AF: Algorithms and software for a new polyhedral polynomial system solver

多面体的方法来解决多项式系统利用稀疏结构，并产生混合符号-数值算法计算的Puiironx级数展开的解决方案。该系列的主导权力是向性和系统的精确牛顿多面体定义，而该系列的系数是初始形式的系统，可能有有限的精度系数的解决方案。新算法将在免费的开源软件系统PHCpack中实现，并与Macaulay 2和Sage等计算机代数系统（将PHCpack库链接为Python模块）接口。由于几乎所有的现代计算机都有多核并行计算，我们设计我们的算法的加速比和质量，使用多核来补偿多精度算术的成本。求解多项式系统是数学中的一个基本问题，在科学和工程的各个领域都有应用。利用稀疏结构的求解器更快地提供更好的解决方案。拟议研究产生的算法和软件将通过Macaulay 2的用户和开发人员社区传播，Macaulay 2是一个用于代数几何和交换代数研究和教学的计算机代数系统。在并行计算机上的实现将使大型计算问题的有效解决方案成为可能。

Polyhedral methods to solve polynomial systems exploit the sparsity structure and give rise to hybrid symbolic-numeric algorithms for the computation of Puiseux series expansions of the solutions. The leading powers of the series are tropisms and are defined by the exact Newton polytopes of the system, while the coefficients of the series are solutions of initial form systems which may have coefficients of limited accuracy. The new algorithms will be implemented in the free open source software system PHCpack, with interfaces to computer algebra systems such as Macaulay 2 and Sage (linking the PHCpack library as a Python module). As almost all modern computers have multiple cores enabling to compute in parallel, we design our algorithms for speedup and quality, using the multiple cores to compensate for the cost of multiprecision arithmetic.Solving polynomial systems is a fundamental problem in mathematics with applications to various fields of science and engineering. Solvers that exploit the sparse structure provide better solutions faster. Algorithms and software produced by the proposed research will be disseminated through the users and developers community of Macaulay 2, a computer algebra system for research and teaching in algebraic geometry and commutative algebra. Implementations on parallel computers will enable the efficient solution of large computational problems.