微分-代数方程组中消元的界
Bounds for elimination of unknowns in systems of differential-algebraic equations
摘要：
从高斯消去法开始，方程组中未知数的消去是一个普遍感兴趣的问题。在微分-代数方程（DAE）系统中，寻找消除未知数的微分数的先验上界是一个重要的挑战，可以追溯到Ritt（1932）。通过渐近分析的第一个特征是由于Grigoriev（1989）关于微分域中量词消除的结果，但挑战仍然存在。在本文中，我们提出了一个新的界限，这是一个重大的改进，以前已知的结果。我们还提出了一个新的下界，这表明我们的上界在低维的渐近紧性，这是经常出现的应用。最后，我们讨论了应用我们的结果来设计新的算法消除未知数的DAE系统。
英文摘要：
Elimination of unknowns in systems of equations, starting with Gaussian elimination, is a problem of general interest. The problem of finding an a priori upper bound for the number of differentiations in elimination of unknowns in a system of differential-algebraic equations (DAEs) is an important challenge, going back to Ritt (1932). The first characterization of this via an asymptotic analysis is due to Grigoriev's result (1989) on quantifier elimination in differential fields, but the challenge still remained. In this paper, we present a new bound, which is a major improvement over the previously known results. We also present a new lower bound, which shows asymptotic tightness of our upper bound in low dimensions, which are frequently occurring in applications. Finally, we discuss applications of our results to designing new algorithms for elimination of unknowns in systems of DAEs.
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。
