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Desingularization in the q -Weyl algebra

Christoph Koutschan, Yi Zhang
<span title="">2018</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cmpnnf7ngjdizknpclbqvvidmm" style="color: black;">Advances in Applied Mathematics</a> </i> &nbsp;

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Moreover, an algorithm is presented for computing a generating set of the first q-Weyl closure of a given q-difference operator.  ...  As an application, we certify that several instances of the colored Jones polynomial are Laurent polynomial sequences by computing the corresponding desingularized operator.  ...  Introduction The desingularization problem has been primarily studied in the context of differential operators, and more specifically, for linear differential operators with polynomial coefficients.  ... 
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Christoph Koutschan, Yi Zhang. "Desingularization in the q -Weyl algebra." Advances in Applied Mathematics (2018) 80-101

Desingularization of First Order Linear Difference Systems with Rational Function Coefficients [article]

Moulay A. Barkatou, Maximilian Jaroschek
<span title="2018-02-04">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>

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It is well known that for a first order system of linear difference equations with rational function coefficients, a solution that is holomorphic in some left half plane can be analytically continued to  ...  We describe two algorithms to (partially) desingularize a given difference system and present a characterization of removable singularities in terms of shifts of the original system.  ...  In this paper we describe the first algorithm to desingularize first order linear difference systems with rational function coefficients.  ... 
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Moulay A. Barkatou, Maximilian Jaroschek. "Desingularization of First Order Linear Difference Systems with Rational Function Coefficients." arXiv (2018)

Desingularization explains order-degree curves for ore operators

Shaoshi Chen, Maximilian Jaroschek, Manuel Kauers, Michael F. Singer
<span title="">2013</span> <i title="ACM Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/p5cu7ybzmnd3jpp7pphgogfzpi" style="color: black;">Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation - ISSAC &#39;13</a> </i> &nbsp;

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Desingularization is the problem of finding a left multiple of a given Ore operator in which some factor of the leading coefficient of the original operator is removed.  ...  An order-degree curve for a given Ore operator is a curve in the (r, d)-plane such that for all points (r, d) above this curve, there exists a left multiple of order r and degree d of the given operator  ...  Desingularization of differential operators is classical [9] , and for difference operators, Abramov and van Hoeij [2, 1] give an algorithm for doing it.  ... 
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Shaoshi Chen, Maximilian Jaroschek, Manuel Kauers, Michael F. Singer. "Desingularization explains order-degree curves for ore operators." Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation - ISSAC '13 (2013) 157-164

Desingularization Explains Order-Degree Curves for Ore Operators [article]

Shaoshi Chen, Maximilian Jaroschek, Manuel Kauers, Michael F. Singer
<span title="2013-01-05">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>

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Desingularization is the problem of finding a left multiple of a given Ore operator in which some factor of the leading coefficient of the original operator is removed.  ...  An order-degree curve for a given Ore operator is a curve in the (r,d)-plane such that for all points (r,d) above this curve, there exists a left multiple of order r and degree d of the given operator.  ...  Desingularization of differential operators is classical [9] , and for difference operators, Abramov and van Hoeij [2, 1] give an algorithm for doing it.  ... 
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Shaoshi Chen, Maximilian Jaroschek, Manuel Kauers, Michael F. Singer. "Desingularization Explains Order-Degree Curves for Ore Operators." arXiv (2013)

Apparent singularities of linear difference equations with polynomial coefficients

S. A. Abramov, M. A. Barkatou, M. van Hoeij
<span title="2006-04-11">2006</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/lnpibodwo5cpnp3qbzbnlmdepu" style="color: black;">Applicable Algebra in Engineering, Communication and Computing</a> </i> &nbsp;

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Let L be a linear difference operator with polynomial coefficients. We consider singularities of L that correspond to roots of the trailing (resp. leading) coefficient of L.  ...  We consider non-commutative operator rings C[z, E] and C(z)[E] (the rings of linear difference operators with polynomial and, resp., rational function coefficients over C).  ...  The operator L ∈ C[z, E] has polynomial coefficients.  ... 
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S. A. Abramov, M. A. Barkatou, M. van Hoeij. "Apparent singularities of linear difference equations with polynomial coefficients." Applicable Algebra in Engineering, Communication and Computing 17.2 (2006) 117-133

Desingularization of Ore operators

Shaoshi Chen, Manuel Kauers, Michael F. Singer
<span title="">2016</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ezljl2d3lzga5efenbxdvvfcpa" style="color: black;">Journal of symbolic computation</a> </i> &nbsp;

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We show that Ore operators can be desingularized by calculating a least common left multiple with a random operator of appropriate order.  ...  Our result generalizes a classical result about apparent singularities of linear differential equations, and it gives rise to a surprisingly simple desingularization algorithm. 1991 Mathematics Subject  ...  Compare coefficients with respect to powers of ∂ on both sides and solve the resulting linear system.  ... 
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Citation

Shaoshi Chen, Manuel Kauers, Michael F. Singer. "Desingularization of Ore operators." Journal of symbolic computation (2016) 617-626

Desingularization of Ore Operators [article]

Shaoshi Chen, Manuel Kauers, Michael F. Singer
<span title="2014-08-23">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>

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We show that Ore operators can be desingularized by calculating a least common left multiple with a random operator of appropriate order.  ...  Our result generalizes a classical result about apparent singularities of linear differential equations, and it gives rise to a surprisingly simple desingularization algorithm.  ...  Compare coefficients with respect to powers of ∂ on both sides and solve the resulting linear system.  ... 
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Citation

Shaoshi Chen, Manuel Kauers, Michael F. Singer. "Desingularization of Ore Operators." arXiv (2014)

Apparent Singularities of Linear Difference Equations with Polynomial Coefficients [article]

S. A. Abramov, M. A. Barkatou, M. van Hoeij
<span title="2004-09-27">2004</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>

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Let L be a linear difference operator with polynomial coefficients. We consider singularities of L that correspond to roots of the trailing (resp. leading) coefficient of L.  ...  We prove that one can effectively construct a left multiple with polynomial coefficients L' of L such that every singularity of L' is a singularity of L that is not apparent.  ...  The operator L ∈ C[z, E] has polynomial coefficients.  ... 
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S. A. Abramov, M. A. Barkatou, M. van Hoeij. "Apparent Singularities of Linear Difference Equations with Polynomial Coefficients." arXiv (2004)

Univariate Contraction and Multivariate Desingularization of Ore Ideals [article]

Yi Zhang
<span title="2017-10-20">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>

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Ore operators with polynomial coefficients form a common algebraic abstraction for representing D-finite functions. They form the Ore ring K(x)[D_x], where K is the constant field.  ...  When L is a linear ordinary differential or difference operator, we design a contraction algorithm for L by using desingularized operators as proposed by Chen, Jaroschek, Kauers and Singer.  ...  The defining property of a D-finite function is that it satisfies a linear differential equation with polynomial coefficients.  ... 
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Citation

Yi Zhang. "Univariate Contraction and Multivariate Desingularization of Ore Ideals." arXiv (2017)

Page 4121 of Mathematical Reviews Vol. , Issue 2002F [page]

<span title="">2002</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;

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(RS-AOS-C; Moscow); van Hoeij, Mark (1-FLS; Tallahassee, FL) Desingularization of linear difference operators with polynomial coefficients.  ...  Let F be a recur- rence operator of the form F = i fy (n)N* with f,(n) being polynomials in n, and where N is the shift operator with respect to n.  ... 
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Mathematical Reviews (2002) 4121

On nonlinear desingularization

M. S. Berger, L. E. Fraenkel
<span title="1980-01-01">1980</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/iwegpttow5b4vhkhs3scejn5fi" style="color: black;">Bulletin of the American Mathematical Society</a> </i> &nbsp;

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Let 12 be a bounded domain in the plane R 2 , with boundary 3£2, and let L be a formally self-adjoint, uniformly elliptic, second order operator, with smooth coefficients a ( j(x) defined on 12 with the  ...  The nonlinear problem and its linear degenerate form.  ...  [V R ]: inf D[U] w ue^x R where 2 R ={U\Ueft 1 >2 (Î2), f a F(x, U-q)=R} with f(x, t) = F t (x, t) and F(x 9 0) = 0, and D[U] is the Dirichlet form associated with the operator L.  ... 
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M. S. Berger, L. E. Fraenkel. "On nonlinear desingularization." Bulletin of the American Mathematical Society 2.1 (1980) 165-168

Formal Desingularization of Surfaces - The Jung Method Revisited - [article]

T. Beck
<span title="2008-01-15">2008</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>

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In this paper we propose the concept of formal desingularizations as a substitute for the resolution of algebraic varieties.  ...  We give a detailed study of the Jung method and show how it facilitates an efficient computation of formal desingularizations for projective surfaces over a field of characteristic zero, not necessarily  ...  This implies that f (z + α) vanishes at z = 0 with multiplicity one or that the constant coefficient of f (z + α) is zero whereas its linear coefficient does not vanish.  ... 
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T. Beck. "Formal Desingularization of Surfaces - The Jung Method Revisited -." arXiv (2008)

Formal desingularization of surfaces: The Jung method revisited

Tobias Beck
<span title="">2009</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ezljl2d3lzga5efenbxdvvfcpa" style="color: black;">Journal of symbolic computation</a> </i> &nbsp;

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In this paper we propose the concept of formal desingularizations as a substitute for the resolution of algebraic varieties.  ...  We give a detailed study of the Jung method and show how it facilitates an efficient computation of formal desingularizations for projective surfaces over a field of characteristic zero, not necessarily  ...  Again E denotes a field of characteristic zero which needs not be algebraically closed. We are dealing with completions of stalks of regular schemes of finite type over E. Therefore  ... 
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Tobias Beck. "Formal desingularization of surfaces: The Jung method revisited." Journal of symbolic computation 44.2 (2009) 131-160

Apparent Singularities of D-finite Systems [article]

Shaoshi Chen, Manuel Kauers, Ziming Li, Yi Zhang
<span title="2017-05-02">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>

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We generalize the notions of singularities and ordinary points from linear ordinary differential equations to D-finite systems.  ...  Ordinary points of a D-finite system are characterized in terms of its formal power series solutions.  ...  Desingularization of linear difference operators with polynomial coefficients. In Proc. of ISSAC’99, 269–275, New York, NY, USA, 1999, ACM. [3] F. Aroca and J. Cano.  ... 
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Citation

Shaoshi Chen, Manuel Kauers, Ziming Li, Yi Zhang. "Apparent Singularities of D-finite Systems." arXiv (2017)

Ore Polynomials in Sage [chapter]

Manuel Kauers, Maximilian Jaroschek, Fredrik Johansson
<span title="">2015</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;

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We present a Sage implementation of Ore algebras.  ...  ; solvers for polynomials, rational functions and (generalized) power series.  ...  f (a(x)) where a(x) is algebraic over the base ring desingularize() computes a left multiple of this operator with polynomial coefficients and lowest possible leading coefficient degree associate_solutions  ... 
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Citation

Manuel Kauers, Maximilian Jaroschek, Fredrik Johansson. "Ore Polynomials in Sage." Lecture Notes in Computer Science (2015) 105-125

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