微分代数中的上界算法
Algorithms yield upper bounds in differential algebra
摘要：
考虑一个算法在一个有几个可交换导子的微分域中进行计算，使得它对域的元素执行的唯一操作是算术运算、微分和零测试。我们表明，如果该算法是保证终止于每个输入，那么有一个可计算的上界的大小输入的大小方面的算法的输出。我们还将其推广到使用足够好的理论模型的算法（例如，包括差场）。然后，我们将其应用到微分代数几何表明，存在一个可计算的统一的上界的数量的组件的任何品种所定义的系统的多项式偏微分方程。然后，我们使用这个界显示存在一个可计算的一致上界的多项式偏微分方程系统中的消除问题的延迟。
英文摘要：
Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that, if the algorithm is guaranteed to terminate on every input, then there is a computable upper bound for the size of the output of the algorithm in terms of the size of the input. We also generalize this to algorithms working with models of good enough theories (including for example, difference fields). We then apply this to differential algebraic geometry to show that there exists a computable uniform upper bound for the number of components of any variety defined by a system of polynomial PDEs. We then use this bound to show the existence of a computable uniform upper bound for the elimination problem in systems of polynomial PDEs with delays.
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