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

多项式系统自然地出现在人类努力的许多领域。这些包括肿瘤生长的建模；机器人装置的设计；从燃烧到凝血等领域产生的化学系统；物理学中的各种问题；加上数学的许多领域。多项式系统的解回答了对这些努力至关重要的问题。这项研究将致力于开发下一代Bertini，这是一个开源软件包，它已经被许多研究人员成功地用于解决许多问题，包括上面提到的所有问题。Bertini将用c++重写为可脚本化和模块化，这将允许它透明地与符号软件接口。新的Bertini将包括允许用户基于Bertini的输出构造和操作同伦的工具。研究的主要焦点将放在微分方程组离散化所产生的多项式系统上。这些非常大的多项式系统的巨大挑战被新的有效和健壮的解决方法将产生的巨大潜在影响所平衡。

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.