution will have.