On the complexity of computing gröbner bases for quasi-homogeneous systems

  • Authors:

  • Jean-Charles Faugère;Mohab Safey El Din;Thibaut Verron

  • Affiliations:

  • Université Pierre et Marie Curie & INRIA, Paris, France;Université Pierre et Marie Curie & INRIA & Institut Universitaire de France, Paris, France;Université Pierre et Marie Curie & INRIA & École Normale Supérieure, Paris, France

  • Venue:
  • Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
  • Year:
  • 2013

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Abstract

Let K be a field and (f1, ..., fn)\subset K[X1, ..., Xn] be a sequence of quasi-homogeneous polynomials of respective weighted degrees (d1, ..., dn) w.r.t a system of weights (w1,...,wn). Such systems are likely to arise from a lot of applications, including physics or cryptography. We design strategies for computing Gröbner bases for quasi-homogeneous systems by adapting existing algorithms for homogeneous systems to the quasi-homogeneous case. Overall, under genericity assumptions, we show that for a generic zero-dimensional quasi homogeneous system, the complexity of the full strategy is polynomial in the weighted Bézout bound Π_{i=1n}di / Π_{i=1nwi. We provide some experimental results based on generic systems as well as systems arising from a cryptography problem. They show that taking advantage of the quasi-homogeneous structure of the systems allow us to solve systems that were out of reach otherwise.