abstract:

The roots of random polynomials and of random polynomial systems have been extensively studied in the past. While the picture is quite complete in the case of polynomials, the case of systems is much harder. Kostlan, Shub, and Smale random polynomial systems were introduced in the nineties and the expectation of the number of their real roots is known since then. Only recently the asymptotic variance, as the degree goes to infinity, was obtained. In the present paper we obtain a Central Limit Theorem for the number of real roots of a square Kostlan-Shub-Smale system of any size as the degree goes to infinity.

中文翻译：

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Kostlan Shub Smale 随机多项式系统实根数的中心极限定理

摘要：

过去对随机多项式和随机多项式系统的根进行了广泛的研究。虽然多项式的情况非常完整，但系统的情况要困难得多。Kostlan、Shub 和 Smale 随机多项式系统是在 90 年代引入的，从那时起就知道它们的实根数的期望。直到最近，随着度数趋于无穷大，才获得了渐近方差。在本文中，我们获得了任何大小的平方 Kostlan-Shub-Smale 系统的实根数的中心极限定理，因为次数趋于无穷大。