2026-09-19T13:33:03Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/1395762022-01-15T03:49:37Zcom_1721.1_7582com_1721.1_7581col_1721.1_131023
Parrilo, Pablo A.
Rao, Sujit
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
2022-01-14T15:20:59Z
2022-01-14T15:20:59Z
2021-06
2021-06-24T19:39:24.256Z
https://hdl.handle.net/1721.1/139576
We introduce fully general Macaulay bases of modules, which are a common generalization of Groebner bases and Macaulay 𝐻-bases to suitably graded modules over a commutative graded k-algebra, where the index sets of the two gradings may differ. The additional generality includes Groebner bases of modules as a special case, in contrast to previous work on Macaulay bases of modules. We show that the standard results on Groebner bases and Macaulay 𝐻-bases generalize in fields of arbitrary characteristic to Macaulay bases, including the reduction algorithm and Buchberger’s criterion and algorithm framework. A key result is that Macaulay bases, in contrast to Groebner bases, respect symmetries when there is a group 𝐺 acting homogeneously on a graded module, in which case the reduction algorithm is 𝐺-equivariant and the k-span of a Macaulay basis is 𝐺-invariant. We also show that some of the standard applications of Groebner bases can be generalized to Macaulay bases, including elimination and computation of syzygy modules, which require the generalization to modules that was not present in previous work.
S.M.
Massachusetts Institute of Technology
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Macaulay Bases of Modules
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Master of Science in Electrical Engineering and Computer Science
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<Abstract>We introduce fully general Macaulay bases of modules, which are a common generalization of Groebner bases and Macaulay 𝐻-bases to suitably graded modules over a commutative graded k-algebra, where the index sets of the two gradings may differ. The additional generality includes Groebner bases of modules as a special case, in contrast to previous work on Macaulay bases of modules. We show that the standard results on Groebner bases and Macaulay 𝐻-bases generalize in fields of arbitrary characteristic to Macaulay bases, including the reduction algorithm and Buchberger’s criterion and algorithm framework. A key result is that Macaulay bases, in contrast to Groebner bases, respect symmetries when there is a group 𝐺 acting homogeneously on a graded module, in which case the reduction algorithm is 𝐺-equivariant and the k-span of a Macaulay basis is 𝐺-invariant. We also show that some of the standard applications of Groebner bases can be generalized to Macaulay bases, including elimination and computation of syzygy modules, which require the generalization to modules that was not present in previous work.</Abstract>
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