Puiseux Power Series Solutions for Systems of Equations

Aroca, Fuensanta; Ilardi, Giovanna; Medrano, Lucía López de

doi:10.1142/s0129167x10006574

Cited by 10 publications

(9 citation statements)

References 28 publications

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“…Later it was established (in full generality) in [SS04]. Extensions to arbitrary codimension ideals and arbitrary valuations have been done subsequently (see for example [AILdM10,JMM08,Aro10]).…”

Section: -Introductionmentioning

confidence: 99%

The fundamental theorem of tropical differential algebraic geometry

2016

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Let I be an ideal of the ring of Laurent polynomials K[x ±1 1 , . . . , x ±1 n ] with coefficients in a real-valued field (K, v). The fundamental theorem of tropical algebraic geometry states the equality trop(V (I)) = V (trop(I)) between the tropicalization trop(V (I)) of the closed subscheme V (I) ⊂ (K * ) n and the tropical variety V (trop(I)) associated to the tropicalization of the ideal trop(I).In this work we prove an analogous result for a differential ideal G of the ring of differential polynomials K[[t]]{x1, . . . , xn}, where K is an uncountable algebraically closed field of characteristic zero. We define the tropicalization trop(Sol(G)) of the set of solutions Sol(G) ⊂ K[[t]] n of G, and the set of solutions Sol(trop(G)) ⊂ P(Z ≥0 ) n associated to the tropicalization of the ideal trop(G). These two sets are linked by a tropicalization morphism trop : Sol(G) −→ Sol(trop(G)).We show the equality trop(Sol(G)) = Sol(trop(G)), answering a question raised by D. Grigoriev earlier this year.

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Section: -Introductionmentioning

confidence: 99%

The fundamental theorem of tropical differential algebraic geometry

2016

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“…The second part of this paper gives an explanation of Newton-Puiseux expansions with a view to applying them to dimensionless equations derived via the method of the previous sections. The development follows [1], which presents explicit formulae for such expansions, extending the classical theory [2,5,14]. A Newton-Puiseux expansion is a fractional power-series solution of an algebraic equation or a system of such equations [14,15].…”

Section: Dimensional Analysis In the Differential Equation Casementioning

confidence: 99%

“…The purpose of this paper is to demonstrate that recent new results [1] in the theory of fractional power series solutions of algebraic equations may be placed in a more general setting, and that this makes possible further developments in the theory of such equations [2] . In particular, it is shown that ideas from algebraic geometry, especially the theory of toric varieties and ideals [3], provide a basis for new computational tools which draw on, and extend, currently available tools within the flourishing research area of computational algebraic geometry [4].…”

Section: Introductionmentioning

confidence: 98%

Non-dimensional Newton-Puiseux Expansions

Chapman¹,

Wynn²,

Atherton³

et al. 2021

Preprint

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Recent results in the theory and application of Newton-Puiseux expansions, i.e. fractional power series solutions of equations, suggest further developments within a more abstract algebraic-geometric framework, involving in particular the theory of toric varieties and ideals. Here, we present a number of such developments, especially in relation to the equations of van der Pol, Riccati, and Schrödinger. Some pure mathematical concepts we are led to are Graver, Gröbner, lattice and circuit bases, combinatorial geometry and differential algebra, and algebraic-differential equations. Two techniques are coordinated: classical dimensional analysis (DA) in applied mathematics and science, and a polynomial and differential-polynomial formulation of asymptotic expansions, referred to here as Newton-Puiseux (NP) expansions. The latter leads to power series with rational exponents, which depend on the choice of the dominant part of the equations, often referred to as the method of "dominant balance". The paper shows, using a new approach to DA based on toric ideals, how dimensionally homogeneous equations may best be non-dimensionalised, and then, with examples involving differential equations, uses recent work on NP expansions to show how non-dimensional parameters affect the results. Our approach finds a natural home within computational algebraic geometry, i.e. at the interface of abstract and algorithmic mathematics.

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“…In [5], we extend Mc Donald's algorithm to algebraic varieties of arbitrary codimension. To do this, we do not work with polygons but with dual fans and, instead of working with equations restricted to edges, we work with initial equations.…”

Section: Introductionmentioning

confidence: 99%

Newton’s lemma for differential equations

Aroca¹,

Ilardi²

2016

Illinois J. Math.

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The Newton method for plane algebraic curves is based on the following remark: the first term of a series, root of a polynomial with coefficients in the ring of series in one variable, is a solution of an initial equation that can be determined by the Newton polygon.Given a monomial ordering in the ring of polynomials in several variables, we describe the systems of initial equations that satisfy the first terms of the solutions of a system of partial differential equations. As a consequence, we extend Mora and Robbiano's Groebner fan to differential ideals.

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Puiseux Power Series Solutions for Systems of Equations

Cited by 10 publications

References 28 publications

The fundamental theorem of tropical differential algebraic geometry

The fundamental theorem of tropical differential algebraic geometry

Non-dimensional Newton-Puiseux Expansions

Newton’s lemma for differential equations

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