Collaborative Research: A Software System for Algebraic Geometry Research

Grayson 这是 Daniel R.Grayson 和 Michael E. Stillman 开展的一个合作项目。 Macaulay 2 是一个免费的计算机代数系统，致力于对多变量多项式方程组进行定性研究。 研究人员继续开发Macaulay 2。他们升级现有算法，安装更多算法，开发和发布新算法，实施发现的新算法，添加符号数字算法支持，改进文档，并进行系统软件更改以帮助贡献者编写、测试和记录代码。 新算法可以产生影响的领域包括数值系统的研究、具有指定类型分母的分数、理想因式分解、变量乘法不满足交换律的系统、几何优化、基因表达水平随时间的观察分析以及生物信息学。 Macaulay 2 是支持涉及多变量多项式方程组的数学研究的基础设施的一部分。 衡量其影响力的一个指标是，至少有 70 篇研究论文引用了麦考利 2，至少有七位数学家贡献了代码，并且书籍和课程材料现在都在使用它。 Macaulay 2 的实验结果有助于提出和发展易于处理的猜想。 现在该软件已接近成熟，研究人员更加重视从研究界招募开发人员，以及与其他科学家在利用麦考利 2 功能的数学和跨学科项目上进行合作。 他们通过参观和在厄巴纳举办为期 5 天的研讨会，向研究生和博士后介绍计算机在数学研究中的使用。 多项式方程组的研究在纯数学和应用数学领域有着悠久而辉煌的历史，最近对密码学和机器人学等领域产生了影响。计算机能力的增强，加上 Macaulay 2 的进一步发展，使科学家能够极大地扩展其研究范围。

Grayson This is a collaborative project carreied out by Daniel R.Grayson and Michael E. Stillman. Macaulay 2 is a free computeralgebra system dedicated to the qualitative investigation ofsystems of polynomial equations in many variables. Theinvestigators continue development of Macaulay 2. They upgradeexisting algorithms, install more algorithms, develop and publishnew algorithms, implement new algorithms as they are discovered,add symbolic-numeric algorithm support, improve thedocumentation, and make systemic software changes that helpcontributors to write, test, and document code. Areas where newalgorithms can make an impact include the study of numericalsystems, fractions with specified types of denominators, idealfactorization, systems where the multiplication of the variablesdoesn't satisfy the commutative law, geometric optimization, theanalysis of observations of gene expression levels over time, andbioinformatics. Macaulay 2 is part of the infrastructure that supportsmathematical research involving systems of polynomial equationsin many variables. A measure of its impact is that at least 70research papers have cited Macaulay 2, at least sevenmathematicians have contributed code, and books and coursematerials are now using it. The experimental results found withMacaulay 2 are helping in the formulation and development oftractable conjectures. Now that the software is approachingmaturity, the investigators place more emphasis on recruitingdevelopers from the research community and on collaborations withother scientists on both mathematical and interdisciplinaryprojects that exploit Macaulay 2's capabilities. They introducegraduate students and postdocs to the use of computers inresearch mathematics through visits and a 5 day workshop inUrbana. The study of systems of polynomial equations has a longand distinguished history in pure and applied mathematics, withrecent impact on such fields as cryptography and robotics.Increasing computer power, combined with further development ofMacaulay 2, enables scientists to expand dramatically its scope.