论文标题
在组理论背景下的多项式系统的真实根源上
On real roots of polynomial systems of equations in the context of group theory
论文作者
论文摘要
随机实际多项式增加程度的零的可能性趋于零。但是，从多项式传递到laurent多项式产生了一个令人惊讶的结果：根为真实的概率倾向于零，而是$ 1/\ sqrt {3} $。对于几个变量中的劳伦（Lourent）多项式系统，也观察到了类似的现象。通过将劳伦（Laurent）多项式视为与圆环表示相关的函数，我们描述了用于任何还原线性群体表示的类似现象。在简单组的情况下，我们为上述限制概率提供了一个公式。
The probability that a zero of a random real polynomial of increasing degree is real tends to zero. However, passing from polynomials to Laurent polynomials yields a surprising result: the probability that a root is real tends not to zero, but to $1/\sqrt{3}$. A similar phenomenon has also been observed for systems of Laurent polynomials in several variables. By considering Laurent polynomials as functions associated with torus representations, we describe an analogous phenomenon for representations of any reductive linear group. In the case of a simple group, we provide a formula for the aforementioned limiting probability.
