Reality, exactness, and computation in numerical algebraic geometry

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。数值代数几何的方法将代数几何的范围扩展到现有的符号方法不太适合的问题，例如，由于变量的数量或系数的不精确。这些方法的价值正在不断得到认可。例如，代数几何软件包Macaulay 2和CoCoA都在积极开发新的同伦模块或与现有数值软件（如Bertini和PHCpack）的接口。尽管这些数值方法有好处（例如，并行性），但也有一些缺点。例如，要用同伦方法求多项式系统的实孤立解，必须先求出所有的复孤立解，然后选出虚部小于预先选定容限的复孤立解。此外，数值代数几何的一个主要好处是，它很容易在代数集合的任何给定的不可约分量上产生许多一般点的近似值。然而，目前没有办法恢复组件的精确定义方程。这个项目有两个方向。其一，一套基于Gale对偶和khovanski - rolle定理的新技术将被开发出来，用于只求多项式系统的实解。另一方面，将利用数值方法在代数集上查找泛型点的简单性。后一个方向将包括通过晶格基约简技术（如LLL或PSLQ）恢复精确定义方程的工作。这两个方向都有望产生新的、免费的软件。多项式方程组在数学、科学和工程中无处不在。一个完整的数学领域——代数几何——就是从寻找这类方程的解的需要中发展出来的。然而，直到20世纪60年代，还没有已知的解决这类方程组的通用技术。然而，在那时开发的方法需要太多的内存，除了相对较小的问题之外，无法有效。最近发展的方法- Sommese， Verschelde和Wampler的数值方法，现在统称为数值代数几何-允许解决更大的多项式系统，将代数几何方法应用于更广泛的问题。然而，这些数值方法仍然存在缺陷。这个项目的目标包括解决其中两个缺点。特别是，PI将致力于开发有效的方法，只找到那些在实际应用中感兴趣的解（即，真正的解而不是复杂的解），并从这些强大的新数值方法的输出提供的近似数据中恢复有价值的精确数据。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The methods of numerical algebraic geometry extend the reach of algebraic geometry to problems for which existing symbolic methods are not well suited, e.g., due to the number of variables or the inexactness of the coefficients. The value of these methods is continuing to gain recognition. For example, the algebraic geometry software packages Macaulay 2 and CoCoA are both actively developing either new homotopy modules or interfaces to existing numerical software, such as Bertini and PHCpack. Despite the benefits of these numerical methods (e.g., parallelizability), there are a few drawbacks. For example, to find the real isolated solutions of a polynomial system using homotopy methods, one must first produce all complex isolated solutions and then sort out those with imaginary part below a pre-chosen tolerance. Also, one major benefit coming from numerical algebraic geometry is that it is simple to produce approximations of many generic points on any given irreducible component of an algebraic set. However, there is currently no way to recover exact defining equations for the component. This project has two directions. In one, a new set of techniques, based on Gale duality and the Khovanskii-Rolle theorem, for finding only the real solutions of polynomial systems will be developed. In the other, the simplicity of finding generic points on algebraic sets via numerical methods will be exploited. The latter direction will include work on recovering exact defining equations via lattice basis reduction techniques such as LLL or PSLQ. Both directions are expected to result in new, freely available software.Polynomial systems of equations are ubiquitous throughout mathematics, science, and engineering. An entire mathematical field - algebraic geometry - grew out of the need to find solutions to these sorts of equations. Until the 1960s, though, there was no known general technique for solving such systems of equations. However, the methods developed at that point require too much memory to be effective except for relatively small problems. More recently developed methods - the numerical methods of Sommese, Verschelde, and Wampler, now collectively known as numerical algebraic geometry - allow for the solution of much larger polynomial systems, opening the application of algebraic geometry methods to a wider class of problems. However, there are still drawbacks to these numerical methods. The goals of this project include addressing two of these drawbacks. In particular, the PI will work on developing efficient methods to find only those solutions that are of interest in real-world applications (i.e., real solutions rather than complex solutions) and on recovering valuable exact data from the approximate data that is provided as the output of these powerful new numerical methods.