We prove that, analogous to the Hilbert–Kunz density function, (used for studying the Hilbert–Kunz multiplicity, the leading coefficient of the Hibert–Kunz function), there exists a \(\beta \)-density function \(g_{R, \mathbf{m}}:[0,\infty )\longrightarrow {\mathbb {R}}\), where \((R, \mathbf{m})\) is the homogeneous coordinate ring associated with the toric pair (X, D), such that$$\begin{aligned} \int _0^{\infty }g_{R, \mathbf{m}}(x)\mathrm{d}x = \beta (R, \mathbf{m}), \end{aligned}$$where \(\beta (R, \mathbf{m})\) is the second coefficient of the Hilbert–Kunz function for \((R, \mathbf{m})\), as constructed by Huneke–McDermott–Monsky. Moreover, we prove, (1) the function \(g_{R, \mathbf{m}}:[0, \infty )\longrightarrow {\mathbb {R}}\) is compactly supported and is continuous except at finitely many points, (2) the function \(g_{R, \mathbf{m}}\) is multiplicative for the Segre products with the expression involving the first two coefficients of the Hilbert polynomials of the rings involved. Here we also prove and use a result (which is a refined version of a result by Henk–Linke) on the boundedness of the coefficients of rational Ehrhart quasi-polynomials of convex rational polytopes.

中文翻译：

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射影复曲面变种的希尔伯特-昆兹函数第二个系数的密度函数

我们证明，类似于Hilbert-Kunz密度函数（用于研究Hilbert-Kunz多重性，Hibert-Kunz函数的前导系数），存在一个\（\ beta \）-密度函数\（g_ { R，\ mathbf {m}}：[0，\ infty）\ longrightarrow {\ mathbb {R}} \），其中\（（R，\ mathbf {m}）\）是与复曲面相关的齐次坐标环对（X，  D），这样$$ \ begin {aligned} \ int _0 ^ {\ infty} g_ {R，\ mathbf {m}}（x）\ mathrm {d} x = \ beta（R，\ mathbf {m}），\ end {aligned} $$其中\（\ beta（R，\ mathbf {m}）\）是\（（R，\ mathbf {m}的Hilbert–Kunz函数的第二个系数）\）由Huneke–McDermott–Monsky构建。此外，我们证明，（1）函数\（g_ {R，\ mathbf {m}}：[0，\ infty）\ longrightarrow {\ mathbb {R}} \）被紧凑地支持并且是连续的，除了有限次数点，（2）函数\（g_ {R，\ mathbf {m}} \）对于Segre乘积是可乘的，其表达式涉及所涉及环的希尔伯特多项式的前两个系数。在这里，我们还证明并使用了一个结果（这是Henk-Linke结果的精简版本），该结果是凸有理多边形的有理Ehrhart拟多项式系数的有界性。