On Hilbertian ideals

Abhyankar, Shreeram ; Kulkarni, Devadatta M. (1989) On Hilbertian ideals Linear Algebra and its Applications, 116 . pp. 53-79. ISSN 0024-3795

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/002437...

Related URL: http://dx.doi.org/10.1016/0024-3795(89)90398-4


Abhyankar defined the index of a monomial in a matrix of indeterminates X to be the maximal size of any minor of X whose principal diagonal divides the given monomial. Using this concept, he characterized a free basis for general type of determinantal ideals formed by the minors coming from a saturated subset of X. In this paper, to a monomial in X of index p we associate a combinatorial object called a superskeleton of latitude p, which can loosely be described as a p-tuple of "almost nonintersecting paths" in a rectangular lattice of points. Using this map, we prove that the ideal generated by the p by p minors of a saturated set in X is hilbertian, i.e., the Hilbert polynomial of this ideal coincides with its Hilbert function for all nonnegative integers.

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