关于由一般线性形式生成的理想的希尔伯特级数
On the Hilbert series of ideals generated by general linear forms
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中文总结 AI 辅助
该研究确定了特定一般线性形式幂生成理想的希尔伯特级数，证明了Iarrobino-Fröberg猜想对足够大n不成立，还给出相关猜想的反例并确定了外代数中特定理想的希尔伯特级数。
中文摘要 AI 辅助
我们确定了n个变量中n+2个一般线性形式的d次幂生成的理想的希尔伯特级数，以此为k>2时n+k个一般线性形式的d次幂生成的理想的希尔伯特级数的次数提供上界。这使我们能够证明，对于足够大的所有n，Iarrobino-Fröberg猜想不成立。我们还针对部分n值，确定了n+3个一般线性形式的d次幂生成的理想的希尔伯特级数的次数，并对“由足够大幂次的一般线性形式生成的理想的弱莱夫谢茨性质失效”这一猜想给出了反例。此外，我们确定了偶数个生成元的外代数中两个 generic 二次形式生成的理想的希尔伯特级数。
英文摘要
We determine the Hilbert series of ideals generated by $d$'th powers of $n+2$ general linear forms in $n$ variables, to give upper bounds on the degree of the Hilbert series of ideals generated by $d$'th powers of $n+k$ general linear forms for $k>2$. This allows us to show that the Iarrobino-Fröberg Conjecture fails for all $n$ large enough. We also determine the degree of the Hilbert series for the ideal generated by $d$'th powers of $n+3$ general linear forms, for some values of $n$, and give counterexamples to a conjecture on the failure of the Weak Lefschetz Property for ideals generated by sufficiently large powers of general linear forms. Moreover, we determine the Hilbert series of the ideal generated by two generic quadratic forms in the exterior algebra on an even number of generators.
