多梯度麦考利对偶空间

Pub Date : 2024-04-05 DOI:10.1142/s0219498825502469
Joseph Cummings, Jonathan D. Hauenstein
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引用次数: 0

摘要

我们描述了一种计算多等级理想的麦考莱对偶空间的算法。对于同阶理想,自然分级是由麦考莱对偶空间继承的,而麦考莱对偶空间已被用于开发计算每个同阶的麦考莱对偶空间的算法。我们的主要理论成果将这一想法扩展到了从多等级理想继承的多等级麦考利对偶空间。这种天然的对偶性使得理想运算可以从同阶理想转化为多阶麦考利对偶空间上的相应运算。我们特别描述了一个具有右逆的线性算子,用于计算多等级多项式的商。通过对麦考莱对偶空间的同质分量进行总排序,我们还描述了如何递归地为每个分量构建一个基础。我们还列举了几个例子来演示这种新方法。
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Multi-graded Macaulay dual spaces

We describe an algorithm for computing Macaulay dual spaces for multi-graded ideals. For homogeneous ideals, the natural grading is inherited by the Macaulay dual space which has been leveraged to develop algorithms to compute the Macaulay dual space in each homogeneous degree. Our main theoretical result extends this idea to multi-graded Macaulay dual spaces inherited from multi-graded ideals. This natural duality allows ideal operations to be translated from homogeneous ideals to their corresponding operations on the multi-graded Macaulay dual spaces. In particular, we describe a linear operator with a right inverse for computing quotients by a multi-graded polynomial. By using a total ordering on the homogeneous components of the Macaulay dual space, we also describe how to recursively construct a basis for each component. Several examples are included to demonstrate this new approach.

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