Quillen-Suslin Theorem Research Articles

The Smith normal form and reduction of weakly linear matrices

The reduction of a multidimensional system is closely related to the reduction of a multivariate polynomial matrix, for which the Smith normal form of the matrix plays a key role. In this paper, we investigate the reduction of weakly linear multivariate polynomial matrices to their Smith normal forms. Using hierarchical-recursive method and Quillen-Suslin Theorem, we derive some necessary and sufficient conditions ensuring that such matrices can be reduced to their Smith normal forms, and these conditions are easily checked by computing the reduced Gröbner bases of the relevant polynomial ideals. Based on the new results, we propose an algorithm for reducing weakly linear multivariate polynomial matrices to their Smith normal forms.

Algebraic curves and foliations

Consider a field k ${\mathsf {k}}$ of characteristic 0 $\hskip.001pt 0$ , not necessarily algebraically closed, and a fixed algebraic curve f = 0 $f=0$ defined by a tame polynomial f ∈ k [ x , y ] $f\in {\mathsf {k}}[x,y]$ with only quasi-homogeneous singularities. We prove that the space of holomorphic foliations in the plane A k 2 $\mathbb {A}^2_{\mathsf {k}}$ having f = 0 $f=0$ as a fixed invariant curve is generated as k [ x , y ] ${\mathsf {k}}[x,y]$ -module by at most four elements, three of them are the trivial foliations f d x , f d y $fdx,fdy$ and d f $df$ . Our proof is algorithmic and constructs the fourth foliation explicitly. Using Serre's GAGA and Quillen–Suslin theorem, we show that for a suitable field extension K ${\mathsf {K}}$ of k ${\mathsf {k}}$ such a module over K [ x , y ] ${\mathsf {K}}[x,y]$ is actually generated by two elements, and therefore, such curves are free divisors in the sense of K. Saito. After performing Groebner basis for this module, we observe that in many well-known examples, K = k ${\mathsf {K}}={\mathsf {k}}$ .

Bounds for degrees of syzygies of polynomials defining a grade two ideal

We make explicit the exponential bound on the degrees of the polynomials appearing in the Effective Quillen-Suslin Theorem, and apply it jointly with the Hilbert-Burch Theorem to show that the syzygy module of a sequence of m polynomials in n variables defining a complete intersection ideal of grade two is free, and that a basis of it can be computed with bounded degrees. In the known cases, these bounds improve previous results.

Elementary Matrix-computational Proof of Quillen-Suslin Theorem for Ore Extensions

Elementary Matrix-computational Proof of Quillen-Suslin Theorem for Ore Extensions

Multi-D Wavelet Filter Bank Design Using Quillen-Suslin Theorem for Laurent Polynomials

In this paper we present a new approach for constructing the wavelet filter bank. Our approach enables constructing nonseparable multidimensional non-redundant wavelet filter banks with FIR filters using the Quillen-Suslin Theorem for Laurent polynomials. Our construction method presents some advantages over the traditional methods of multidimensional wavelet filter bank design. First, it works for any spatial dimension and for any sampling matrix. Second, it does not require the initial lowpass filters to satisfy any additional assumption such as interpolatory condition. Third, it provides an algorithm for constructing a wavelet filter bank from a single lowpass filter so that its vanishing moments are at least as many as the accuracy number of the lowpass filter.

Computation of the Smith Form for Multivariate Polynomial Matrices Using Maple

In this paper we show how the transformations associated with the reduction to the Smith form of some classes of mul-tivariate polynomial matrices are computed. Using a Maple implementation of a constructive version of the Quillen-Suslin Theorem, we present two algorithms for the reduction to a particular Smith form often associated with the simplification of linear systems of multidimensional equations.

Hidden constructions in abstract algebra (6): The theorem of Maroscia and Brewer & Costa

We give a constructive deciphering for a generalization of the Quillen–Suslin theorem due to Maroscia and Brewer & Costa stating that finitely generated projective modules over R [ X 1 , … , X n ] , where R is a Prüfer domain with Krull dimension ≤ 1 , are extended from R .

An algorithm for unimodular completion over Noetherian rings

We give an algorithm for the well-known result asserting that if R is a polynomial ring in a finite number of variables over a Noetherian ring A of Krull dimension d < ∞ , then for n ⩾ max ( 3 , d + 2 ) , SL n ( R ) acts transitively on Um n ( R ) . For technical reasons we demand that the Noetherian ring A has a theory of Gröbner bases and contains an infinite set E = { y 1 , y 2 , … } such that y i − y j ∈ A × for each i ≠ j . The most important guiding examples are affine rings K [ x 1 , … , x m ] / I and localizations of polynomial rings S −1 K [ x 1 , … , x m ] , with K an infinite field. Moreover, we give an algorithmic proof of Suslin's stability theorem over these rings. For the purpose to prepare the ground for this algorithmic generalizations of the Quillen–Suslin theorem (corresponding to the particular case A is a field), we will give in the first section a constructive proof of an important lemma of Suslin which is the only nonconstructive step in Suslin's second elementary solution of Serre's conjecture. This lemma says that for a commutative ring A, if 〈 v 1 ( X ) , … , v n ( X ) 〉 = A [ X ] where v 1 is monic and n ⩾ 3 , then there exist γ 1 , … , γ ℓ ∈ E n − 1 ( A [ X ] ) such that 〈 Res ( v 1 , e 1 . γ 1 ( v 2 , … , v n ) ) t , … , Res ( v 1 , e 1 . γ ℓ ( v 2 , … , v n ) ) t 〉 = A . Thanks to this constructive proof, Suslin's second proof of Serre's conjecture becomes fully constructive.

Exotic embeddings of smooth affine varieties

We find examples of exotic embeddings of smooth affine varieties into C n in large codimensions. We show also examples of affine smooth, rational algebraic varieties X, for which there are algebraically exotic embeddings ϕ : X → X × C l , which are holomorphically trivial. Using this we construct an infinite family { C 2 p + 3 } ( p is a prime number) of complex manifolds, such that every C 2 p + 3 has at least two different algebraic (quasi-affine) structures. We show also that there is a natural connection between Abhyankar–Sathaye Conjecture and the famous Quillen–Suslin Theorem.

A generalized Serre problem

A possible generalization of the Serre problem (Quillen–Suslin theorem) on the freeness of projective modules over polynomial rings, which seems to be of interest in systems theory applications, is shown to follow from the solution to the original Serre problem.

Optimal design of synthesis filters in multidimensional perfect reconstruction FIR filter banks using Grobner bases

For a given low-pass analysis filter in an n-dimensional (n-D) perfect reconstruction (PR) FIR filter bank, there exists certain degree of freedom in designing a synthesis filter to make the overall filter bank have PR property, and Quillen-Suslin Theorem can be used to precisely determine the degree of freedom. Designing a PR synthesis filter with optimal frequency response amounts to minimizing the deviation between the frequency responses of the synthesis filter and an ideal filter under the PR constraint. In this paper, a method based on Grobner bases computation is developed for optimally designing critically sampled n-D PR FIR filter banks when one analysis filter is known. This design method can be easily combined with various existing design methods including minimax optimization and weighted least squares optimization, and can incorporate additional design goals including linear phase and bandpass characteristic.

Constructions in R[ x1,…, xn]: applications to K-theory

A classical result in K-theory about polynomial rings like the Quillen–Suslin theorem admits an algorithmic approach when the ring of coefficients has some computational properties, associated with Gröbner bases. There are several algorithms when we work in K[x 1,…,x n] , K a field. In this paper we compute a free basis of a finitely generated projective module over R[ x 1,…, x n ], R a principal ideal domain with additional properties, test the freeness for projective modules over D[ x 1,…, x n ], with D a Dedekind domain like Z[ −5 ] and for the one variable case compute a free basis if there exists any.

An Algorithm for the Quillen–Suslin Theorem for Quotients of Polynomial Rings by Monomial Ideals

This paper presents an algorithm for the Quillen–Suslin Theorem for quotients of polynomial rings by monomial ideals, that is, quotients of the form A=k [ x0, . ,xn ]/I, with I a monomial ideal and k a field. Vorst proved that finitely generated projective modules over such algebras are free. Given a finitely generated module P, described by generators and relations, the algorithm tests whether P is projective, in which case it computes a free basis forP .

On the degrees of bases of free modulesover a polynomial ring

Let k be an infinite field, A the polynomial ring \(k[x_1,...,x_n]\) and \(F\in A^{N\times M}\) a matrix such that \({\rm Im}\,F\subset A^N\) is A-free (in particular, Quillen-Suslin Theorem implies that \({\rm Ker}\,F\) is also free). Let D be the maximum of the degrees of the entries of F and s the rank of F. We show that there exists a basis \(\{ v_1,\ldots,v_{M} \}\) of \(A ^M\) such that \(\{ v_1,\ldots ,v_{M-s} \}\) is a basis of \({\rm Ker}\,F\), \(\{ F(v_{M-s+1}), \ldots , F(v_M) \}\) is a basis of \({\rm Im}\,F\) and the degrees of their coordinates are of order \(((M-s)sD)^{O(n^4)}\).

Hermite reduction methods for generation of a complete class of linear-phase perfect reconstruction filter banks-Part I: Theory

Motivated by the possibility of extensions to two-dimensions, we address the problem constructing a linear-phase multiband perfect reconstruction finite impulse response filter bank by constructing the polyphase matrix associated with it. The equivalent problem of construction of linear phase, i.e., symmetric or antisymmetric compactly supported wavelets are, thus, also considered. Our approach rests on the fact that if any proper subset of the set of linear-phase analysis filters is almost arbitrarily specified, then the complete set of linear-phase analysis filters can always be constructed. The solution to this problem is obtained by solving the problem of completing an incompletely specified analysis polyphase matrix having the structure mandated by the linear-phase property. Symmetric versions of matrix reduction algorithms akin to the Hermite reduction algorithm well known in system theory are used in our method of construction. The technique closely follows the proof of (nonsymmetric) Quillen-Suslin theorem for the completion of multivariable polynomial matrices, and, thus, in addition, has the potential for extension to the multidimensional case. Examples are given to demonstrate the procedure.

An algorithm for the Quillen-Suslin theorem for monoid rings

Let k be a field, and let M be a commutative, seminormal, finitely generated monoid, which is torsionfree, cancellative, and has no nontrivial units. Gubeladze [8] proved that finitely generated projective modules over kM are free. This paper contains an algorithm for finding a free basis for a finitely generated projective module over kM. As applications one obtains new algorithms for the Quillen-Suslin Theorem for polynomial rings and Laurent polynomial rings, based on Quillen's proof.

Some Applications of Artamonov-Quillen-Suslin Theorems to Metabelian Inner Rank and Primitivity

AbstractFor any variety of groups, the relative inner rank of a given groupG is defined to be the maximal rank of the -free homomorphic images of G. In this paper we explore metabelian inner ranks of certain one-relator groups. Using the well-known Quillen-Suslin Theorem, in conjunction with an elegant result of Artamonov, we prove that if r is any "Δ-modular" element of the free metabelian group Mn of rank n &gt; 2 then the metabelian inner rank of the quotient group Mn/(r) is at most [n/2]. As a corollary we deduce that the metabelian inner rank of the (orientable) surface group of genus k is precisely k. This extends the corresponding result of Zieschang about the absolute inner ranks of these surface groups. In continuation of some further applications of the Quillen-Suslin Theorem we give necessary and sufficient conditions for a system g = (g1,..., gk) of k elements of a free metabelian group Mn, k ≤ n, to be a part of a basis of Mn. This extends results of Bachmuth and Timoshenko who considered the cases k = n and k &lt; n — 3 respectively.

Algorithms for the Quillen-Suslin theorem

We give an algorithm for computing a free basis of a projective C [ x 1,… x n ,-module which is presented as the image, kernel, or cokernel of a polynomial matrix. Our method can be implemented using Gröbner bases, and it provides an elementary, constructive proof of the Quillen-Suslin theorem.

Nullstellensatz effectif et Conjecture de Serre (Théorème de Quillen‐Suslin) pour le Calcul Formel

AbstractLet k be an arbitrary field, X1,….,Xn indeterminates over k and F1…, F3 ε ∈ k[X1…,Xn] polynomials of maximal degree \documentclass{article}\pagestyle{empty}\begin{document}$ d: = \mathop {\max }\limits_{1 \le i \le a} \deg $\end{document} (Fi). We give an elementary proof of the following effective Nullstellensatz: Assume that F1,…,F have no common zero in the algebraic closure of k. Then there exist polynomials P1…, P3 ε ∈ k[X1…,Xn] such that \documentclass{article}\pagestyle{empty}\begin{document}$ 1: = \mathop \Sigma \limits_{1 \le i \le a} $\end{document} PiFi and This result has many applications in Computer Algebra. To exemplify this, we give an effective quantitative and algorithmic version of the Quillen‐Suslin Theorem baaed on our effective Nullstellensatz.

A glance back at the Quillen-Suslin theorem and the recent status o f the Bass-Quillen conjecture

In his famous FAC-article J.P. Serre asked whether all finitely generated projective modules over K[X1,...,Xn] (K a field) are free. This question soon became known in the mathematical society as “Serre’s Conjecture”. This conjecture was proved independently in 1976 by A. A. Suslin and D. Quillen. Important further developments followed from the ideas in Quillen’s proof. In this article we discuss the geometric motivation of the original problem as well as certain aspects of Quillen’s proof. Then we discuss further developments, in particular a proof that finitely generated projective modules over R[X1,...,Xn], R a Bezout domain, are free. We also give attention to new results on the existence of free summands in projective modules. In the case of polynomial rings, this work strengthens Serre’s famous free summand theorem: if R is a Noetherian ring and P is a finitely generated projective R-module such that rank P &gt; dim R, then there exists a free rank one summand in P. Finally we discuss the present status of the most important open problem in this field, namely the Bass-Quillen Conjecture: if R is a regular local ring, is every finitely generated projective R[X1,...,Xn] module free?