Computational tropical geometry and its applications

热带几何是一个年轻的数学领域，它研究由多项式方程产生的组合对象。这些所谓的热带变种自然出现在许多数学领域和其他领域，如生物学中的遗传学，物理学中的天体力学和经济学中的拍卖理论。无论它们出现在哪里，热带品种往往允许新的计算方法来解决现有的问题。在英国，自金融危机以来，英格兰银行一直在使用热带几何来向英国金融体系分配资金。在法国，热带几何被用于优化移动的网络的负载平衡和紧急呼叫中心的性能分析。该研究项目旨在将热带几何建立为应用科学和工业中超出优化的计算问题的强大而多功能的工具。为此，我们追求具体的应用以及计算方法的改进。最后交付的是一个全面的开放源码软件系统的热带代数几何与强烈强调其广泛的应用。我们集中在三个主要问题，这是选择，以最大限度地提高影响和技术，他们所涵盖的范围。第一个问题围绕系统的多项式方程，这是无处不在的应用科学。它们描述了化学反应网络的稳定状态，机器人手臂的运动范围，或生化系统中配体的结合行为。在过去的二十年里，解决这类系统的最新技术一直是同伦延拓，它的工作原理是通过仔细地将一个容易启动的系统变形到目标系统，同时沿着沿着追踪所有的解决方案。虽然已经研究了将热带几何应用于同伦延拓的想法，但由于效率问题，所有过去的方法都失败了。然而，在过去的几年里，热带几何在算法上取得了重大突破，我们将在此基础上加以利用和发展。第二个问题涉及p进数，这是数论中不可或缺的一类领域。这不仅使它们对热带几何在数论中的应用很重要，而且还需要大量的数论工具专门用于它们。因此，很好地掌握p-adics数上的热带几何对于理论和实践都是必要的。也就是说，在计算上，p-adic数上的热带几何由于其独特的算法挑战而被忽视。我们试图纠正这种情况，并探讨热带几何的计算方面，特别是在p-adic数，促进了最近的趋势，在计算机algebra. Third问题涉及Gröbner基地，这在计算代数几何和相邻领域，如密码学有很长的历史。此外，在过去的十年中，代数几何技术在数学以外的领域出现了爆炸式的发展。因此，Gröbner基作为研究多项式系统的工具和感兴趣的对象本身都获得了牵引力，例如，代数统计中的马尔可夫基然而，Gröbner基是出了名的难以计算，这严重限制了它们在实际应用中的应用。一般来说，多项式未知数表示系数域的任意元素，并且Gröbner基计算中的所有运算都考虑这种模糊性。在实践中，人们通常只对特定的解感兴趣，例如严格正的真实的解。饱和Gröbner基算法是一种符号算法，它能够利用在许多应用中丰富的这种数值信息，并使用它来加速其性能。

Tropical geometry is a young area of mathematics which studies combinatorial objects arising from polynomial equations. These so-called tropical varieties arise naturally in many areas of mathematics and beyond, such as phylogenetics in biology, celestial mechanics in physics, and auction theory in economics. Wherever they arise, tropical varieties often allow new computational approaches to existing problems. In the UK, the Bank of England has been using tropical geometry since the financial crises to allocate money to the UK financial system. In France, tropical geometry is used for optimisation of load balancing of mobile networks, and performance analysis of emergency call centres.This research projects aims at establishing tropical geometry as a powerful and versatile tool for computational questions in applied sciences and industry beyond optimisation. To this end, we pursue concrete applications as well as improvements of computational methods. The final deliverable is a comprehensive open source software system for tropical algebraic geometry with strong emphasis on its wide spectrum of applications. We focus on three main problems, which were chosen to maximise the impact and the range of techniques that they encompass.The first problem revolves around systems of polynomial equations, which are ubiquitous in applied science. They describe the steady states of chemical reaction networks, the range of movement of a robot arm, or the binding behaviour of ligands in a biochemical system. For over two decades, the state of the art for solving such systems has been homotopy continuation, which works by carefully deforming an easy start system to the target system while tracing all solutions along the way.We seek to improve the existing capabilities, in particular for the type of polynomial systems which arise in the aforementioned applications. While ideas to apply tropical geometry to homotopy continuation have already been studied, all past approaches have failed due to questions of efficiency. However, the last couple of years have seen significant algorithmic breakthroughs in tropical geometry, which we will exploit and build upon.The second problem involves p-adic numbers, which are an indispensable class of fields for number theory. This not only makes them important for the applications of tropical geometry in number theory, but also entails a vast array of number theoretic tools available exclusively over them. Hence a good grasp on tropical geometry over p-adics numbers is an imperative for both theory and practice.That being said, computationally, tropical geometry over p-adic numbers has been neglected due to the unique algorithmic challenges they pose. We seek to remedy this situation and explore computational aspects of tropical geometry specifically over p-adic numbers, facilitated by recent trends in computer algebra.The third problem involves Gröbner bases, which have long history in computational algebraic geometry and adjacent fields such as cryptography. Furthermore, the past decade featured an explosion of algebro-geometric techniques in areas outside of mathematics. As such, Gröbner bases have gained traction both as tool for studying polynomial systems and as object of interest themselves, e.g., as Markov bases in algebraic statistics. However, Gröbner bases are notoriously hard to compute, which severely inhibits their use in practical applications.We will investigate so-called saturating Gröbner bases. In general, polynomial unknowns represent arbitrary elements of the coefficient field, and all operations within a Gröbner basis computation respect this ambiguity. In practice, one is often only interested in specific solutions, e.g. strictly positive real solutions. Saturating Gröbner basis algorithm are symbolic algorithms which are capable of exploiting this numerical information that is abundant in many applications and use it to speed up its performance.