Mathematical Sciences: The Pierce-Birkhoff Conjecture and Hilbert's 17th Problem

本文主要研究实代数几何中的两个问题，即1956年的皮尔斯-伯克霍夫猜想和德尔泽尔猜想。第一个证明了从实n维空间到实空间的每个连续分段多项式函数都是有限个多项式的下确界的逐点上确界。第二个将改进Delzell对Hilbert关于平方和的第17个问题的连续解。这项研究属于实代数几何的一般领域。这是一个对纯数学具有基本重要性的领域。此外，该领域与机器人学有接口，因此，将对理论机器人学、科学和工程产生积极影响。

This research will focus on two problems in real algebraic geometry, namely the Pierce-Birkhoff conjecture of 1956 and Delzell's conjecture. The first would show that every continuous piecewise-polynomial function from real n-space into the reals is a pointwise supremum of infima of finitely many polynomials. The second would improve Delzell's continuous solution of Hilbert's 17th problem on sums of squares. This research is in the general area of real algebraic geometry. This is an area which is of basic importance to pure mathematics. In addition, this field has an interface with robotics and consequently, will have a positive impact on theoretical robotics, science and engineering.