论文标题
KAC-RICE公式和多项式方程参数化系统的解决方案数量
Kac-Rice formulas and the number of solutions of parametrized systems of polynomial equations
论文作者
论文摘要
KAC-RICE公式表达了预期的元素数量，而随机场的光纤具有多元积分。我们在这里考虑多项式方程的参数化系统，该系统在足够的参数中是线性的，并在参数遵循连续分布时为系统的预期解决方案提供了KAC-RICE公式。结合蒙特卡洛集成，我们将公式根据解决方案数量进行分区或在系统中具有最大溶液数量的参数空间中的区域进行分配。动机源于对化学反应网络的稳态的研究，并为确定网络至少具有两个阳性稳态的参数区域提供了新的工具。我们用大量示例说明了我们的方法成功处理大量参数比精确方法。
Kac-Rice formulas express the expected number of elements a fiber of a random field has in terms of a multivariate integral. We consider here parametrized systems of polynomial equations that are linear in enough parameters, and provide a Kac-Rice formula for the expected number of solutions of the system when the parameters follow continuous distributions. Combined with Monte Carlo integration, we apply the formula to partition the parameter region according to the number of solutions or find a region in parameter space where the system has the maximal number of solutions. The motivation stems from the study of steady states of chemical reaction networks and gives new tools for the open problem of identifying the parameter region where the network has at least two positive steady states. We illustrate with numerous examples that our approach successfully handles a larger number of parameters than exact methods.
