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Suslin's algorithms for reduction of unimodular rows

2005
*
Journal of symbolic computation
*

These efficient elementary operations enable us to give new and simple

doi:10.1016/j.jsc.2005.01.004
fatcat:tr6xt5p7o5fm7fbp7dfcncormm
*algorithms**for*reducing*unimodular**rows*with entries in K[X 1 , . . . , X k ] to t (1, 0, . . . , 0) using elementary operations in ... A well-known lemma*of*Suslin says that*for*a commutative ring A if (v 1 (X), . . . , v n (X)) ∈ (A[X]) n is*unimodular*where v 1 is monic and n ≥ 3, then there exist γ 1 , . . . , γ ∈ E n−1 (A[X]) such ... Precise bounds have been computed by some authors concerning*algorithms**for*the Quillen-Suslin theorem based on*Suslin's*proof*of*Serre's Conjecture. ...##
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Contents

2005
*
Journal of symbolic computation
*

*for*

*reduction*

*of*

*unimodular*

*rows*. . . . . . . . . . . . . . . . . . . . . 707 Alcázar, J.G. and Sendra, J.R., Computation

*of*the topology

*of*real algebraic space curves . . . . . . . . . . . . . . ... . . . . . . . . 671 Chen, F., Cox, D. and Liu, Y., The µ-basis and implicitization

*of*a rational parametric surface . . . . . . . . . . . . . . . . . 689 Lombardi, H. and Yengui, I.,

*Suslin's*

*algorithms*...

##
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An Algorithmic Proof of Suslin's Stability Theorem over Polynomial Rings
[article]

1994
*
arXiv
*
pre-print

*Suslin's*stability theorem states that the same is true

*for*the multivariate polynomial ring SL_n(k[x_1,... ,x_m]) with n≥ 3. ... Then Gaussian elimination over k and the Euclidean division

*algorithm*

*for*the univariate polynomial ring k[x] allow us to write any matrix in SL_n(k) or SL_n(k[x]), n≥ 2, as a product

*of*elementary matrices ... Vetterli

*for*all the valuable support, insightful discussions and encouragement. ...

##
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Unimodular rows over monoid rings
[article]

2018
*
arXiv
*
pre-print

*For*a commutative Noetherian ring R

*of*dimension d and a commutative cancellative monoid M, the elementary action on

*unimodular*n-

*rows*over the monoid ring R[M] is transitive

*for*n>=max(d+2,3). ... The main result completes a project, initiated in the early 1990s, and suggests a new direction in the study

*of*K-theory

*of*monoid rings. ... Also, if R is a field, the proof

*of*Theorem 1.1 can be converted, along the lines in [17] , into an

*algorithm*that,

*for*a given

*unimodular*

*row*over R[M], finds elementary transformations, leading to ( ...

##
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Computing bases of complete intersection rings in Noether position

2001
*
Journal of Pure and Applied Algebra
*

We show that there exists a well parallelizable

doi:10.1016/s0022-4049(00)00135-3
fatcat:zuewn5jhwjdbdieerp6l7jz5cm
*algorithm*which computes bases*of*the kernel and the image*of*F in time (nL) O(1) (MD) O(n) : By means*of*this result we obtain a single exponential*algorithm*... xn]=(f1; : : : ; fn−r) such that S is integral over R; we show that there exists an*algorithm*running in time O(n)'d O(n 2 ) which computes a basis*of*S over R: Also, as a consequence*of*our techniques ... Acknowledgements We thank Alicia Dickenstein, Joos Heintz and Guillermo Matera*for*many helpful suggestions and remarks. ...##
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Bounded generation and commutator width of Chevalley groups: function case
[article]

2022
*
arXiv
*
pre-print

The paper contains also a broader discussion

arXiv:2204.10951v2
fatcat:nu3jwejs6baefnihvwedoe3b4u
*of*the bounded generation problem*for*groups*of*Lie type, some applications and a list*of*unsolved problems in the field. ... Under some additional assumptions, we prove similar results*for*other classes*of*Chevalley groups over Dedekind rings*of*arithmetic rings in positive characteristic. ... This paper started jointly with her as a discussion*of*the bounded commutator width*of*various classes*of*Kac Moody groups over finite fields, and*for*a long time it was supposed to be a work*of*four authors ...##
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On the degrees of bases of free modulesover a polynomial ring

1999
*
Mathematische Zeitschrift
*

*For*this purpose it is convenient to consider the matrix UC in Proposition 10 as a k(x 1 , . . . , x n−1 )[x n ] -

*unimodular*matrix in order to apply

*Suslin's*

*reduction*procedure following [17] and ...

*For*example, if the matrix is

*unimodular*(i.e. the

*rows*can be extended to a basis

*of*the whole space), a single exponential upper bound

*for*the degree

*of*a basis

*of*its kernel is given in [5, Corollary ...

##
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Clifford Quantum Cellular Automata: Trivial group in 2D and Witt group in 3D
[article]

2021
*
arXiv
*
pre-print

Further, we prove that

arXiv:1907.02075v4
fatcat:fkbyul52xbbqtflaj64ewucltu
*for*any D the fourth power*of*any TI Clifford QCA is trivial. ... We present explicit examples*of*nontrivial TI Clifford QCA*for*D=3 and any odd prime p, and show that the Witt group*of*the finite field 𝔽_p is a subgroup*of*the group ℭ(D = 3, p)*of*all TI Clifford QCA ... Hence, we see that the first*row**of*EQ is*unimodular*and self-orthogonal, and has 1 at the first position. ...##
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The Method of Virtual Components in the Multivariate Setting

2009
*
Journal of Fourier Analysis and Applications
*

Two examples

doi:10.1007/s00041-009-9108-7
fatcat:rjyxo62hezhinp6vmaqov3nxby
*of*the virtual components*for*tight wavelet frames based on bivariate box splines on three or four direction mesh are given. ... We describe the so-called method*of*virtual components*for*tight wavelet framelets to increase their approximation order and vanishing moments in the multivariate setting. ... The authors would like to thank the reviewers*for*their comments which enable the authors to clarify a few ambiguous points and improve the readability*of*this paper. ...##
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Reversible interger-to-interger wavelet transforms for image coding

2002

*Of*the transforms considered, several were found to perform particularly well, with the best choice

*for*a given application depending on the relative importance

*of*the preceding criteria. ... The applicability

*of*the GRITIT framework

*for*block transforms is also demonstrated. Throughout all

*of*this work, particularly close attention is paid to rounding operators and their characteristics. ... Hong Man

*for*his assistance in collecting some

*of*the experimental results reported in this appendix. ...