Polynomial ensembles are a sub-class of probability measures within determinantal point processes. Examples include products of independent random matrices, with applications to Lyapunov exponents, and random matrices with an external field, that may serve as schematic models of quantum field theories with temperature. We first analyse expectation values of ratios of an equal number of characteristic polynomials in general polynomial ensembles. Using Schur polynomials, we show that polynomial ensembles constitute Giambelli compatible point processes, leading to a determinant formula for such ratios as in classical ensembles of random matrices. In the second part, we introduce invertible polynomial ensembles given, e.g. by random matrices with an external field. Expectation values of arbitrary ratios of characteristic polynomials are expressed in terms of multiple contour integrals. This generalises previous findings by Fyodorov, Grela, and Strahov. for a single ratio in the context of eigenvector statistics in the complex Ginibre ensemble.

多项式合奏是确定点过程中概率测度的子类。例子包括应用到Lyapunov指数的独立随机矩阵的乘积，以及具有外部场的随机矩阵，它们可以用作温度场的量子场理论模型。我们首先分析一般多项式集合中相等数量的特征多项式的比率的期望值。使用Schur多项式，我们证明了多项式合奏构成了Giambelli兼容点过程，从而为诸如随机矩阵的经典合奏之类的比率得出了一个行列式。在第二部分中，我们介绍给定的可逆多项式集合，例如由具有外部场的随机矩阵给出。特征多项式的任意比率的期望值用多个轮廓积分表示。这归纳了Fyodorov，Grela和Strahov的先前发现。在复杂的Ginibre集合中，在特征向量统计的背景下获得单个比率。