加法复杂性的直接超度量方法与Shub-Smale Tau猜想
A Direct Ultrametric Approach to Additive Complexity and the Shub-Smale Tau Conjecture
摘要：
Shub-Smale Tau猜想（英语：Shub-Smale Tau Conjecture）是一个关于多项式f在一个变量中的整根个数和f的直线程序（Straight-Line Program，SLP）复杂度的假设。这个猜想的真实性的一个推论是，对于复数上的Blum-Shub-Smale模型，P不同于NP。我们证明了两个弱版本的Tau猜想，并在这样做表明，Tau猜想遵循一个更合理的假设。我们的研究结果如下从一个新的p-adic模拟早期的工作有关真正的代数几何添加剂的复杂性。例如，我们可以证明一个非零的一元多项式的加法复杂度s不能超过15+s^3（s+1）（7.5）^s！=O（e^{s \log s}）在2-adic有理数Q_2中的根，从而极大地改进了作者早先的一个结果.这直接意味着普通有理根的个数有相同的界，而通过实代数几何的早期技术得到的最佳前上界是Ω（（22.6）^{s^2}）中的一个量。本文介绍了作者的程序中的另一个步骤，即建立一个算法的算术形式的多项式理论。
英文摘要：
The Shub-Smale Tau Conjecture is a hypothesis relating the number of integral roots of a polynomial f in one variable and the Straight-Line Program (SLP) complexity of f. A consequence of the truth of this conjecture is that, for the Blum-Shub-Smale model over the complex numbers, P differs from NP. We prove two weak versions of the Tau Conjecture and in so doing show that the Tau Conjecture follows from an even more plausible hypothesis. Our results follow from a new p-adic analogue of earlier work relating real algebraic geometry to additive complexity. For instance, we can show that a nonzero univariate polynomial of additive complexity s can have no more than 15+s^3(s+1)(7.5)^s s! =O(e^{s\log s}) roots in the 2-adic rational numbers Q_2, thus dramatically improving an earlier result of the author. This immediately implies the same bound on the number of ordinary rational roots, whereas the best previous upper bound via earlier techniques from real algebraic geometry was a quantity in Omega((22.6)^{s^2}). This paper presents another step in the author's program of establishing an algorithmic arithmetic version of fewnomial theory.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
