Generating non-Noetherian modules efficiently.

Heitmann, Raymond C.

doi:10.1307/mmj/1029003021

Cited by 33 publications

(52 citation statements)

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“…No assumption on the existence of a doubly coprime factorization for P has been made. Moreover, parametrization (15) only requires the knowledge of a stabilizing controller C and the explicit computation of the three transfer matrices (I q − P C ) −1 , C (I q − P C ) −1 and C (I q − P C ) −1 P as we have the following relations (see (10) for more details):…”

Section: Theorem 3 Let P ∈ K Q×r Be a Stabilizable Plant C A Stabilimentioning

confidence: 99%

“…Let us note that there exist various refinements of Theorem 6 due to Eisenbud and Evans [7], Heitmann [10] and Coquand et al [4]. A version of Theorem 6 exists using the concept of the j-dimension j-dim A instead of the Krull dimension dim A.…”

Section: Definition 4 We Havementioning

confidence: 99%

“…Using Heitmann's generalization of Forster-Swan's theorem [4,7,10], we give an upper bound on the minimal number of free parameters appearing in the general Q-parametrization.…”

mentioning

confidence: 99%

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On a generalization of the Youla–Kučera parametrization. Part II: the lattice approach to MIMO systems

Quadrat

2005

Math. Control Signals Syst.

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Within the lattice approach to analysis and synthesis problems recently developed in Quadrat (Signal Syst, to appear), we obtain a general parametrization of all stabilizing controllers for internally stabilizable multi input multi output (MIMO) plants which do not necessarily admit doubly coprime factorizations. This parametrization is a linear fractional transformation of free parameters and the set of arbitrary parameters is characterized. This parametrization generalizes for MIMO plants the parametrization obtained in Quadrat (Syst Control Lett 50:135-148, 2003) for single input single output plants. It is named general Q-parametrization of all stabilizing controllers as we show that some ideas developed in this paper can be traced back to the pioneering work of Zames and Francis (IEEE Trans Automat control 28:585-601, 1983) on H ∞ -control. Finally, if the plant admits a doubly coprime factorization, we then prove that the general Q-parametrization becomes the well-known Youla-Kučera parametrization of all stabilizing controllers (Desoer et al. IEEE Trans Automat control 25:399-412, 1980; Vidyasagar, Control system synthesis: a factorization approach MIT Press, Cambridge 1985).

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Section: Theorem 3 Let P ∈ K Q×r Be a Stabilizable Plant C A Stabilimentioning

confidence: 99%

Section: Definition 4 We Havementioning

confidence: 99%

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On a generalization of the Youla–Kučera parametrization. Part II: the lattice approach to MIMO systems

Quadrat

2005

Math. Control Signals Syst.

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“…The previous example of Serre's theorem may involve too complicated computations, and we shall analyse a simpler statement, the abstract version of a theorem of Kronecker [17,7]. In this case, it is possible to get from an abstract proof a concrete algorithm that could have been formulated by Kronecker [14].…”

Section: Kronecker's Theoremmentioning

confidence: 99%

A Logical Approach to Abstract Algebra

Coquand

2005

New Computational Paradigms

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Abstract. Recent work in constructive mathematics show that Hilbert's program works for a large part of abstract algebra. Using in an essential way the ideas contained in the classical arguments, we can transform a large number of abstract non effective proofs of "concrete" statements into elementary proofs. Surprisingly the arguments we get are not only elementary but also mathematically clearer and not necessarily longer. We present an example where the simplification was significant enough to suggest an improved version of a classical theorem.

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“…We use the following fact, which is the culmination of work of Forster [16], Swan [37], Eisenbud and Evans [15] (for Noetherian rings) and Heitmann [21], [22] (in the general case): Let M be a finitely generated R-module, and k ∈ N. If for each prime ideal p of R, the R p -module M p = M ⊗ R R p can be generated by k elements, then the R-module M can be generated by d + k elements. The first part of the proposition now follows immediately.…”

Section: Coherent Modules and Coherent Ringsmentioning

confidence: 99%

Bounds and definability in polynomial rings

Aschenbrenner

2005

The Quarterly Journal of Mathematics

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Abstract. We study questions around the existence of bounds and the dependence on parameters for linear-algebraic problems in polynomial rings over rings of an arithmetic flavor. In particular, we show that the module of syzygies of polynomials f 1 , . . . , fn ∈ R[X 1 , . . . , X N ] with coefficients in a Prüfer domain R can be generated by elements whose degrees are bounded by a number only depending on N , n and the degree of the f j . This implies that if R is a Bézout domain, then the generators can be parametrized in terms of the coefficients of f 1 , . . . , fn using the ring operations and a certain division function, uniformly in R.

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Generating non-Noetherian modules efficiently.

Cited by 33 publications

References 0 publications

On a generalization of the Youla–Kučera parametrization. Part II: the lattice approach to MIMO systems

On a generalization of the Youla–Kučera parametrization. Part II: the lattice approach to MIMO systems

A Logical Approach to Abstract Algebra

Bounds and definability in polynomial rings

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