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

Vladimir Dragovic, Renat Gontsov, Irina Goryuchkina
2021 arXiv   pre-print
The existence, uniqueness and convergence of 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 + . . . .  ... 
arXiv:2008.02982v2 fatcat:d4jwmcszzncmjfmu2roidb7sg4

Puiseux Series and Algebraic Solutions of First Order Autonomous AODEs – A MAPLE Package [article]

Francois Boulier, Jose Cano, Sebastian Falkensteiner, Rafael Sendra
2021 arXiv   pre-print
More precisely, all 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.  ... 
arXiv:2103.03646v1 fatcat:v3bwhoret5ho3gcra3l6vwb3fi

Existence and convergence of Puiseux series solutions for autonomous first order differential equations [article]

Jose Cano, Sebastian Falkensteiner, J. Rafael Sendra
2020 arXiv   pre-print
Given an autonomous first order algebraic ordinary differential equation F(y,y')=0, we prove that every 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.  ... 
arXiv:1908.09196v2 fatcat:amdgb3tz5re7hbkj2z6twoi3gy

Algebraic, Rational and Puiseux Series Solutions of Systems of Autonomous Algebraic ODEs of Dimension One

José Cano, Sebastian Falkensteiner, J. Rafael Sendra
2020 Mathematics in Computer Science  
In this paper, we study the algebraic, rational and 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) = ∞.  ... 
doi:10.1007/s11786-020-00478-w fatcat:6rvtd7orkrcijdnyhgira4qznu

Algebraic, rational and Puiseux series solutions of systems of autonomous algebraic ODEs of dimension one [article]

Jose Cano and Sebastian Falkensteiner and J. Rafael Sendra
2020 arXiv   pre-print
In this paper, we study the algebraic, rational and 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) = ∞  ... 
arXiv:2001.10992v1 fatcat:3ngrl76c2ffr5obhutxnzxprti

On the number of Puiseux exponents of an invariant branch of a vector field [article]

Pedro Fortuny Ayuso
2017 arXiv   pre-print
We show that the multiplicity of a plane analytic 1-form is a bound for the number of 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  ... 
arXiv:1709.09411v1 fatcat:x7n2epualjcvbijykd3n6akoai

Jacobian pairs and Hamiltonian flows

L.Andrew Campbell
1997 Journal of Pure and Applied Algebra  
Both i and I; can be computed 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).  ... 
doi:10.1016/0022-4049(95)00176-x fatcat:nbt3fa75gvdjxh5hbstujc3vyi

On a Leibnitz-type fractional derivative [article]

V. V. Kobelev
2017 arXiv   pre-print
The ring of polynomials with fractional power 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 ] .  ... 
arXiv:1202.2714v3 fatcat:qs3lboibjzb2pk2qehy2mrfemu

Decompositions of the higher order polars of plane branches [article]

Evelia R. García Barroso, Janusz Gwoździewicz
2016 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 < +∞.  ... 
arXiv:1602.01143v1 fatcat:7ie76itqdnayxgmeyhzaa2k6ri

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

Vladimir Dragović, Irina Goryuchkina
2020 Bulletin of the American Mathematical Society  
These results generalize the famous Newton-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] ).  ... 
doi:10.1090/bull/1684 fatcat:3vfheasrnja7pmka2glvpr5wgu

Two Notes on Formal Power Series

Shreeram Abhyankar
1956 Proceedings of the American Mathematical Society  
This paper consists of two more or less disjoint notes, the first on integral 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.  ... 
doi:10.2307/2033558 fatcat:3juypqhr75fcvhrvzk77gh3hai

Two notes on formal power series

Shreeram Abhyankar
1956 Proceedings of the American Mathematical Society  
This paper consists of two more or less disjoint notes, the first on integral 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.  ... 
doi:10.1090/s0002-9939-1956-0080647-9 fatcat:fs657ahpurdqjpjeb4qatbvxmy

Asymptotic and convergent factorial series in the solution of linear ordinary differential equations

Robert L. Evans
1954 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 formal solutions about x = oo.  ...  This value of m will also change the Puiseux slopes to integers.  ... 
doi:10.1090/s0002-9939-1954-0059424-9 fatcat:be4zrewuvbc6fpuca77par3c5m

Linear Programs and Convex Hulls Over Fields of Puiseux Fractions [chapter]

Michael Joswig, Georg Loho, Benjamin Lorenz, Benjamin Schröter
2016 Lecture Notes in Computer Science  
We describe the implementation of a subfield of the field of 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.  ... 
doi:10.1007/978-3-319-32859-1_37 fatcat:gehk3rs6yvetrga36ojumh4y7i

Algebraic and Puiseux series solutions of systems of autonomous algebraic ODEs of dimension one in several variables [article]

Jose Cano and Sebastian Falkensteiner and Daniel Robertz and Rafael Sendra
2021 arXiv   pre-print
For such systems of algebraic dimension one, we show that all 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.  ... 
arXiv:2110.05558v1 fatcat:qjippu5c5zhmdlc2gjxhzbcuny
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