Resolving Decompositions for Polynomial Modules

We introduce the novel concept of a resolving decomposition of a polynomial module as a combinatorial structure that allows for the effective construction of free resolutions. It provides a unifying framework for recent results of the authors for different types of bases.

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Published in:Mathematics (Basel) Vol. 6; no. 9; p. 161
Main Authors: Albert, Mario, Seiler, Werner M.
Format: Journal Article
Language:English
Published: Basel MDPI AG 01.09.2018
Subjects:
Algebra
Algorithms
combinatorial decompositions
Decomposition
free resolutions
Geometry
Mathematics
polynomial modules
Polynomials
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United States > US
Switzerland
Germany
ISSN:2227-7390, 2227-7390
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Abstract We introduce the novel concept of a resolving decomposition of a polynomial module as a combinatorial structure that allows for the effective construction of free resolutions. It provides a unifying framework for recent results of the authors for different types of bases.
AbstractList We introduce the novel concept of a resolving decomposition of a polynomial module as a combinatorial structure that allows for the effective construction of free resolutions. It provides a unifying framework for recent results of the authors for different types of bases.
[...]one can say that the definition of resolving decompositions evolved from an abstraction and combination of these two basic examples: the emphasis on Stanley decompositions and unique normal forms represents a key feature of involutive bases and the somewhat convoluted last condition in Definition 1 stems from the theory of marked bases where it allows for the introduction of a Noetherian reduction relation without having head terms selected by a term order. Multiplying with some variables, we obtain the following chain of equations: xi1 ⋯xit hm(hk1 )=xi2 ⋯xit xμ2 hm(hk2 )=xi3 ⋯xit xμ2 xμ3 hm(hk3 )⋮=xit xμ2 ⋯xμt hm(hkt )=xμ1 ⋯xμt hm(hk1 ) which implies that xi1 ⋯xit =xμ1 ⋯xμt xi1 ⋯xit =xμ1 ⋯xμt . Furthermore, Condition (v) of Definition 1 implies in ??s ??sthe following chain: xi1 ⋯xit ek1(1) ⪰B xi2 ⋯xit xμ2 ek2(1) ⪰B⋯⪰B xμ1 ⋯xμt ek1(1). Because of xi1 ⋯xit =xμ1 ⋯xμt xi1 ⋯xit =xμ1 ⋯xμt , we must have throughout equality entailing that k1=⋯=kt k1=⋯=ktwhich contradicts our assumptions. ☐ The following two results provide a converse of this proposition for the special case of a monomial generating set B B by showing that whenever the B B-graph of such a set is acyclic, then there exists a term order satisfying Condition (v). For the last condition in Definition 1, we observe that the head term xi eα(1) xi eα(1) is the leading term of the syzygy Sα;i Sα;i for the module term order ≺B(0) ≺B(0) . [...]the used Schreyer order indeed satisfies Condition (v). ☐ As is the case for the classical Schreyer theorem, this construction can now be iterated to obtain resolving decompositions of the second and higher syzygy modules. [...]the two resolutions considered possess the same shape, meaning that the homogeneous components of the contained free modules satisfy dimk (Fi)j=dimk (Gi)j dimk (Fi)j=dimk (Gi)j. The made assertion is now a trivial consequence of Theorem 3. ☐ This proof already indicates that the two considered resolutions actually possess very similar differentials.
Author Seiler, Werner M.
Albert, Mario
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10.1007/978-3-642-23568-9
10.1007/978-3-319-24021-3
10.1016/j.jsc.2014.09.008
10.1007/s00200-009-0098-0
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10.1007/978-1-4612-5350-1
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10.1007/978-1-4757-2181-2
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Snippet We introduce the novel concept of a resolving decomposition of a polynomial module as a combinatorial structure that allows for the effective construction of...
[...]one can say that the definition of resolving decompositions evolved from an abstraction and combination of these two basic examples: the emphasis on...
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StartPage 161
SubjectTerms Algebra
Algorithms
combinatorial decompositions
Decomposition
free resolutions
Geometry
Mathematics
polynomial modules
Polynomials
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