Abstract

We conjecture a formula supported by computations for the valuation of Kac polynomials of a quiver, which only depends on the number of loops at each vertex. We prove a convergence property of renormalized Kac polynomials of quivers when increasing the number of arrows: they converge in the ring of power series, with a linear rate of convergence. Then, we propose a conjecture concerning the global behavior of the coefficients of Kac polynomials. All computations were made using SageMath.

中文翻译：

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Kac 多项式的渐近行为

摘要

我们推测一个由计算支持的公式来评估箭袋的 Kac 多项式，该公式仅取决于每个顶点的循环数。我们证明了箭袋重正化 Kac 多项式在增加箭头数量时的收敛性：它们以线性收敛速率收敛于幂级数环中。然后，我们提出了关于 Kac 多项式系数的全局行为的猜想。所有计算均使用 SageMath 进行。