Gröbner Bases Applied to Finitely Generated Field Extensions

Müller-Quade, Jörn; Steinwandt, Rainer

doi:10.1006/jsco.1999.0417

Cited by 5 publications

(5 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, r which is closed with respect to certain derivations (see Definition 7) and then we build a field of fractions of its elements. The algorithms for computing the transcendence degree of field extensions of a field are presented in [27] and (of not necessarily purely transcendental field extensions) in [28]. There are other algorithms for the same problem which can be found in the references therein.…”

Section: Procedures For Checking Algebraic Controllability Of a Rationmentioning

confidence: 98%

See 1 more Smart Citation

Realization Theory for Rational Systems: Minimal Rational Realizations

Němcová

Schuppen

2009

Acta Appl Math

View full text Add to dashboard Cite

The study of realizations of response maps is a topic of control and system theory. Realization theory is used in system identification and control synthesis.A minimal rational realization of a given response map p is a rational realization of p such that the dimension of its state space equals the transcendence degree of the observation field of p. We relate minimality of rational realizations with their rational observability, algebraic controllability and canonicity. We show that the existence of a minimal rational realization is implied by the existence of a rational realization. We also specify the relation between birational equivalence of rational realizations and the properties of being canonical and minimal. Furthermore, we briefly discuss the procedures for checking various properties of rational realizations.

show abstract

Section: Procedures For Checking Algebraic Controllability Of a Rationmentioning

confidence: 98%

“…The third step of the procedure above could be executed element-wise. The algorithms for checking whether an element of B (and therefore an element of the field Q of rational functions on X) is also an element of the field Q obs ( ) are described in [27] and [28].…”

Section: Computational Algebra For Realizationmentioning

confidence: 99%

Realization Theory for Rational Systems: Minimal Rational Realizations

Němcová

Schuppen

2009

Acta Appl Math

View full text Add to dashboard Cite

show abstract

“…See (Zariski and Samuel, 1958, I, §17, p.28). An extension not described here because of lack of space is to define a transcedence basis for the algebraic structure used which then allows the use of the algorithms of Müller-Quade and Steinwandt (2000) for the computation of such a basis. Problem D.2.…”

Section: Appendix C Polynomialsmentioning

confidence: 99%

An Algorithm for System Identification of a Discrete-Time Polynomial System without Inputs

Němcová,

Petreczky,

van Schuppen

2015

Preprint

View full text Add to dashboard Cite

A subalgebraic approximation algorithm is proposed to estimate from a set of time series the parameters of the observer representation of a discrete-time polynomial system without inputs which can generate an approximation of the observed time series. A major step of the algorithm is to construct a set of generators for the polynomial function from the past outputs to the future outputs. For this singular value decompositions and polynomial factorizations are used. An example is provided.

show abstract

“…Computing the transcendence degree of Q(x, y, z)/Q(~) by means of GrSbner bases as discussed in [7,4,6,5] is significantly more expensive than the above solution. For the method described in [6,5] a GrSbner basis of the ideal (nl (2) -gl" d, (2) . .…”

Section: E X a M P L E Smentioning

confidence: 99%

“…In [7,4], solutions to this problem based on GrSbner basis techniques using tag variables are given; the approach in [6,5] uses GrSbner basis techniques which use, instead of tag variables, computations in an extension field of k.…”

Section: Introductionmentioning

confidence: 99%

On computing a separating transcendence basis

Steinwandt¹

2000

SIGSAM Bull.

Self Cite

View full text Add to dashboard Cite

R a i n e r S t e i n w a n d t I n s t i t u t fiir A l g o r i t h m e n u n d K o g n i t i v e S y s t e m e , Prof. Dr. T h . B e t h , A r b e i t s g r u p p e C o m p u t e r~l g e b r a , F a k u l t~t fiir I n f o r m a t i k , Universit/it K a r l s r u h e , G e r m a n y AbstractLet k(x~,..., x,~)/k be a finitely generated field extension, and gl,... ,gr e k(g). For k ( x l , . . . ,x,~)/k(g:,... ,g~) being separably generated (which in particular includes the case char(k) = 0) we give a method to compute the transcendence degree and a separating transcendence basis of this extension by means of simple linear algebra techniques.

show abstract

Gröbner Bases Applied to Finitely Generated Field Extensions

Cited by 5 publications

References 15 publications

Realization Theory for Rational Systems: Minimal Rational Realizations

Realization Theory for Rational Systems: Minimal Rational Realizations

An Algorithm for System Identification of a Discrete-Time Polynomial System without Inputs

On computing a separating transcendence basis

Contact Info

Product

Resources

About