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Differential equations for algebraic functions
 ISSAC’07: Proceedings of the 2007 international symposium on Symbolic and algebraic computation
, 2007
"... Abstract. It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose deg ..."
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Cited by 13 (5 self)
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Abstract. It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose degree is cubic in the degree of the function. We also show that there exists a linear differential equation of order linear in the degree whose coefficients are only of quadratic degree. Furthermore, we prove the existence of recurrences of order and degree close to optimal. We study the complexity of computing these differential equations and recurrences. We deduce a fast algorithm for the expansion of algebraic series. 1.
Effective scalar products of Dfinite symmetric series
 Journal of Combinatorial Theory Series A, 112:1
"... Abstract. Many combinatorial generating functions can be expressed as combinations of symmetric functions, or extracted as subseries and specializations from such combinations. Gessel has outlined a large class of symmetric functions for which the resulting generating functions are Dfinite. We ext ..."
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Cited by 13 (10 self)
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Abstract. Many combinatorial generating functions can be expressed as combinations of symmetric functions, or extracted as subseries and specializations from such combinations. Gessel has outlined a large class of symmetric functions for which the resulting generating functions are Dfinite. We extend Gessel’s work by providing algorithms that compute differential equations these generating functions satisfy in the case they are given as a scalar product of symmetric functions in Gessel’s class. Examples of applications to kregular graphs and Young tableaux with repeated entries are given. Asymptotic estimates are a natural application of our method, which we illustrate on the same model of Young tableaux. We also derive a seemingly new formula for the Kronecker product of the sum of Schur functions with itself. (This article completes the extended abstract published in the proceedings of FPSAC’02 under the title “Effective DFinite Symmetric Functions”.)
Desingularization Explains OrderDegree Curves for Ore Operators
"... 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 orderdegree curve for a given Ore operator is a curve in the (r, d)plane such that for all points (r, d) above this curve, the ..."
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Cited by 3 (3 self)
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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 orderdegree 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. We give a new proof of a desingularization result by Abramov and van Hoeij for the shift case, and show how desingularization implies orderdegree curves which are extremely accurate in examples. Categories and Subject Descriptors
Génération automatique de procédures numériques pour les fonctions Dfinies
"... L’évaluation numérique à grande précision de constantes comme π, e, γ, ln 2, etc., de fonctions élémentaires comme exp et arctan, puis de fonctions spéciales comme Γ ou ζ est un problème classique. D’un point de vue informatique, « grande précision » s’oppose à précision fixe, mais sousentend auss ..."
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Cited by 2 (1 self)
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L’évaluation numérique à grande précision de constantes comme π, e, γ, ln 2, etc., de fonctions élémentaires comme exp et arctan, puis de fonctions spéciales comme Γ ou ζ est un problème classique. D’un point de vue informatique, « grande précision » s’oppose à précision fixe, mais sousentend aussi que l’on cherche des algorithmes asymptotiquement efficaces quand le nombre de chiffres demandés grandit. Le développement d’algorithmes de complexité quasilinéaire en le nombre de chiffres du résultat remonte aux années 1970, avec par exemple les travaux de Richard Brent, Eugene Salamin ou R. William Gosper. Les fonctions holonomes sont les solutions d’équations différentielles linéaires à coefficients polynomiaux. Leurs propriétés élémentaires sont bien connues depuis le dixneuvième siècle, mais elles ont pris une place importante en combinatoire (comme séries génératrices) et en calcul formel (en tant que classe de fonctions bénéficiant de propriétés algorithmiques agréables, tant du point de vue de la calculabilité que de celui de la complexité) depuis les années 1980. Parmi les responsables de ce regain d’intérêt, on peut citer Richard Stanley, Leonard Lipshitz et Doron Zeilberger. Mon travail de stage s’incrit dans une démarche générale du projet Algo de développer pour toute la classe des fonctions holonomes une algorithmique efficace utilisant
Algorithms for Algebraic Analysis
"... Doctor of Philosophy in Mathematics One of the major goals in the field of symbolic computation of differential equations is to develop algorithms for exact or closedform solutions. This thesis studies symbolic computation of maximally overdetermined systems of linear partial differential equati ..."
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Cited by 1 (0 self)
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Doctor of Philosophy in Mathematics One of the major goals in the field of symbolic computation of differential equations is to develop algorithms for exact or closedform solutions. This thesis studies symbolic computation of maximally overdetermined systems of linear partial differential equations by using constructions in the corresponding ring of differential operators vith polynomial coefficients, vhich is called the Weyl algebra D. We develop algorithms to find polynomial solutions, rational function solutions, and more generally holonomic solutions. By holonomic solutions, we mean the following: sometimes the best way to specify a function F is as the solution of a system of differential equations  this is for instance how many special functions are classically described. Our algorithm takes as input the differential equations describing F as vell as the system S that we wish to solve, and returns as output any solutions to S existing within the Dmodule generated by F. We also study aspects of the opposite problem, namely given a function F, hov can differential equations describing F be produced? We introduce the Weyl closure of an ideal I of the Weyl algebra, vhich is the set of all differential operators annihilating the common holomorphic solutions of I at a generic point. We give an algorithm to compute Weyl closure, vhich has applications to symbolic integration, and vhich ve also use to make a detailed study of ideals in the first Weyl algebra.
Abstract
, 2005
"... Many combinatorial generating functions can be expressed as combinations of symmetric functions, or extracted as subseries and specializations from such combinations. Gessel has outlined a large class of symmetric functions for which the resulting generating functions are Dfinite. We extend Gessel ..."
Abstract
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Many combinatorial generating functions can be expressed as combinations of symmetric functions, or extracted as subseries and specializations from such combinations. Gessel has outlined a large class of symmetric functions for which the resulting generating functions are Dfinite. We extend Gessel’s work by providing algorithms that compute differential equations these generating functions satisfy in the case they are given as a scalar product of symmetric functions in Gessel’s class. Examples of applications to kregular graphs and Young tableaux with repeated entries are given. Asymptotic estimates are a natural application of our method, which we illustrate on the same model of Young tableaux. We also derive a seemingly new formula for the Kronecker product of the sum of Schur functions with itself. (This article completes the extended abstract published in the proceedings of FPSAC’02 under the title “Effective DFinite Symmetric Functions”.)
DIFFERENTIAL EQUATIONS FOR ALGEBRAIC FUNCTIONS
, 2007
"... Abstract. It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose deg ..."
Abstract
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Abstract. It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose degree is cubic in the degree of the function. We also show that there exists a linear differential equation of order linear in the degree whose coefficients are only of quadratic degree. Furthermore, we prove the existence of recurrences of order and degree close to optimal. We study the complexity of computing these differential equations and recurrences. We deduce a fast algorithm for the expansion of algebraic series. 1.
Linear differential equations on P1 and root systems by
, 2011
"... We consider a linear differential operators on P 1 having unramified irregular singular points. For this operator, we attach the root lattice of a KacMoody Lie algebra and the certain element in this lattice. Then we study the Euler transform for this differential operator and show that this transl ..."
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We consider a linear differential operators on P 1 having unramified irregular singular points. For this operator, we attach the root lattice of a KacMoody Lie algebra and the certain element in this lattice. Then we study the Euler transform for this differential operator and show that this translation by the Euler transform can be understand as the Weyl group action on the root lattice. Moreover we show that if the differential operator is irreducible, then the corresponding element becomes a root of this root system.