Daniel Lazard

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Daniel Lazard (born December 10, 1941 in Carpentras) is a French mathematician and computer scientist. He is emeritus professor at Université Pierre et Marie Curie.

He began his academic career by working in commutative algebra, especially on flat modules. Around 1970, he began to work in computer algebra, which, soon after, became his main research area. In this field, he is specially interested in multivariate polynomials and more generally in computational algebraic geometry, with emphasis on polynomial system solving.

Selected contributions[edit]

  • Lazard (1969) noted that a module M is flat if and only if it is a direct limit of finitely-generated free modules. As a consequence, one can deduce that every finitely-presented flat module is projective. (See flat module#Categorical colimits)
  • In computer algebra, the resultant of two polynomials can be used to analyze modular images of the greatest common divisor of integer polynomials where the coefficients are taken modulo some prime number p. The resultant of two polynomials is frequently computed in the Lazard-Rioboo-Trager method of finding the integral of a ratio of polynomials.
  • Lazard (1992) introduced the lextriangular algorithm to obtain the triangular decomposition of a polynomial system. See System of polynomial equations#Regular chains.


  • Lazard, D. (1969), "Autour de la platitude", Bulletin de la Société Mathématique de France 97: 81–128 
  • Faugère, J. C.; Gianni, P.; Lazard, D.; Mora, T. Efficient computation of zero-dimensional Gröbner bases by change of ordering. J. Symbolic Comput. 16 (1993), no. 4, 329–344.
  • D. Lazard (2009), "Thirty years of Polynomial System Solving, and now?", Journal of Symbolic Computation, special issue in honor of Daniel Lazard 44 (3): 222–231 
  • Daniel Lazard (1981), "Résolution des systèmes d'équations algébriques", Theoretical Computer Science 15: 77–110 
  • Lazard, D. (1992), "Solving zero-dimensional algebraic systems", Journal of Symbolic Computation 13 .
  • Michael Abramson (2001), "Solving systems of algebraic equations, translation of (Lazard 1981)", ACM SIGSAM Bulletin 35 (3): 11–37, doi:10.1145/569746.569750 
  • ICPSS, International Conference on Polynomial System Solving, and special issue of Journal of Symbolic Computation, in honor of Daniel Lazard 

External links[edit]