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

本文主要研究真实的代数中的两个问题 1956年的Pierce-Birkhoff猜想， Delzell猜想 第一个例子表明， 从真实的n-空间到实数的分段多项式函数是 多项式下确界的点态上确界 的 第二个是改进希尔伯特的Delzell连续解 第17题关于平方和。 本文的研究属于真实的代数的一般性研究 几何 这是一个对纯粹的人来说至关重要的领域。 数学 此外，此字段还具有与 因此，机器人技术将对 理论机器人科学和工程

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.