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.

中文翻译：

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微分代数方程组中未知数的消去边界

消除方程组中的未知数，从高斯消除开始，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量的先验上限的问题是一个重要的挑战，这可以追溯到 Ritt (1932)。通过渐近分析对此进行的第一个表征是由于 Grigoriev 的结果（1989）关于微分场中的量词消除，但挑战仍然存在。在本文中，我们提出了一个新的界限，这是对先前已知结果的重大改进。我们还提出了一个新的下界，它显示了我们的上界在低维中的渐近紧密性，这在应用程序中经常出现。最后，