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From formal to actual Puiseux series solutions of algebraic differential equations of first order
[article]

2021
*
arXiv
*
pre-print

The existence, uniqueness and convergence of

arXiv:2008.02982v2
fatcat:d4jwmcszzncmjfmu2roidb7sg4
*formal**Puiseux**series*solutions of non-autonomous algebraic differential equations of first order at a nonsingular point of the equation is studied, including ...*formal**Puiseux**series*solutions in Theorem 1. ... Actually at this point there is a*formal**Puiseux**series*solution y = x 4 k 0 a k x k/2 = x 4 + 8x 9/2 + 108x 5 + 1863x 11/2 + 37665x 6 + . . . . ...##
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Puiseux Series and Algebraic Solutions of First Order Autonomous AODEs – A MAPLE Package
[article]

2021
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arXiv
*
pre-print

More precisely, all

arXiv:2103.03646v1
fatcat:v3bwhoret5ho3gcra3l6vwb3fi
*formal**Puiseux**series*and algebraic solutions, including the generic and singular solutions, are computed and described uniquely. ... The authors have presented in previous works a method to overcome this problem for autonomous first order algebraic ordinary differential equations and*formal**Puiseux**series*solutions and algebraic solutions ...*Formal**Puiseux**Series*Solutions.*Formal**Puiseux**series*can either be expanded around a finite point or at infinity. ...##
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Existence and convergence of Puiseux series solutions for autonomous first order differential equations
[article]

2020
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arXiv
*
pre-print

Given an autonomous first order algebraic ordinary differential equation F(y,y')=0, we prove that every

arXiv:1908.09196v2
fatcat:amdgb3tz5re7hbkj2z6twoi3gy
*formal**Puiseux**series*solution, expanded around any finite point or at infinity, is convergent. ... The proof is constructive and we provide an algorithm to describe all such*Puiseux**series*solutions. ... Hence,*formal**Puiseux**series*solutions expanded around infinity. ...##
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Algebraic, Rational and Puiseux Series Solutions of Systems of Autonomous Algebraic ODEs of Dimension One

2020
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Mathematics in Computer Science
*

In this paper, we study the algebraic, rational and

doi:10.1007/s11786-020-00478-w
fatcat:6rvtd7orkrcijdnyhgira4qznu
*formal**Puiseux**series*solutions of certain type of systems of autonomous ordinary differential equations. ... Using results on such equations, we prove the convergence of the*formal**Puiseux**series*solutions of the system, expanded around a finite point or at infinity, and we present an algorithm to describe them ... (x) = ± √ 2 x 1/2 .There is no*formal**Puiseux**series*solution with the initial value y(0) = ∞. ...##
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Algebraic, rational and Puiseux series solutions of systems of autonomous algebraic ODEs of dimension one
[article]

2020
*
arXiv
*
pre-print

In this paper, we study the algebraic, rational and

arXiv:2001.10992v1
fatcat:3ngrl76c2ffr5obhutxnzxprti
*formal**Puiseux**series*solutions of certain type of systems of autonomous ordinary differential equations. ... Using results on such equations, we prove the convergence of the*formal**Puiseux**series*solutions of the system, expanded around a finite point or at infinity, and we present an algorithm to describe them ... We obtain all the*formal**Puiseux**series*solutions, expanded around zero, by the one-parameter family of solutions y(x) = y 0 + There is no*formal**Puiseux**series*solution with the initial value y(0) = ∞ ...##
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On the number of Puiseux exponents of an invariant branch of a vector field
[article]

2017
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arXiv
*
pre-print

We show that the multiplicity of a plane analytic 1-form is a bound for the number of

arXiv:1709.09411v1
fatcat:x7n2epualjcvbijykd3n6akoai
*Puiseux*exponents of a (*formal*or convergent) branch. ... Let Γ = f k x k/m be a*formal*power*series*with k ∈ N for k ≥ m and m ∈ N (i.e. Γ is a*Puiseux*expansion of a*formal*branch transverse to x = 0). ... In the most general case we shall need, we consider a*formal*1−form (1) ω = a(x, y)dx + b(x, y)dy where a(x, y) and b(x, y) are power*series*in y whose coefficients belong to some ring of*formal*power ...##
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Jacobian pairs and Hamiltonian flows

1997
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Journal of Pure and Applied Algebra
*

Both i and I; can be computed

doi:10.1016/0022-4049(95)00176-x
fatcat:nbt3fa75gvdjxh5hbstujc3vyi
*formally*by differentiating the*Puiseux**series*. ... A*Puiseux*flow of X over k is a pair of*formal**Puiseux**series*in t, x = a&" + higher-order terms (a, # 0), v = bat" + higher-order terms (bfi # O), that satisfy (*formally*) X = Y(X, y), j = s(x, y). ...##
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On a Leibnitz-type fractional derivative
[article]

2017
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arXiv
*
pre-print

The ring of polynomials with fractional power

arXiv:1202.2714v3
fatcat:qs3lboibjzb2pk2qehy2mrfemu
*series*is known as*Puiseux**series*[ 14 ] . As we show in the next section, α-derivative of*Puiseux**series*is*Puiseux**series*. ... or the field of*Puiseux**series*[ 18 , 19 ] . ...##
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Decompositions of the higher order polars of plane branches
[article]

2016
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arXiv
*
pre-print

*Formal*

*Puiseux*power

*series*Denote by C [[x] ] * the set of

*formal*

*Puiseux*power

*series*. The order of any nonzero

*formal*

*Puiseux*power

*series*is the minimal degree of its terms. ... Hereinafter, for brevity,

*formal*

*Puiseux*power

*series*will be called

*Puiseux*

*series*. 3 Newton-

*Puiseux*roots of higher order polars y] ] be such that 1 < ord f (0, y) = n < +∞. ...

##
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About the cover: The Fine–Petrović Polygons and the Newton–Puiseux Method for Algebraic Ordinary Differential Equations

2020
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Bulletin of the American Mathematical Society
*

These results generalize the famous Newton-

doi:10.1090/bull/1684
fatcat:3vfheasrnja7pmka2glvpr5wgu
*Puiseux*polygonal method and apply to algebraic ODEs rather than algebraic equations. ... of terms in the expansion of*formal*solutions (which have a form of*Puiseux**series*) of algebraic ODEs in a neighborhood of the point x = 0. ... Fine also treated the question of the convergence of*formal**series*. Fine proved the following result. Theorem 3 (Fine, 1889, [8] ). ...##
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Two Notes on Formal Power Series

1956
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Proceedings of the American Mathematical Society
*

This paper consists of two more or less disjoint notes, the first on integral

doi:10.2307/2033558
fatcat:3juypqhr75fcvhrvzk77gh3hai
*formal*power*series*in several variables and the second concerning the generalized*Puiseux*expansion of a certain algebraic ... Analytically independent*formal*(integral) power*series*. ... Let k be an arbitrary field and let Ln be the ring of*formal**series*k[[xu x2, • ■ • , xn]] in n variables xi, x2, ■ ■ ■ , xn with coefficients in k. ...##
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Two notes on formal power series

1956
*
Proceedings of the American Mathematical Society
*

This paper consists of two more or less disjoint notes, the first on integral

doi:10.1090/s0002-9939-1956-0080647-9
fatcat:fs657ahpurdqjpjeb4qatbvxmy
*formal*power*series*in several variables and the second concerning the generalized*Puiseux*expansion of a certain algebraic ... Analytically independent*formal*(integral) power*series*. ... Let k be an arbitrary field and let Ln be the ring of*formal**series*k[[xu x2, • ■ • , xn]] in n variables xi, x2, ■ ■ ■ , xn with coefficients in k. ...##
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Asymptotic and convergent factorial series in the solution of linear ordinary differential equations

1954
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Proceedings of the American Mathematical Society
*

«o» * 0 and 0 = 0, 1, • ■ ■ , n) converges for all |x| >x0 (and |x| >x0 contains no finite singular points of (1)), Fabry2 has presented n linearly independent

doi:10.1090/s0002-9939-1954-0059424-9
fatcat:be4zrewuvbc6fpuca77par3c5m
*formal*solutions about x = oo. ... This value of m will also change the*Puiseux*slopes to integers. ...##
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Linear Programs and Convex Hulls Over Fields of Puiseux Fractions
[chapter]

2016
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Lecture Notes in Computer Science
*

We describe the implementation of a subfield of the field of

doi:10.1007/978-3-319-32859-1_37
fatcat:gehk3rs6yvetrga36ojumh4y7i
*formal**Puiseux**series*in polymake. This is employed for solving linear programs and computing convex hulls depending on a real parameter. ... It is our opinion that this is a subfield of the*formal**Puiseux**series*which is particularly well suited for exact computations with (some)*Puiseux**series*; see [20] for an entirely different approach ... Hilbert's ordered field of rational functions is a subfield of the field of*formal**Puiseux**series*R{ {t} } with real coefficients. ...##
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Algebraic and Puiseux series solutions of systems of autonomous algebraic ODEs of dimension one in several variables
[article]

2021
*
arXiv
*
pre-print

For such systems of algebraic dimension one, we show that all

arXiv:2110.05558v1
fatcat:qjippu5c5zhmdlc2gjxhzbcuny
*formal**Puiseux**series*solutions can be approximated up to an arbitrary order by convergent solutions. ... We show that the existence of*Puiseux**series*and algebraic solutions can be decided algorithmically. Moreover, we present a symbolic algorithm to compute all algebraic solutions. ... Note that since the field of*formal**Puiseux**series*is algebraically closed, all algebraic solutions can be represented as (*formal*)*Puiseux**series*. ...
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