Illinois Journal of Mathematics

Newton’s lemma for differential equations

Fuensanta Aroca and Giovanna Ilardi

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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.

Article information

Illinois J. Math., Volume 60, Number 3-4 (2016), 859-867.

Received: 10 January 2017
Revised: 18 April 2017
First available in Project Euclid: 22 September 2017

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Zentralblatt MATH identifier

Primary: 14J17: Singularities [See also 14B05, 14E15] 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx] 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] 14Q15: Higher-dimensional varieties 13P99: None of the above, but in this section


Aroca, Fuensanta; Ilardi, Giovanna. Newton’s lemma for differential equations. Illinois J. Math. 60 (2016), no. 3-4, 859--867. doi:10.1215/ijm/1506067296.

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  • F. Aroca and J. Cano, Formal solutions of linear PDEs and convex polyhedra, J. Symbolic Comput. (Effective methods in rings of differential operators) 32 (2001), no. 6, 717–737.
  • F. Aroca, J. Cano and F. Jung, Power series solutions for non-linear PDEs, Proceeding of the 2003 international symposium on symbolic and algebraic computation, ACM, New York, 2003, pp. 15–22.
  • F. Aroca, C. Garay and Z. Toghani, The fundamental theorem of tropical differential algebraic geometry, Pacific J. Math. 283 (2016), no. 2, 257–270.
  • F. Aroca and G. Ilardi, Some algebraically closed fields containing polynomial in several variables, Comm. Algebra 37 (2009), no. 4, 1284–1296.
  • F. Aroca, G. Ilardi and L. Lopez de Medrano, Puiseux power series solutions for systems of equations, Internat. J. Math. 21 (2010), no. 11, 1439–1459.
  • A. Assi, F. J. Castro-Jiménez and M. Grange, The analytic standard fan of a D-module, J. Pure Appl. Algebra 164 (2001), no. 1, 3–21.
  • C. Briot and J. Bouquet, Propriétés des fonctions définies par des équations différentielles, J. Éc. Polytech. 36 (1856), 133–198.
  • J. Cano, On the series defined by differential equations, with an extension of the Puiseux polygon construction to these equations, Analysis (Munich) 13 (1993), nos. 1–2, 103–119.
  • H. B. Fine, On the functions defined by differential equations, with an extension of the Puiseux polygon construction to these equations, Amer. J. Math. XI (1889), 317–328.
  • D. Y. Grigoriev, Tropical differential equations, Adv. in Appl. Math. 82 (2017), 120–128.
  • R. Kolchin, Differential algebra and algebraic groups, Pure and Applied Mathematics, vol. 54, Academic Press, New York, 1973.
  • T. Mora and L. Robbiano, The Groebner fan of an ideal, J. Symbolic Comput. 6 (1988), nos. 2–3, 183–208.
  • I. Newton, The mathematical papers of Isaac Newton. Vol III: 1670–1673 (D. T. Whiteside, ed.), Cambridge University Press, London, 1969. With the assistance in publication of M. A. Hoskin and A. Prag.
  • V. Puiseux, Recherches sur les fonctions algébriques, J. Math. Pures Appl. 15 (1850), 365–480.
  • J. F. Ritt, Differential algebra, American Mathematical Society Colloquium Publications, vol. XXXIII, Am. Math. Soc., New York, 1950.