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Suslin's algorithms for reduction of unimodular rows

Henri Lombardi, Ihsen Yengui
2005 Journal of symbolic computation  
These efficient elementary operations enable us to give new and simple 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.  ... 
doi:10.1016/j.jsc.2005.01.004 fatcat:tr6xt5p7o5fm7fbp7dfcncormm


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  ... 
doi:10.1016/s0747-7171(05)00160-4 fatcat:a3hcsoc2orad3fh2g2mvgsxm6q

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

H. Park, C. Woodburn
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.  ... 
arXiv:alg-geom/9405003v1 fatcat:ozs4snxuhjfdvdyak2fnmyje3q

Unimodular rows over monoid rings [article]

Joseph Gubeladze
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 (  ... 
arXiv:1706.04364v3 fatcat:44qfgwyotzgmdfhngqjn3ejvrq

Computing bases of complete intersection rings in Noether position

Marcela Almeida, Manuela Blaum, Lisi D'Alfonso, Pablo Solernó
2001 Journal of Pure and Applied Algebra  
We show that there exists a well parallelizable 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.  ... 
doi:10.1016/s0022-4049(00)00135-3 fatcat:zuewn5jhwjdbdieerp6l7jz5cm

Bounded generation and commutator width of Chevalley groups: function case [article]

Boris Kunyavskii, Eugene Plotkin, Nikolai Vavilov
2022 arXiv   pre-print
The paper contains also a broader discussion 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  ... 
arXiv:2204.10951v2 fatcat:nu3jwejs6baefnihvwedoe3b4u

On the degrees of bases of free modulesover a polynomial ring

Marcela Almeida, Lisi D'Alfonso, Pablo Solernó
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  ... 
doi:10.1007/pl00004746 fatcat:ihgrq24zg5b3tmqcitutfl5xdq

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

Jeongwan Haah
2021 arXiv   pre-print
Further, we prove that 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.  ... 
arXiv:1907.02075v4 fatcat:fkbyul52xbbqtflaj64ewucltu

The Method of Virtual Components in the Multivariate Setting

Ming-Jun Lai, Alexander Petukhov
2009 Journal of Fourier Analysis and Applications  
Two examples 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.  ... 
doi:10.1007/s00041-009-9108-7 fatcat:rjyxo62hezhinp6vmaqov3nxby

Reversible interger-to-interger wavelet transforms for image coding

Michael David Adams
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.  ... 
doi:10.14288/1.0065047 fatcat:v475sqkmebbd5gzcdyczst4dzi