Hilbert Series of Generic Ideals in Products of Projective Spaces

IF 0.7 4区数学 Q2 MATHEMATICS Experimental Mathematics Pub Date : 2022-10-02 DOI:10.1080/10586458.2021.1925999

R. Fröberg

引用次数: 0

Abstract

Abstract If , k a field, is a standard graded algebra, then the Hilbert series of R is the formal power series . It is known already since Macaulay which power series are Hilbert series of graded algebras. A much harder question is which series are Hilbert series if we fix the number of generators of I and their degrees, say for ideals , . In some sense “most” ideals with fixed degrees of their generators have the same Hilbert series. There is a conjecture for the Hilbert series of those “generic” ideals, see below. In this article we make a conjecture, and prove it in some cases, in the case of generic ideals of fixed degrees in the coordinate ring of , which might be easier to prove.

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射影空间积中一般理想的Hilbert级数

如果域k是一个标准的分级代数，则R的Hilbert级数就是形式幂级数。自麦考利以来，人们已经知道哪些幂级数是分级代数的希尔伯特级数。一个更难的问题是哪个级数是希尔伯特级数如果我们确定I的生成函数的个数和它们的度数，比如说理想值。在某种意义上，“大多数”理想具有固定度的发生器具有相同的希尔伯特级数。对于那些“一般”理想的希尔伯特级数有一个猜想，见下文。本文提出了一个猜想，并在某些情况下证明了它，在定度的一般理想的情况下，这可能更容易证明。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Experimental Mathematics 数学-数学

CiteScore

1.70

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Experimental Mathematics publishes original papers featuring formal results inspired by experimentation, conjectures suggested by experiments, and data supporting significant hypotheses. Experiment has always been, and increasingly is, an important method of mathematical discovery. (Gauss declared that his way of arriving at mathematical truths was "through systematic experimentation.") Yet this tends to be concealed by the tradition of presenting only elegant, fully developed, and rigorous results. Experimental Mathematics was founded in the belief that theory and experiment feed on each other, and that the mathematical community stands to benefit from a more complete exposure to the experimental process. The early sharing of insights increases the possibility that they will lead to theorems: An interesting conjecture is often formulated by a researcher who lacks the techniques to formalize a proof, while those who have the techniques at their fingertips have been looking elsewhere. Even when the person who had the initial insight goes on to find a proof, a discussion of the heuristic process can be of help, or at least of interest, to other researchers. There is value not only in the discovery itself, but also in the road that leads to it.

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